For the Continuous Mountain Car Solution Discretization

The Finite Element Method for Elliptic Problems

In Studies in Mathematics and Its Applications, 1978

2.1. Basic aspects of the finite element method

The Galerkin and Ritz methods

Consider the linear abstract variational problem: Find u ∈ V such that

(2.1.1) $\forall v\in V,a\left(u,v\right)=f\left(v\right),$

where the space V, the bilinear form a(·, ·), and the linear form f are assumed to satisfy the assumptions of the Lax–Milgram lemma (Theorem 1.1.3). Then the Galerkin method for approximating the solution of such a problem consists in defining similar problems in finite-dimensional subspaces of the space V. More specifically, with any finite-dimensional subspace V_h of V, we associate the discrete problem:Find u_h ∈ V_h such that

(2.1.2) $\forall v_h\in V_h,a\left(u_h,v_h\right)=f\left(v_h\right).$

Applying the Lax–Milgram lemma, we infer that such a problem has one and only one solution u_h , which we shall call a discrete solution .

Remark 2.1.1.

In case the bilinear form is symmetric, the discrete solution is also characterized by the property (Theorem 1.1.2)

(2.1.3) $J\left(u_h\right)=\underset{v_h\in V_h}{\inf }J\left(v_h\right),$

where the functional J is given by $J\left(v\right)=\frac{1}{2}a\left(v,v\right)-f\left(v\right).$ This alternate definition of the discrete solution is known as the Ritz method.

The three basic aspects of the finite element method. Conforming finite element methods

Let us henceforth assume that the abstract variational problem (2.1.1) corresponds to a second-order or to a fourth-order elliptic boundary value problem posed over an open subset Ω of R ⁿ , with a Lipschitz-continuous boundary Γ. Typical examples of such problems have been studied in Section 1.2.

In order to apply Galerkin method, we face, by definition, the problem of constructing finite-dimensional subspaces V_h of spaces V such as $H_0^1\left(\Omega \right)$ , H ¹(Ω), $H_0^2\left(\Omega \right)$ , etc…

The finite element method, in its simplest form, is a specific process of constructing subspaces V_h , which shall be called finite element spaces. This construction is characterized by three basic aspects, which for convenience shall be recorded as (FEM 1), (FEM 2) and (FEM 3), respectively, and which shall be described in this section.

(FEM 1) The first aspect, and certainly the most characteristic, is that a triangulation T _h is established over the set $\overline{\Omega },$ i.e., the set $\overline{\Omega }$ is subdivided into a finite number of subsets K, called finite elements, in such a way that the following properties are satisfied:

$\left({\mathbb{T}}_{h}1\right)\overline{\Omega }={\cup }_{K\in T_h}K.$

(T _h 2) For each K ∈ T _h , the set K is closed and the interior $\stackrel{\circ }{K}$ is non empty.

(T _h 3) For each distinct K ₁, K ₂ ∈ ∈ T _h , one has ${\stackrel{\circ }{K}}_1\cap {\stackrel{\circ }{K}}_2=\phi .$

(T _h 4) For each K ∈ T _h , the boundary ∂K is Lipschitz-continuous.

Remark 2.1.2.

A fifth condition (T _h , 5) relating "adjacent" finite elements, will be introduced in the next section.

Once such a triangulation T _h is established over the set $\overline{\Omega },$ one defines a finite element space X _h through a specific process, which will be illustrated by many examples in the next section and subsequently. We shall simply retain for the moment that X_h is a finite-dimensional space of functions defined over the set $\overline{\Omega }$ (we shall deliberately ignore at this stage instances of finite element spaces whose "functions" may have two definitions across "adjacent" finite elements; see Section 2.3).

Given a finite element space X_h , we define the (finite-dimensional) spaces

$P_K=\left\{v_{h|K};v_h\in X_h\right\}$

spanned by the restrictions v_h/K of the functions v_h ∈ X_h to the finite elements K ∈ T _h . Without specific assumptions concerning the spaces P_K, K ∈ T _h , there is no reason for an inclusion such as X_h ⊂ H ¹(Ω) – let alone an inclusion such as X_h ⊂ H ²(Ω) – to hold.

In order to obtain such inclusions, we need additional conditions of a particularly simple nature, as we show in the next theorems (converses of these results hold, as we shall show in Theorems 4.2.1 and 6.2.1).

Remark 2.1.3.

Here and subsequently, we shall comply with the use of the notation H ^m (K), in lieu of $H_m\left(\stackrel{\circ }{K}\right).$

Theorem 2.1.1.

Assume that the inclusions P _K ⊂ H ¹(K) for all K ∈ T _h and $X_h\subset C^0\left(\overline{\Omega }\right)$ hold. Then the inclusions

$\begin{array}{l}{X}_h\subset H^1\left(\Omega \right),\\ X_{oh}=\left\{v_h\in X_h;v_h=0\text{on}\Gamma \right\}\subset H_0^1\left(\Omega \right),\end{array}$

hold.

Proof. Let a function v ∈ X_h be given. We already know that it is in the space L ²(Ω). Therefore, by definition of the space H ¹(Ω), we must find for each i = 1, …, n, a function v_i , ∈ L ²(Ω) such that

$\forall \phi \in D\left(\Omega \right),{\int }_{\Omega }{v}_i\phi \text{d}x=-{\int }_{\Omega }v{\partial }_i\phi \text{d}x.$

For each i, a natural candidate is the function whose restriction to each finite element K is the function ∂ _i (v| _K ). Since each finite element K has a Lipschitz-continuous boundary ∂K, we may apply Green's formula (1.2.4): For each K ∈ T _h ,

${\int }_K{\partial }_i\left(v{|}_K\right)\phi \text{d}x=-{\int }_{K}v{|}_K{\partial }_i\phi \text{d}x+{\int }_{\partial K}v{|}_K\phi v_{i,K}\text{d}\gamma ,$

where v _i,K is the i-th component of the unit outer normal vector along ∂K. By summing over all finite elements, we obtain

${\int }_{\Omega }{v}_i\phi \text{d}x=-{\int }_{\Omega }v{\partial }_i\phi \text{d}x+\sum _{K\in J_h}{\int }_{\partial K}v{|}_K\phi v_{i,K}\text{d}\gamma ,$

and the proof follows if we notice that the sum ${\sum }_{K\in T_h}{\int }_{\partial K}v{|}_K\phi v_{i,K}\text{d}\gamma$ vanishes: Either a portion of ∂K is a portion of the boundary Γ of Ω in which case φ = 0 along this portion, or the contribution of adjacent elements is zero.

The boundary Γ being Lipschitz-continuous by assumption, the second inclusion follows from the characterization

$H_0^1\left(\Omega \right)=\left\{v\in H^1\left(\Omega \right),v=0\text{on}\Gamma \right\},$

which was mentioned in Section 1.2.

Assuming Theorem 2.1.1 applies, we shall therefore use the finite element space V_h = X _oh if we are solving a second-order homogeneous Dirichlet problem, or V_h = X_h if we are solving a second-order homogeneous or nonhomogeneous Neumann problem.

The proof of the next theorem is similar to that of Theorem 2.1.1 and, for this reason, is left to the reader as an exercise (Exercise 2.1.1).

Theorem 2.1.2.

Assume that the inclusions P _K ⊂ H ²(K) for all K ∈ T _h and $X_h\subset C^1\left(\overline{\Omega }\right)$ hold. Then the inclusions

$\begin{array}{l}{X}_h\subset H^2\left(\Omega \right),\\ X_{oh}=\left\{v_h\in X_h;v_h=0\text{on}\Gamma \right\}\subset H^2\left(\Omega \right)\cap H_0^1\left(\Omega \right),\\ X_{ooh}=\left\{v_h\in X_h;v_h={\partial }_v{v}_h=0\text{on}\Gamma \right\}\subset H_0^2\left(\Omega \right),\end{array}$

hold.

Thus if we are to solve a simply supported plate problem, or a clamped plate problem, we shall use the finite element space V_h = X_oh , or the finite element space V_h = X_ooh , respectively, as given in the previous theorem.

Let us return to the description of the finite element method.

(FEM 2) The second basic aspect of the finite element method is that the spaces P _K , K ∈ T _h , contain polynomials, or, at least, contain functions which are "close to" polynomials. At this stage, we cannot be too specific about the underlying reasons for this aspect of the method but at least, we can say that

(i)

it is the key to all convergence results as we shall see, and

(ii)

it yields simple computations of the coefficients of the resulting linear system (see (2.1.4) below).

Let us now briefly examine how the discrete problem (2.1.2) is solved in practice. Let ${(w_k)}_{k=1}^M$ be a basis in the space V_h . Then the solution ${\text{u}}_h={\sum }_{k=1}^M{u}_k{w}_k$ of problem (2.1.2) is such that the coefficients u _k are solutions of the linear system

(2.1.4) $\sum _{k=1}^{M}a\left(w_k,w_l\right)u_k=f\left(w_l\right),1\le l\le M,$

whose matrix is always invertible, since the bilinear form, being assumed to be V-elliptic, is a fortiori V _h -elliptic. By reference to the elasticity problem, the matrix (a(w_k , w_l )) and the vector (f(w_l )) are often called the stiffness matrix and the load vector, respectively.

In the choice of the basis ${\left({\text{w}}_k\right)}_{k=1}^M$ , it is of paramount importance, from a numerical standpoint, that the resulting matrix possess as many zeros as possible.

For all the examples which were considered in Section 1.2 the coefficients a(w_k , w_l ) are integrals of a specific form: For instance, in the case of the first examples, one has

$a\left(w_k,w_l\right)={\int }_{\Omega }\left(\sum _{i=1}^n{\partial }_i{w}_k{\partial }_i{w}_l+a{w}_k{w}_l\right)dx,$

so that a coefficient a(w_k , w_l ) vanishes whenever the dx-measure of the intersection of the supports of the basis functions w _k and w_l , is zero.

(FEM 3) As a consequence, we shall consider as the third basic aspect of the finite element method that there exists at least one "canonical" basis in the space V_h whose corresponding basis functions have supports which are as "small" as possible, it being implicitly understood that these basis functions can be easily described.

Remark 2.1.4.

When the bilinear form is symmetric, the matrix (a(w_k , w_l )) is symmetric and positive definite, which is an advantage for the numerical solution of the linear system (2.1.4). By contrast, this is not generally the case for standard finite-difference methods, except for rectangular domains.

Assuming again the symmetry of the bilinear form, one could conceivably start out with any given basis, and, using the Gram-Schmidt orthonormalization procedure, construct a new basis ${\left({\text{w}}_{{}_k}^{*}\right)}_{k=1}^M$ which is orthonormal with respect to the inner product a(·, ·). This is indeed an efficient way of getting a sparse matrix since the corresponding matrix $(a(w_k^{*},w_l^{*}))$ is the identity matrix! However, this process is not recommended from a practical standpoint: For comparable computing times, it yields worse results than the solution by standard methods of the linear system corresponding to the "canonical" basis.

It was mentioned earlier that the three basic aspects were characteristic of the finite element method in its simplest form. Indeed, there are more general finite element methods:

(i)

One may start out with more general variational problems, such as variational inequalities (see Section 5.1) or various nonlinear problems (see Sections 5.2 and 5.3), or different variational formulations (see Chapter 7).

(ii)

The space V_h , in which one looks for the discrete solution, may no longer be a subspace of the space V. This may happen when the boundary of the set Ω is curved, for instance. Then it cannot be exactly triangulated in general by standard finite elements and thus it is replaced by an approximate set Ω _h (see Section 4.4). This also happens when the functions in the space V_h lack the proper continuity across adjacent finite elements (see the "nonconforming" methods described in Section 4.2 and Section 6.2).

(iii)

Finally, the bilinear form and the linear form may be approximated. This is the case for instance when numerical integration is used for computing the coefficients of the linear system (2.1.4) (see Section 4.1), or for the shell problem (see Section 8.2).

Nevertheless, it is characteristic of all these more general finite element methods that the three basic aspects are again present.

To conclude these general considerations, we shall reserve the terminology conforming finite element methods for the finite element methods described at the beginning of this section, i.e., for which V_h is a subspace of the space V, and the bilinear form and the linear form of the discrete problem are identical to the original ones.

Exercises

2.1.1.

Prove Theorem 2.1.2.

2.1.2.

The purpose of this problem is to give another proof of the Lax–Milgram lemma (Theorem 1.1.3; see also Exercise 1.1.2) in case the Hilbert space V is separable. Otherwise the bilinear form and the linear form satisfy the same assumptions as in Theorem 1.1.3.

(i)

Let V_h be any finite-dimensional subspace of the space V, and let u_h be the discrete solution of the associated discrete problem (2.1.2). Show that there exists a constant C independent of the subspace V_h such that ||u_h || ≤ C (as usual, there is a simpler proof when the bilinear form is symmetric).

(ii)

The space V being separable, there exists a nested sequence (V _v ) _v∈N of finite-dimensional subspaces such that (∪ _v∈N V _v )⁻ = V. Let (u _v ) _v∈N be the sequence of associated discrete solutions. Show that there exists a subsequence of the sequence (u _v ) _v∈N which weakly converges to a solution u of the original variational problem.

(iii)

Show that the whole sequence converges in the norm of V to the solution u.

(iv)

Show that the Sobolev spaces H ^m (Ω) are separable.

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Real and Numerical Analysis

Sylvie Boldo , Guillaume Melquiond , in Floating-Point Algorithms and Formal Proofs, 2017

9.3.8 Convergence

The convergence of a numerical scheme expresses the fact that the discrete solution gets closer to the continuous solution as the discretization steps decrease to zero. More precisely, it is formally proved that when the continuous solution of the wave equation [9.2]–[9.5] is sufficiently regular of order 4 uniformly on [x _min; x _max] × [0; t _max], and under the CFL(ξ) condition [9.15], the numerical scheme [9.7]–[9.10] is convergent of order (2, 2) uniformly on interval [0; t _max] (see definition [9.12] in section 9.3.3).

First, we prove that the convergence error e _h is itself the discrete solution to a numerical scheme of the same form but with different input data (as in section 9.3.3). ⁵ In particular, p _0h is zero and p _1h corresponds to some initialization error: p _1h = e _h ¹/Δt. The interesting value is the source term: it is the truncation error ε _h associated with the initial numerical scheme for p _h. Then, the previous stability result holds, and gives an overestimation of the square root of the discrete energy associated with the convergence error E _h(c)(e _h) that involves a sum of the corresponding source terms, i.e., the truncation error. Finally, the consistency result also makes this sum a big $\mathcal{O}$ of Δx ² + Δt ² and a few more technical steps conclude the proof.

Theorem 9.1 Convergence

${‖e_h^{k_{\mathrm{\Delta }t}\left(t\right)}‖}_{\mathrm{\Delta }x}={\mathcal{O}}_{\left|\begin{array}{l}t\in \left[0;t_{\max }\right]\hfill \\ \left(\mathrm{\Delta }x,\mathrm{\Delta }t\right)\to 0\hfill \\ 0<\mathrm{\Delta }x\wedge 0<\mathrm{\Delta }t\wedge \hfill \\ c\frac{\mathrm{\Delta }t}{\mathrm{\Delta }x}\le 1-\xi \hfill \end{array}\right.}\left(\mathrm{\Delta }{x}^2+\mathrm{\Delta }{t}^2\right).$

We have sketched the proof of the convergence of this simple numerical scheme. This proof means that formulas[9.7]–[9.10] compute something closer to the exact solution to the wave equation when the step sizes tend to zero [BOL 10a, BOL 13b]. This is a first step. The second step is to take into account the FP computations and the rounding errors.

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Handbook of Numerical Methods for Hyperbolic Problems

S. Gottlieb , D.I. Ketcheson , in Handbook of Numerical Analysis, 2016

4.3 Optimal Implicit Methods

If we have a spatial discretization that satisfies (6 ) for some norm (or convex functional) then the fully discrete solution will also be strongly stable, in the same norm (or convex functional), for the implicit Euler method, without any time step restriction (Higueras, 2004; Hundsdorfer et al., 2003). However, all Runge–Kutta, linear multistep, and other general linear methods of greater than first order have finite SSP coefficient (Spijker, 1983). The question becomes whether the allowable step size is large enough to offset the additional computational cost of solving an algebraic system at each step.

For linear multistep methods, the results of Lenferink (1991) imply that any linear multistep method of order p > 1 has SSP coefficient no greater than $\mathcal{C}=2$ (Hundsdorfer et al., 2003). Even when special starting procedures are used the step-size restrictions for the implicit multistep methods are hardly better than those of explicit methods. Furthermore, methods of this type with order p > 1 are subject to the same maximal SSP coefficient of two (Hundsdorfer and Ruuth, 2005). Thus, although no order barrier on implicit SSP multistep methods exists (Sand, 1986), these step-size restrictions are too severe to make the use of these methods efficient for typical cases.

For implicit Runge–Kutta methods, the size of this finite time step has not been established analytically. However, extensive numerical searches suggest that the time step restrictions for implicit SSP methods are limited to twice the number of stages, so that their effective SSP coefficient is bounded by ${\mathcal{C}}_{\text{eff}}\le 2$ (Ketcheson et al., 2009). Implicit SSP Runge–Kutta methods of order greater than six do not exist. Methods of up to order five were found in Kraaijevanger (1991). Later, Ferracina and Spjker (2008) showed that optimal singly diagonally implicit SSP Runge–Kutta methods may exist for up to fourth order and found such methods. In this work, they also conjectured the form of optimal second and third-order singly diagonally implicit SSP Runge–Kutta methods with any number of stages. Fully implicit SSP Runge–Kutta methods were investigated via numerical optimization in Ketcheson et al. (2009), and the first sixth-order method was found in that work. Results therein showed that the second- and third-order optimal methods among the class of all fully implicit methods were the same singly diagonally implicit methods found in Ferracina and Spjker (2008). The optimal methods of fourth through sixth order were found to be diagonally implicit.

The results on the SSP coefficient bound for implicit methods have been disappointing. When the time step is limited by a linear stability requirement, or even by a nonlinear stability requirement involving classical stability properties (such as absolute stability of B-stability), there exist some well-known classes of implicit methods that allow the use of arbitrarily large time steps. However, the SSP condition does not allow such large time steps. Table 1 (bottom) contains the optimal effective SSP coefficients of implicit Runge–Kutta methods. Furthermore, this bound of ${\mathcal{C}}_{\text{eff}}\le 2$ is observed to be practically relevant in even the simplest applications, indicating that implicit SSP methods are unlikely to be efficient enough to out-perform the explicit methods.

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Multiscale Wavelet Methods for Partial Differential Equations

Peter Oswald , in Wavelet Analysis and Its Applications, 1997

Abstract

We study to which extent the geometric multilevel approach based on dyadic scales of shift-invariant subspaces on ℝ ^d can be used to produce accurate discrete solutions of elliptic boundary value problems of positive order on nonrectangular domains. We also deal with the construction of optimal preconditioners, including the case of nested refinement. Sufficient geometric conditions on a domain are given such that a robust and asymptotically optimal algorithm can be expected. In contrast to other approaches which emphasize biorthogonal wavelet decompositions, we are satisfied with a simpler frame concept which incorporates recent experience with finite element multilevel solvers.

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Nonlinear Partial Differential Equations and their Applications

F. Collino , B. Despres , in Studies in Mathematics and Its Applications, 2002

4 Variational formulation and well-posedness. The penalized systems

Variational formulation are useful for minimization problems. It provides a good framework for the study of uniqueness and existence, and also for discretization and convergence of the discrete solution. The natural variational formulation of our problem is as follows

(58) $\{\begin{array}{ll}x\in V_x,y\in V_y,\hfill & \hfill \\ \left(x,\tilde{x}\right)+\left(A_{\infty }x,A_{\infty }\tilde{x}\right)-\left(T^{\ast }y,\tilde{x}\right)=\left(g,\tilde{x}\right),\hfill & \forall \tilde{x}\in V_x,\hfill \\ \left(x,T^{\ast }\tilde{y}\right)+\left(A_{\infty }y,A_{\infty }\tilde{y}\right)=0,\hfill & \forall \tilde{y}\in V_y.\hfill \end{array}$

It remains to define the functional spaces V_x and V_y . Let us remark that both smooth analytic operators A _∞ y, A _∞ are not involved in the coerciveness of the system. Due to the L ² coerciveness of the formulation it is clear that the only choice for V_x is

$V_x=X=L^2\left(T\text{Γ}\right)\times L^2\left(T\text{Γ}\right).$

It may seem at first sight impossible to define a compatible space V_y such that the inf-sup condition of Babuska-Brezzi holds, [2],

(59) $\begin{array}{ccc}\max \frac{\left(x,T^{\ast }y\right)}{{\Vert x\Vert }_{L^2}}\ge k{\Vert y\Vert }_{W_y},& k>0,& with{W}_y=\frac{V_y}{Ker{T}^{\ast }},\end{array}$

just because standard functional spaces in which T is continuous are known to be based on

$H^{-\frac{1}{2}}\left(div,\text{Γ}\right)\text{and}{H}^{-\frac{1}{2}}\left(curl,\text{Γ}\right).$

Nevertheless it is at least possible to provide an abstract framework in which the inf-sup condition holds. Let us take V_x = X, V_y = Y, defined in (51).

Definition 6

We define the quotient Hilbertian space

$W=\frac{Y}{Ker{T}^{\ast }}$

equipped with the norm

$\underset{y_0\in Ker{T}^{\ast }}{\inf }{\Vert y-y_0\Vert }_Y={\Vert T^{\ast }y\Vert }_X+\underset{y_0\in Ker{T}^{\ast }}{\inf }{\Vert y-y_0\Vert }_X.$

System (57) is well-posed as soon as the inf-sup condition

(60) $\underset{x\in X}{\text{sup}}\frac{\left(x,T^{\ast }y\right)x}{{\Vert x\Vert }_X}\ge C{\Vert y\Vert }_W$

holds for some strictly positive constant C > 0. This inequality can be derived as follows. Picking x = T*y in (60), we find

(61) $\underset{x\in X}{\text{sup}}\frac{\left(x,T^{\ast }y\right)x}{{\Vert x\Vert }_X}\ge {\Vert T^{\ast }y\Vert }_X.$

Let y ∈ Y. In appendix it is proved that ‒ ΠT with Π defined in (12), is a projector (it is a Calderon Projector):

(62) $T\Pi T=-T.$

Let y ₁= ‒ Π*T*y. We have that T*(y ‒ y₁ ) = T*y + T*Π*T*y = 0. So y ₀ = y ‒ y ₁ ∈ Ker T* hence

(63) $\underset{y_0\in Ker{T}^{\ast }}{\inf }{\Vert y-y_0\Vert }_X\le {\Vert y_1\Vert }_X={\Vert {\Pi }^{\ast }{T}^{\ast }y\Vert }_X={\Vert T^{\ast }y\Vert }_X,$

where we have used the isometric property Π*Π = I. So we have

(64) $\underset{x\in X}{\sup }\frac{{\left(x,T^{\ast }y\right)}_X}{{\Vert x\Vert }_X}\ge \frac{1}{2}{\Vert T^{\ast }y\Vert }_X+\frac{1}{2}\underset{y_0\in Ker{T}^{\ast }}{\inf }{\Vert y-y_0\Vert }_X=\frac{1}{2}{\Vert y\Vert }_W.$

Thus it gives

Lemma 2

The inf-sup condition (60) in spaces = X × W is true with $C=\frac{1}{2}$ .

Since the natural continuity T* : Y → X holds and one has the bound (98) ${\Vert A_{\infty }^{\ast }{A}_{\infty }y\Vert }_X\le C^{\prime }{\Vert y\Vert }_W$ for some C′ >0, it proves following [2]

Theorem 3

The variational system (58) is well posed, that is, for every pair $\left(g,g_y\right)\in \left({V^{\prime }}_x,{V^{\prime }}_y\right)=\left(V_x,{V^{\prime }}_y\right)$ there exists a unique (x, y) ∈ V_x × W such that

(65) $\{\begin{array}{ll}\left(x,\tilde{x}\right)+\left(A_{\infty }x,A_{\infty }\tilde{x}\right)-\left(T^{\ast }y,\tilde{x}\right)=\left(g,\tilde{x}\right),\hfill & \forall \tilde{x}\in V_x,\hfill \\ \left(x,T^{\ast }\tilde{y}\right)+\left(A_{\infty }y,A_{\infty }\tilde{y}\right)=w^{\prime }{〈g_y,\tilde{y}〉}_W,\hfill & \forall \tilde{y}\in V_y.\hfill \end{array}$

However when discretization is considered a difficulty arises with the use of (X, Y) = (V_x , V_y ) spaces. The reason is that we want to avoid the construction of some discrete space compatible with the L ² based space (X, Y). We would like to take a classical integral code based on the duality $H^{-\frac{1}{2}}\left(div,\text{Γ}\right)$ and $H^{-\frac{1}{2}}\left(curl,\text{Γ}\right)$ , rearrange all the routines, and use the iterative algorithms described further in order to solve our new discrete integral system. Then the question of the convergence of the discrete solution to the exact solution arises. All our efforts in order to prove the convergence using this strategy failed. The reason seems to be that the classical discretization of integral operator is based on $H^{-\frac{1}{2}}\left(div,\text{Γ}\right)$ and $H^{-\frac{1}{2}}\left(curl,\text{Γ}\right)$ , and not on (X, Y), [4]. Moreover some numerical results in 2D for the Helmholtz equation, [1], show that this problem may be a real one. It seems that there are cases where the discrete solution obtained through the strategy described above does not converge to the exact solution, even in some very simple and regular cases. Of course this conclusion has to be re-evaluated if the discretization of the integral operators are compatible with (X, Y). Note also that the kernel Ker T* is a large space of infinite dimension. Some algorithms are very sensitive to the dimension of this kernel. It is our purpose now to modify the system and to present what we will call the penalized problem, with much stronger coercivity.

Let β some positive penalisation parameter (for instance β = 1). Remembering that y = ix, we modify system (15) to obtain the penalized system

(66) $\{\begin{array}{l}\left(1+\beta \right)x+{\left(A^{\infty }\right)}^{\ast }{A}^{\infty }x-T^{\ast }y+i\beta y=g\hfill \\ +Tx-i\beta x+\beta y+{\left(A^{\infty }\right)}^{\ast }{A}^{\infty }y=0.\hfill \end{array}$

which appears to be a system of the form

(67) ${\mathcal{A}}_{\beta }\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}g\\ 0\end{array}\right].$

A simple calculation shows that

(68) $\{\begin{array}{c}\Re e{\left({\mathcal{A}}_{\beta }\left[\begin{array}{c}x\\ y\end{array}\right],\left[\begin{array}{c}x\\ y\end{array}\right]\right)}_{X\times X}\\ ={\Vert x\Vert }_X^2+\beta {\Vert x+iy\Vert }_X^2+{\Vert A^{\infty }y\Vert }_Z^2+{\Vert A^{\infty }x\Vert }_Z^2\\ \ge \min \left(\frac{\beta }{2},\frac{1}{3}\right)\left({\Vert x\Vert }_X^2+{\Vert y\Vert }_X^2\right),\end{array}$

and the system appears now as coercive in the x variable and also in the y variable.

Another possibility is to modify system (15) according to

(69) $\{\begin{array}{l}\left(1-\beta \right)x+{\left(A^{\infty }\right)}^{\ast }{A}^{\infty }x-\left(T^{\ast }y+i\beta y\right)=g\hfill \\ \left(Tx-i\beta x\right)+\beta y+{\left(A^{\infty }\right)}^{\ast }{A}^{\infty }y=0,\hfill \end{array}$

where β is now some positive number less than 1 (let $\beta =\frac{1}{2}$ ). The interest of (69) is that it corresponds to a saddle point for the Lagrangian

(70) $\begin{array}{c}{\mathcal{L}}_{\beta }\left(x,\text{γ},y\right)=\left(1-\beta \right)\frac{1}{2}{\Vert x\Vert }_X^2-\beta \frac{1}{2}{\Vert y\Vert }_X^2-\Re e{\left(g,x\right)}_X\\ +\frac{1}{2}{\Vert \text{γ}\Vert }_X^2+\frac{1}{2}{\Vert A^{\infty }x\Vert }_Z^2-\Re e\left({\left(x,T^{\ast }y+i\beta y\right)}_X-i{\left(\text{γ},\left(A^{\infty }\right)y\right)}_Z\right).\end{array}$

The problem appears as a penalized saddle point problem.

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Nonlinear Partial Differential Equations and their Applications

V. Girault , L.R. Scott , in Studies in Mathematics and Its Applications, 2002

1 Introduction

A fluid of grade two is a particular non-Newtonian Rivlin-Ericksen fluid (cf. [28]) whose equation of motion is

(1.1) $\begin{array}{l}\frac{\partial }{\partial t}\left(u-{\alpha }_1\text{Δ}u\right)-\nu \text{Δ}u+curl\left(u-\left(2{\alpha }_1+{\alpha }_2\right)\text{Δ}u\right)\times u\hfill \\ -\left({\alpha }_1+{\alpha }_2\right)\text{Δ}\left(u\cdot \nabla u\right)+2\left({\alpha }_1+{\alpha }_2\right)u\cdot \nabla \left(\text{Δ}u\right)+\nabla \tilde{p}=f.\hfill \end{array}$

This system of equations is completed by the condition of incompressibility:

(1.2) $\text{div}u=0.$

and suitable initial and boundary conditions. Here f is an an external force (usually gravity), u is the velocity, $\tilde{p}$ is the pressure, ν is the viscosity and α ₁ and α ₂ are material stress moduli, the three parameters being constant and divided by the density. It is considered an appropriate model for the motion of a water solution of polymers.

Dunn and Fosdick prove in [13] that, to be consistent with thermodynamics, the viscosity and normal stress moduli must satisfy

$\begin{array}{lll}\nu \ge 0,\hfill & {\alpha }_1\ge 0\hfill & \text{and}{\alpha }_1+{\alpha }_2=0.\hfill \end{array}$

The reader can refer to [14] for a thorough discussion on the sign of α ₁.With these assumptions, setting α = α ₁, (1.1) simplifies and leads to the equation of motion

$\frac{\partial }{\partial t}\left(u-\alpha \text{Δ}u\right)-\nu \text{Δ}u+curl\left(u-\alpha \text{Δ}u\right)\times u+\nabla p=f,$

where the modified pressure p is related to $\tilde{p}$ by

$p=\tilde{p}+\frac{1}{2}u\cdot u-\alpha \left(u\cdot \text{Δ}u+\frac{1}{4}\text{tr}{A}_1^2\right),$

where A ₁ = ∇ u + (∇ u) ^t is the symmetric gradient tensor. Interestingly, in [21, 22], Holm, Marsden and Ratiu derive these equations with ν = 0 and α ₁ + α ₂ = 0, as a model of turbulence. They are called averaged-Euler equations, and α ₁ is an averaged length scale. It can also be interpreted as a measure of dispersion and in this respect, these equations describe a dispersive fluid model (cf. [18, 21]).

The equations of a grade-two fluid model have been studied by many authors (Videman gives in [31] a very extensive list of references), but the best construction of solutions for the problem, with homogeneous Dirichlet boundary conditions and mildly smooth data, is given by Ouazar in [26] and by Cioranescu and Ouazar in [7, 8]. These authors prove existence of solutions, with H ³ regularity in space, by looking for a velocity u such that

(1.3) $z=curl\left(u-\alpha \text{Δ}u\right),$

has L ² regularity in space, introducing z as an auxiliary variable and discretizing the equations of motion (in variational form) by Galerkin's method in the basis of the eigenfunctions of the operator curl curl(u – α Δ u). This excellent choice of basis allows one to recover estimates from the transport equation

(1.4) $\alpha \frac{\partial z}{\partial t}+\nu z+\alpha \left(u\cdot \nabla z-z\cdot \nabla u\right)=\nu curlu+\alpha curlf.$

Whenever curl f belongs to L ²(Ω)³, this construction is optimal because, in contrast to fixed-point arguments, it uses all the information conveyed by (1.1)–(1.4). Thus, it allows one to derive global existence of solutions with minimal restrictions on the size of the data, cf. [3] and [9]. A fixed-point argument cannot use all four equations because they are redundant. It is particularly important to preserve (1.1) since it implies that the energy is bounded without restrictions on the data. This point will be crucial for the numerical analysis of schemes discretizing (1.1).

The transport equation (1.4) substantially simplifies in two dimensions since the second nonlinear term z · ∇u vanishes. In this case, z = (0, 0, z) with z = curl(u – α Δ u), where curl is the operator

$\text{curl}v=\frac{\partial {\upsilon }_1}{\partial x_2}-\frac{\partial {\upsilon }_2}{\partial x_1}.$

Hence z is necessarily orthogonal to u = (u ₁, u ₂, 0). This vanishing term has a very important consequence: all the analysis can be performed without having to derive an a priori estimate for u in W^1,∞(Ω)². The same property will hold in the discrete case, provided the discrete scheme is suitably chosen.

In this article, we propose finite-element schemes for solving numerically the equations of a steady two dimensional grade-two fluid model, with a non-homogeneous tangential boundary condition. Defining z as above, the equation of motion becomes

(1.5) $\begin{array}{ll}-\nu \text{Δ}u+z\times u+\nabla p=f\hfill & \text{in}\text{Ω}\hfill \end{array},$

the incompressibility condition is unchanged:

(1.6) $\begin{array}{ll}\text{div}u=0\hfill & \text{in}\text{Ω}\hfill \end{array},$

the boundary condition is

(1.7) $\begin{array}{ll}u=g\hfill & \text{on}\partial \text{Ω}\text{with}g\cdot n=0\hfill \end{array},$

where n denotes the unit exterior normal to ∂Ω, and the transport equation becomes

(1.8) $\nu z+\alpha u\cdot \nabla z=\nu \text{curl}u+\alpha \text{curl}f.$

Girault and Scott in [16] prove that (1.5)–(1.8) always has a solution u in H ¹(Ω)² and p in L ²(Ω), on a Lipschitz-continuous domain, without restriction on the size of the data, provided curl f belongs to L ²(Ω), thus extending to rough data a result of Ouazar [26]. This unconditional existence result relies entirely on the fact that u does not need to be bounded in W^1,∞(Ω)². Similarly, our finite-element schemes are chosen so the numerical analysis can be performed without having to derive a uniform W ^1,∞ estimate for the discrete velocity. As expected, the difficulties arise from the transport equation (1.8). As is observed in [17] and [10], a straightforward argument shows that either the discrete velocity must have exactly zero divergence, or its non-zero divergence must be compensated by an extra stabilizing term in the transport equation or by a compatibility condition between the spaces of discrete pressure and discrete auxiliary variable z. Roughly speaking, let X_h , M_h and Z_h be discrete spaces for the velocity, pressure and variable z and, as usual, let us discretize (1.6) by

(1.9) $\forall q_h\in M_h,\left(q_h,\text{div}{u}_h\right)=0.$

Clearly, if we want to derive an unconditional a priori estimate from the discrete analogue of (1.8), we must be able to eliminate the nonlinear term. But even in the simplest case, Green's formula gives

${\displaystyle {\int }_{\text{Ω}}\left(u_h\cdot \nabla z_h\right)z_{h}dx=-\frac{1}{2}}{\displaystyle {\int }_{\text{Ω}}\left(\text{div}{u}_h\right){\left(z_h\right)}^{2}dx.}$

Hence, we can eliminate this right-hand side either by adding to the left-hand side of (1.8) a stabilizing, consistent term, so that it becomes

(1.10) $\nu z+\alpha u\cdot \nabla z+\frac{1}{2}\left(\text{div}u\right)z=\nu \text{curl}u+\alpha \text{curl}f,$

or by asking that

(1.11) ${\left(z_h\right)}^2\in M_h,$

and applying (1.9). Keeping this in mind, we propose to discretize (1.10) or (1.8) by an upwind scheme based on the discontinuous Galerkin method of degree one introduced by Lesaint and Raviart in [23]. This means that in each element of the triangulation, z _h is a polynomial of degree one, without continuity requirement on interelement boundaries. On one hand, if the form (1.10) is used for discretizing the transport equation, then we can approximate the velocity and pressure by the standard ℙ ₂ – ℙ ₁ Hood-Taylor scheme, where ℙ_k denotes the space of polynomials of degree k in two variables (cf. for example [15]). On the other hand, if we discretize the transport equation in the form (1.8), then (1.11) implies that p_h must be a polynomial of degree two, discontinuous across elements. In addition, the fact that the pressure and velocity spaces must satisfy a uniform discrete inf-sup condition implies that each component of u _h can be a polynomial of degree three plus two bubble functions of degree four, with continuity requirement on interelement boundaries (cf. [15]). Thus, denoting by c(u _h ; z_h , θ_h ) the discrete nonlinear part of (1.10) in variational form (cf. (3.5)), our scheme is: Find u _h in X_h + g _h , p_h in M_h and z_h in Z_h such that

(1.12) $\forall v_h\in X_h,\nu \left(\nabla u_h,\nabla v_h\right)+\left(z_h\times u_h,v_h\right)-\left(p_h,\text{div}{v}_h\right)=\left(f,v_h\right),$

(1.13) $\forall q_h\in M_h,\left(q_h,\text{div}{u}_h\right)=0,$

(1.14) $\begin{array}{r}\hfill \forall {\theta }_h\in Z_h,\nu \left(z_h,{\theta }_h\right)+c\left(u_h;z_h,{\theta }_h\right)=\nu \left(\text{curl}{u}_h,{\theta }_h\right)\\ \hfill +\alpha \left(\text{curl}f,{\theta }_h\right).\end{array}$

Here g _h is a suitable approximation of g and the functions of X_h vanish on ∂ Ω. Without restriction on the size of the data, we establish that this scheme always has a discrete solution in a Lipschitz polygonal domain and that this solution converges strongly to a solution of the exact problem. Furthermore, if the domain is convex and the data small, this solution can be computed by a converging successive approximation algorithm, with arbitrary starting guess. In addition, we prove an error inequality that leads to error estimates when the solution is sufficiently smooth. For both velocity-pressure discretizations, the error is of the order of h ^3/2, a result that remains valid as α tends to zero. With the ℙ ₂ – ℙ ₁ Hood-Taylor scheme, this is the best that can be achieved, considering that the discretization of the transport equation loses inevitably a factor h ^1/2. For the ℙ ₃ – ℙ ₂ scheme with discontinuous pressure, whose transport equation is simpler, this result is disappointing considering that the interpolation error for the velocity and pressure is of order h ³. These two results complete those of Girault and Scott in [17].

Remark 1.1

Another possibility is that z _h be constant in each element of the triangulation. Then (1.11) implies that p_h must also be a piecewise constant and we can associate with it the incomplete ℙ ₂ finite-element of Bernardi and Raugel [15] for the velocity, or even the non-conforminig element of Crouzeix and Raviart [12]. Otherwise, if we use the stabilizing term of (1.10), we can discretize the velocity and pressure with the "minielement" of Arnold, Brezzi and Fortin [15]. The analysis below extends to these examples and it can be shown that their error is the order of h ^1/2.

Remark 1.2

The results presented here are much more valuable than what Baia and Sequeira derive in [2]. Their analysis is of very limited use because, in order to guarantee the convergence of their algorithm (or even any algorithm), they must start with a first guess that has an error of order h ^3/2. And since they prove no a priori estimate, they cannot construct this first guess, which in fact amounts to solving their problem directly.

The remainder of this paper is divided into three sections. Section 1 briefly recalls the analysis of the exact problem (1.5)–(1.8) and compares it with the formulation proposed by [2]. The finite-element schemes are described in Section 2 and their error is estimated in Section 3.

We end this introduction by recalling some notation and basic functional results. For any non-negative integer m and number r ≥ 1, recall the classical Sobolev space (cf. Adams [1] or Nečas [25])

$W^{m,r}\left(\text{Ω}\right)=\left\{\upsilon \in L^r\left(\text{Ω}\right);{\partial }^k\upsilon \in L^r\left(\text{Ω}\right)\forall \left|k\right|\le m\right\},$

equipped with the seminorm

${\left|\upsilon \right|}_{W^{m,r}\left(\text{Ω}\right)}={\left[{\displaystyle \sum _{\left|k\right|=m}{\displaystyle {\int }_{\text{Ω}}{\left|{\partial }^k\upsilon \right|}^{r}dx}}\right]}^{1/r},$

and norm (for which it is a Banach space)

${\left|\upsilon \right|}_{W^{m,r}\left(\text{Ω}\right)}={\left[{\displaystyle \sum _{\left|k\right|=m}{\left|\upsilon \right|}_{W^{k,r}\left(\text{Ω}\right)}^r}\right]}^{1/r},$

with the usual extension when r = ∞. The reader can refer to [20] and [24] for extensions of this definition to non-integral values of m. When r = 2, this space is the Hilbert space H^m (Ω). In particular, the scalar product of L ²(Ω) is denoted by (·, ·). The definitions of these spaces are extended straightforwardly to vectors, with the same notation, but with the following modification for the norms in the non-Hilbert case. Let u = (u ₁, u ₂); then we set

${\Vert u\Vert }_{L^r\left(\text{Ω}\right)}={\left[{\displaystyle {\int }_{\text{Ω}}{\Vert u\left(x\right)\Vert }^{r}dx}\right]}^{1/r},$

where || · || denotes the Euclidean vector norm.

For vanishing boundary values, we define

$H_0^1\left(\text{Ω}\right)=\left\{\upsilon \in H^1\left(\text{Ω}\right);\upsilon |{}_{\partial \text{Ω}}=0\right\}.$

We shall often use Sobolev's imbeddings: for any real number p ≥ 1, there exists a constant S_p such that

(1.15) $\begin{array}{ll}\forall \upsilon \in H_0^1\left(\text{Ω}\right),\hfill & {\Vert \upsilon \Vert }_{L^p\left(\text{Ω}\right)}\le S_p{\left|\upsilon \right|}_{H^1\left(\text{Ω}\right)}.\hfill \end{array}$

When p = 2, this reduces to Poincaré's inequality and S ₂ is Poincaré's constant. For tangential boundary values, we define

(1.16) $H_T^1\left(\text{Ω}\right)=\left\{v\in H^1{\left(\text{Ω}\right)}^2;v\cdot n=0\text{on}\partial \text{Ω}\right\}.$

A straightforward application of Peetre-Tartar's Theorem (cf. [27] and [30] or [15]) shows that the analogue of Sobolev's imbeddings holds in $H_T^1\left(\text{Ω}\right)$ for any real number p ≥ 1:

(1.17) $\begin{array}{ll}\forall v\in H_T^1\left(\text{Ω}\right),\hfill & {\Vert v\Vert }_{L^p\left(\text{Ω}\right)}\le {\overline{S}}_p{\left|v\right|}_{H^1\left(\text{Ω}\right)}.\hfill \end{array}$

In particular, for p = 2, the mapping $v\mapsto {\left|v\right|}_{H^1\left(\text{Ω}\right)}$ is a norm on $H_T^1\left(\text{Ω}\right)$ equivalent to the H ¹ norm and ${\tilde{S}}_2$ is the analogue of Poincaré's constant. We shall also use the standard spaces for Navier-Stokes equations

$\begin{array}{l}V=\left\{v\in H_0^1{\left(\text{Ω}\right)}^2;\text{div}v=0\text{in}\text{Ω}\right\},\hfill \\ W=\left\{v\in H_T^1\left(\text{Ω}\right);\text{div}v=0\text{in}\text{Ω}\right\},\hfill \\ L_0^2\left(\text{Ω}\right)=\left\{q\in L^2\left(\text{Ω}\right);{\displaystyle {\int }_{\text{Ω}}qdx=0}\right\},\hfill \end{array}$

and also the space

$H\left(\text{curl},\text{Ω}\right)=\left\{v\in L^2{\left(\text{Ω}\right)}^2;\text{curl}v\in L^2\left(\text{Ω}\right)\right\}.$

Finally, recall the standard lifting w_g in W of g: it is the solution of the non-homogeneous Stokes problem:

(1.18) $\begin{array}{lll}-\text{Δ}{w}_g+\nabla p_g=0\text{and}\text{div}{w}_g=0\text{in}\text{Ω}\hfill & ,\hfill & w_g=g\text{on}\partial \text{Ω}.\hfill \end{array}$

It satisfies the bound, with a constant T that depends only on the domain (cf. for instance [15])

(1.19) ${\left|w_g\right|}_{H^1\left(\text{Ω}\right)}\le T{\Vert g\Vert }_{H^{1/2}\left(\text{Ω}\right)}.$

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Design Optimization Applications with Implicit Functions

Jasbir S. Arora , in Introduction to Optimum Design (Second Edition), 2004

14.7.2 Discrete Variable Optimization

We use the adaptive numerical optimization procedure described in Section 2.11.4 to obtain a discrete variable solution. The procedure is to use the program IDESIGN in an interactive mode. Design conditions are monitored and decisions made to fix design variables that are not changing. The interactive facilities used include design variable histories, maximum constraint violation, and the cost function.

Table 14-12 contains a snapshot of the design conditions at various iterations and the decisions made. It can be seen that for the first five iterations the constraint violations are

TABLE 14-12. Interactive Solution for a 10-Member Structure with Discrete Variables

Iteration no.	Maximum violation (%)	Cost function	Algorithm used	Variables fixed to value shown in parentheses
1	1.274E+04	679.83	CC	All free
2	4.556E+03	1019.74	CC	All free
3	1.268E+03	1529.61	CC	All free
4	4.623E+02	2294.42	CC	All free
5	1.144E+02	3441.63	CC	All free
6	2.020E+01	4722.73	CC	5 (1.62), 10 (1.62)
7	2.418E+00	5389.28	CCC	2 (1.80)
11	1.223E-01	5402.62	SQP	1 (30.0), 6 (3.84), 7 (13.5)
13	5.204E-04	5411.13	SQP	3 (26.5), 9 (19.9)
14	1.388E+00	5424.69	—	4 (13.5), 8 (16.9)

CC, constraint correction algorithm; CCC, constraint correction at constant cost; SQP, sequential quadratic programming.

very large, so the constraint correction (CC) algorithm is used to correct the constraints. At the sixth iteration, it is determined that design variables 5 and 10 are not changing, so they are fixed to their current value, Similarly, at other iterations, variables are assigned values from the available set. At the 14th iteration, variables have discrete values, the constraint violation is about 1.4 percent and the structural weight is 5424.69, which is an increase of less than 1 percent from the true optimum. This is a reasonable final solution.

It should be noted that with the discrete variables, several solutions near the true optimum point are possible. A different sequence of fixing variables can give a different solution. For example, starting from the optimum solution with continuous variables, the following acceptable discrete solutions are obtained interactively:

1.

30.0, 1.62, 26.5, 13.9, 1.62, 4.18, 13.5, 18.8, 18.8, 1.62; cost = 5485.6, max. viol. = 4.167 percent for stress in member 2.

2.

Same as (1) except the eighth design variable is 16.9; cost = 5388.9 and max. viol. = 0.58 percent.

3.

Same as (1) except design variables 2 and 6 are 2.38 and 2.62; cost = 5456.8, max. viol. = 3.74 percent for stress in member 2

4.

Same as (3) except design variable 2 is 2.62; cost = 5465.4; all constraints are satisfied.

It can be seen that the interactive facilities described in Chapter 13 can be exploited to obtain practical engineering designs.

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The Finite Element Method for Elliptic Problems

In Studies in Mathematics and Its Applications, 1978

Introduction

In this chapter, we consider the problem of determining estimates in various norms of the difference (u − u _h ), where u ∈ V is the solution of a second-order boundary value problem and u _h ∈ V _h is the discrete solution obtained in a subspace V _h of V.

From Céa's lemma (Theorem 2.4.1), the best error estimate would result in exhibiting the element θ _h u ∈ V _h which is such that inf _vh ∈V _h ||u − v _h ||_1,Ω = ||u - θ _h u||_1,Ω, i.e., the projection of the solution u on the space V _h . However, such a projection is not particularly easy to work with, and it turns out that it is much more convenient to use the X _h -interpolant Π _h u of the solution u, so that we shall get instead the error estimate ||u − u _h ||_1,Ω ≤ C||u - Π _h u||_1,Ω.

Since we shall assume in this chapter that the set $\overline{\Omega }$ is polygonal, it can be written as a union $\overline{\Omega }={\cup }_{K\in {\mathcal{T}}_h}K$ of polygonal finite elements K, such as the ones which have been heretofore described. This in turn implies that the corresponding spaces V _h will be contained in the space V (the domains of definition of their functions are identical), i.e., that the corresponding finite element method is indeed conforming.

Taking into account that we are using the norm ||·||_1,Ω and that (π _h u)| _K = Π _K u for all K ∈ C _h (Theorem 2.3.2), we can write

${\left|\left|u-{\Pi }_{h}u\right|\right|}_{1,\Omega }={\left(\sum _{K\in {\mathcal{T}}_h}{\left|\left|u-{\Pi }_{K}u\right|\right|}_{1,K}^2\right)}^{1/2}.$

Therefore, the problem of finding an estimate for the error ||u − u _h ||_1,Ω is reduced to the problem of evaluating quantities such as ||u - Π _K u||_1,K and the solution of such "local" interpolation problems is the object of Section 3.1. In view of other future needs, we shall in fact estimate the difference (u - π _K u) with respect to more general norms and seminorms.

A typical – and crucial – result in this direction is that, for a finite element (K, P _K , Σ _K ) which can be imbedded in an affine family and whose P _K -interpolation operator leaves invariant the polynomials of degree ≤ k (equivalently, the inclusions P _k (K)⊂ P _K hold), there exists a constant C independent of K such that

$\forall v\in H^{k+1}\left(K\right),{\left|v-{\Pi }_{K}v\right|}_{m,K}\le C\frac{h_K^{k+1}}{{\rho }_K^m}{\left|v\right|}_{k+1,K},0\le m\le k+1,$

where

$\begin{array}{l}{h}_K=\text{diameter}\text{of}K,\\ {\rho }_K=\text{supremum}\text{of\hspace{0.17em}the\hspace{0.17em}diameters}\text{of\hspace{0.17em}the}\text{sphers}\text{inscribed}\text{in}\text{K}\end{array}$

Such a result is proved (in a more general form) in Theorem 3.1.5.

One key idea in the process of getting such estimates is to go from any finite element K of an affine family to the reference finite element of the family and then back to the finite element K.

Another key is to use a basic result about Sobolev spaces, due to J. Deny and J.L. Lions, which pervades the mathematical analysis of the finite element method: Over the quotient space H ^k+1(Ω)/P _k (Ω), the semi-norm |·| _k+1,Ω is a norm equivalent to the quotient norm. This result is proved in Theorem 3.1.1, for the more general Sobolev spaces W ^m,p (Ω).

In practice, one often considers a regular family of finite elements, in the sense that the diameters h _K approach zero, and that there exists a constant σ independent of K such that h _K ≤ σρ _K . For such a regular family, the previous interpolation error estimate becomes (Theorem 3.1.6)

${\left|v-{\Pi }_{K}v\right|}_{m,K}=\text{O}\left(h_K^{k+1-m}\right),0\le m\le k+1.$

Using Céa's lemma, we obtain in Section 3.2 the error estimates (Theorem 3.2.2).

${\left|\left|u-u_h\right|\right|}_{1,\Omega }\le C{\left|\left|u-{\Pi }_{h}u\right|\right|}_{1,\Omega }=\text{O}\left(h^k\right),withh=\underset{K\in {\mathcal{J}}_h}{\max }{h}_K,$

under basically the same assumptions as before about the family of finite elements which make up the finite element spaces. It is worth mentioning here that, although the above error estimate is not the best, it is generally possible to show that the order of convergence is the best possible: In other words, it would not be improved by replacing Π _h u by θ _h u.

Nevertheless the range of applicability of the above results is limited inasmuch as the X _h -interpolant of the solution u is defined only if some smoothness is assumed on the solution u, and likewise, the above error estimates are obtained provided the solution is sufficiently smooth (u ∈ H ^k+1(Ω)). Fortunately, we show in Theorem 3.2.3 that, with the minimal assumptions that the solution u is in the space H ¹(Ω) and that the spaces P _K contain the space P ₁(K), K ∈

_h , convergence still holds, i.e., one has lim _h→0 ||u − u _h ||_1,Ω = 0.

Next, using a method due to J.P. Aubin and J.A. Nitsche (cf. the Aubin – Nitsche lemma; Theorem 3.2.4), we show that there is in most cases an improvement in the error estimate in the norm |·|_0,Ω in the sense that (Theorem 3.2.5)

${\left|u-u_h\right|}_{0,\Omega }=\text{O}\left(h^{k+1}\right).$

Section 3.2 ends up with the so-called inverse inequalities (Theorem 3.2.6).

Finally, in Section 3.3, we follow the penetrating method of weighted norms of J.A. Nitsche, who has recently shown that, if u ∈ W ^k+1,∞(Ω),

$\begin{array}{l}{\left|u-u_h\right|}_{0,\infty ,\Omega }=\{\begin{array}{l}\text{O}\left(h^{2-\in }\right)\text{for}\text{any}\in >0ifk=1,\\ \text{O}\left(h^{k+1}\right)ifk\ge 2,\end{array}\\ {\left|u-u_h\right|}_{1,\infty ,\Omega }=\{\begin{array}{l}\text{O}\left(h^{1-\in }\right)\text{for}\text{any}\in >0ifk=1,\\ \text{O}\left(h^k\right)ifk\ge 2,\end{array}\end{array}$

where |·|_0,∞,Ω and ||·||_1,∞,Ω stand for the norms of the spaces L ^∞(Ω) and W ^1,∞(Ω), respectively. Restricting ourselves for brevity to the case k = 1, the corresponding error estimates are obtained in Theorem 3.3.7.

It is worth pointing out that all the error estimates found in Section 3.2 and 3.3 are optimal in the sense that, with the same regularity assumptions on the function u, one gets the same asymptotic estimates (or "almost" the same for the norms |·|_0,∞,Ω and ||·||_1,∞,Ω when k = 1) when the discrete solution u _h ∈ V _h is replaced by the X _h -interpolant Π _h u ∈ V _h .

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Discrete Variable Optimum Design Concepts and Methods

Jasbir S. Arora , in Introduction to Optimum Design (Third Edition), 2012

15.10.2 Discrete Variable Optimization

We use the adaptive numerical optimization procedure that was described in Section 2.11.8 to obtain a discrete variable solution. The procedure is to use the IDESIGN program in the interactive mode. Design conditions are monitored and decisions made to fix design variables that are not changing. The interactive facilities used include design variable histories, maximum constraint violation, and the cost function.

Table 15.6 contains a snapshot of the design conditions at various iterations and the decisions made. It can be seen that for the first five iterations the constraint violations are very large, so the constraint correction (CC) algorithm is used to correct the constraints. At the 6th iteration, it is determined that design variables 5 and 10 are not changing, so they are fixed to their current value. Similarly, at other iterations, variables are assigned values from the available set. At the 14th iteration, all variables have discrete values, the constraint violation is about 1.4 percent, and the structural weight is 5424.69, which is an increase of less than 1 percent from the optimum with continuous variables. This is a reasonable final solution.

Table 15.6. Interactive solution for a 10-Member structure with discrete variables

Iteration no.	Maximum violation (%)	Cost function	Algorithm used	Variables fixed to value shown in parentheses
1	1.274E+04	679.83	CC	All free
2	4.556E+03	1019.74	CC	All free
3	1.268E+03	1529.61	CC	All free
4	4.623E+02	2294.42	CC	All free
5	1.144E+02	3441.63	CC	All free
6	2.020E+01	4722.73	CC	5(1.62), 10(1.62)
7	2.418E+00	5389.28	CCC	2(1.80)
11	1.223E−01	5402.62	SQP	1(30.0), 6(3.84), 7(13.5)
13	5.204E−04	5411.13	SQP	3(26.5), 9(19.9)
14	1.388E+00	5424.69	—	4(13.5), 8(16.9)

CC=constraint correction algorithm; CCC=constraint correction at constant cost; SQP=sequential quadratic programming.

It should be noted, that with the discrete variables, several solutions near the true optimum point are possible. A different sequence of fixing variables can give a different solution. For example, starting from the optimum solution with continuous variables, the following acceptable discrete solutions are obtained interactively:

1.

30.0, 1.62, 26.5, 13.9, 1.62, 4.18, 13.5, 18.8, 18.8, 1.62; cost=5485.6, maximum violation=4.167 percent for stress in member 2.

2.

Same as (1) except the 8th design variable is 16.9; cost=5388.9, and maximum violation=0.58 percent.

3.

Same as (1) except design variables 2 and 6 are 2.38 and 2.62; cost=5456.8, maximum violation=3.74 percent for stress in member 2

4.

Same as (3) except design variable 2 is 2.62; cost=5465.4; all constraints are satisfied.

Exercises for Chapter 15*

15.1

Solve Example 15.1 with the available discrete values for the variables as x₁∈{0, 1, 2, 3}, and x₂∈{0, 1, 2, 3, 4, 5, 6}. Assume that the functions of the problem are not differentiable.

15.2

Solve Example 15.1 with the available discrete values for the variables as x₁∈{0, 1, 2, 3}, and x₂∈{0, 1, 2, 3, 4, 5, 6}. Assuming that the functions of the problem are differentiable, use a continuous variable optimization procedure to solve for discrete variables.

15.3

Formulate and solve Exercise 3.34 using the outside diameter d ₀ and the inside diameter d _i as design variables. The outside diameter and thickness must be selected from the following available sets:

$d_0\in \left\{0.020\text{,}0.022\text{,}0.024\text{,}\dots \text{,}0.48\text{,}0.50\right\}\mathrm{m};t\in \left\{5\text{,}7\text{,}9\text{,}\dots \text{,}23\text{,}25\right\}\text{mm}$

Check your solution using the graphical method of Chapter 3. Compare continuous and discrete solutions.

15.4

Consider the minimum mass tubular column problem formulated in Section 2.7. Find the optimum solution for the problem using the following data: P=100   kN, length, l=5   m, Young's modulus, E=210   GPa, allowable stress, σ _a =250   MPa, mass density, ρ=7850   kg/m³, R≤0.4   m, t≤0.05   m, and R, t≥0. The design variables must be selected from the following sets:

$R\in \left\{0.01\text{,}0.012\text{,}0.014\text{,}\dots \text{,}0.38\text{,}0.40\right\}\mathrm{m};t\in \left\{4\text{,}6\text{,}8\text{,}\dots \text{,}48\text{,}50\right\}\text{mm}$

Check your solution using the graphical method of Chapter 3. Compare continuous and discrete solutions.

15.5

Consider the plate girder design problem described and formulated in Section 6.6. The design variables for the problem must be selected from the following sets

$h\text{,}b\in \left\{0.30\text{,}0.31\text{,}0.32\text{,}\dots \text{,}2.49\text{,}2.50\right\}\mathrm{m};t_w\text{,}{t}_f\in \left\{10\text{,}12\text{,}14\text{,}\dots \text{,}98\text{,}100\right\}\text{mm}$

Assume that the functions of the problem are differentiable and a continuous variable optimization program can be used to solve subproblems, if needed. Solve the discrete variable optimization problem. Compare the continuous and discrete solutions.

15.6

Consider the plate girder design problem described and formulated in Section 6.6. The design variables for the problem must be selected from the following sets

$h\text{,}b\in \left\{0.30\text{,}0.31\text{,}0.32\text{,}\dots \text{,}2.49\text{,}2.50\right\}\mathrm{m};t_w\text{,}{t}_f\in \left\{10\text{,}12\text{,}14\text{,}\dots \text{,}98\text{,}100\right\}\text{mm}$

Assume functions of the problem to be nondifferentiable. Solve the discrete variable optimization problem. Compare the continuous and discrete solutions.

15.7

Consider the plate girder design problem described and formulated in Section 6.6. The design variables for the problem must be selected from the following sets

$h\text{,}b\in \left\{0.30\text{,}0.31\text{,}0.32\text{,}\dots \text{,}2.48\text{,}2.50\right\}\mathrm{m};t_w\text{,}{t}_f\in \left\{10\text{,}14\text{,}16\text{,}\dots \text{,}96\text{,}100\right\}\text{mm}$

Assume that the functions of the problem are differentiable and a continuous variable optimization program can be used to solve subproblems, if needed. Solve the discrete variable optimization problem. Compare the continuous and discrete solutions.

15.8

Consider the plate girder design problem described and formulated in Section 6.6. The design variables for the problem must be selected from the following sets

$h\text{,}b\in \left\{0.30\text{,}0.32\text{,}0.34\text{,}\dots \text{,}2.48\text{,}2.50\right\}\mathrm{m};t_w\text{,}{t}_f\in \left\{10\text{,}14\text{,}16\text{,}\dots \text{,}96\text{,}100\right\}\text{mm}$

Assume functions of the problem to be nondifferentiable. Solve the discrete variable optimization problem. Compare the continuous and discrete solutions.

15.9

Solve the problems of Exercises 15.3 and 15.5. Compare the two solutions, commenting on the effect of the size of the discreteness of variables on the optimum solution. Also, compare the continuous and discrete solutions.

15.10

Consider the spring design problem formulated in Section 2.9 and solved in Section 6.5. Assume that the wire diameters are available in increments of 0.01 in, the coils can be fabricated in increments of 1/16th of an inch, and the number of coils must be an integer. Assume functions of the problem to be differentiable. Compare the continuous and discrete solutions.

15.11

Consider the spring design problem formulated in Section 2.9 and solved in Section 6.5. Assume that the wire diameters are available in increments of 0.01 in, the coils can be fabricated in increments of 1/16th of an inch, and the number of coils must be an integer. Assume the functions of the problem to be nondifferentiable. Compare the continuous and discrete solutions.

15.12

Consider the spring design problem formulated in Section 2.9 and solved in Section 6.5. Assume that the wire diameters are available in increments of 0.015 in, the coils can be fabricated in increments of 1/8th of an inch, and the number of coils must be an integer. Assume functions of the problem to be differentiable. Compare the continuous and discrete solutions.

15.13

Consider the spring design problem formulated in Section 2.9 and solved in Section 6.5. Assume that the wire diameters are available in increments of 0.015 in, the coils can be fabricated in increments of 1/8th of an inch, and the number of coils must be an integer. Assume the functions of the problem to be nondifferentiable. Compare the continuous and discrete solutions.

15.14

Solve problems of Exercises 15.8 and 15.10. Compare the two solutions, commenting on the effect of the size of the discreteness of variables on the optimum solution. Also, compare the continuous and discrete solutions.

15.15

Formulate the problem of optimum design of prestressed concrete transmission poles described in Kocer and Arora (1996a). Use a mixed variable optimization procedure to solve the problem. Compare the solution to that given in the reference.

15.16

Formulate the problem of optimum design of steel transmission poles described in Kocer and Arora (1996b). Solve the problem as a continuous variable optimization problem.

15.17

Formulate the problem of optimum design of steel transmission poles described in Kocer and Arora (1996b). Assume that the diameters can vary in increments of 0.5 in and the thicknesses can vary in increments of 0.05 in. Solve the problem as a discrete variable optimization problem.

15.18

Formulate the problem of optimum design of steel transmission poles using standard sections described in Kocer and Arora (1997). Compare your solution to the solution given there.

15.19

Solve the following mixed variable optimization problem (Hock and Schittkowski, 1981):

Minimize

$f={(x_1-10)}^2+5{(x_2-12)}^2+3{(x_4-11)}^2+10x_5^6+7x_6^2+x_7^4-4x_6{x}_7-10x_6-8x_7$

subject to

$g_1=2x_1^2+3x_2^4+x_3+4x_4^2+5x_5\le 127$

$g_2=7x_1+3x_2+10x_3^2+x_4-x_5\le 282$

$g_3=23x_1+x_2^2+6x_6^2-8x_7\le 196$

$g_4=4x_1^2+x_2^2-3x_1{x}_2+2x_3^2+5x_6-11x_7\le 0$

The first three design variables must be selected from the following sets

$x_1\in \left\{1\text{,}2\text{,}3\text{,}4\text{,}5\right\};x_2\text{,}{x}_3\in \left\{1\text{,}2\text{,}3\text{,}4\text{,}5\right\}$

15.20

Formulate and solve the three-bar truss of Exercise 3.50 as a discrete variable problem where the cross-sectional areas must be selected from the following discrete set:

$A_i\in \left\{50\text{,}100\text{,}150\text{,}\dots \text{,}4950\text{,}5000\right\}{\text{mm}}^2$

Check your solution using the graphical method of Chapter 3. Compare continuous and discrete solutions.

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URL:

https://www.sciencedirect.com/science/article/pii/B9780123813756000152

Discrete Variable Optimum Design Concepts and Methods

Jasbir S. Arora , in Introduction to Optimum Design (Second Edition), 2004

Exercises for Chapter 15 ^*

15.1

Solve Example 15.1 with the available discrete values for the variables as x₁ ∈ {0, 1, 2, 3}, and x₂ ∈ {0, 1, 2, 3, 4, 5, 6}. Assume that the functions of the problem are not differentiable.

15.2

Solve Example 15.1 with the available discrete values for the variables as x₁ ∈ {0, 1, 2, 3}, and x₂ ∈ {0, 1, 2, 3, 4, 5, 6}. Assuming that the functions of the problem are differentiable, use a continuous variable optimization procedure to solve for discrete variables.

15.3

Formulate and solve Exercise 3.34 using the outside diameter d ₀ and the inside diameter d _i as design variables. The outside diameter and thickness must be selected from the following available sets:

d ₀ ∈ {0.020, 0.022, 0.024, …, 0.48, 0.50} m;t ∈ {5, 7, 9, …, 23,25} mm

Check your solution using the graphical method of Chapter 3 . Compare continuous and discrete solutions.

15.4

15.4 Consider the minimum mass tubular column problem formulated in Section 2.7. Find the optimum solution for the problem using the following data: P = 100 kN, length, l = 5 m, Young's modulus, E = 210 GPa, allowable stress, σ _a = 250 MPa, mass density, ρ = 7850 kg/m³, R ≤ 0.4 m, t ≤ 0.05 m, and R, t ≥ 0. The design variables must be selected from the following sets:

R ∈ {0.01, 0.012, 0.014, …, 0.38, 0.40} m; t ∈ {4, 6, 8, …, 48, 50} mm

Check your solution using the graphical method of Chapter 3. Compare continuous and discrete solutions.

15.5

Consider the plate girder design problem described and formulated in Section 10.6. The design variables for the problem must be selected from the following sets

h, b ∈ {0.30, 0.31, 0.32, …, 2.49, 2.50} m; t _w , t _f ∈ {10, 12, 14, …, 98, 100} mm

Assume that the functions of the problem are differentiable and a continuous variable optimization program can be used to solve subproblems, if needed. Solve the discrete variable optimization problem. Compare the continuous and discrete solutions.

15.6

Consider the plate girder design problem described and formulated in Section 10.6. The design variables for the problem must be selected from the following sets

h, b ∈ {0.30, 0.31, 0.32, …, 2.49, 2.50} m; t _w , t _f ∈ {10, 12, 14, …, 98, 100} mm

Assume functions of the problem to be nondifferentiable. Solve the discrete variable optimization problem. Compare the continuous and discrete solutions.

15.7

Consider the plate girder design problem described and formulated in Section 10.6. The design variables for the problem must be selected from the following sets

h, b ∈ {0.30, 0.31, 0.32, …, 2.48,2.50} m; t _w , t _f ∈ {10, 14, 16, …, 96, 100} mm

Assume that the functions of the problem are differentiable and a continuous variable optimization program can be used to solve subproblems, if needed. Solve the discrete variable optimization problem. Compare the continuous and discrete solutions.

15.8

Consider the plate girder design problem described and formulated in Section 10.6. The design variables for the problem must be selected from the following sets

h, b ∈ {0.30, 0.32, 0.34, …, 2.48, 2.50} m; t _w , t _f ∈ {10, 14, 16, …, 96, 100} mm

Assume functions of the problem to be nondifferentiable. Solve the discrete variable optimization problem. Compare the continuous and discrete solutions.

15.9

Solve the problems of Exercises 15.3 and 15.5. Compare the two solutions, commenting on the effect of the size of the discreteness of variables on the optimum solution. Also, compare the continuous and discrete solutions.

15.10

Consider the spring design problem formulated in Section 2.9 and solved in Section 13.5. Assume that the wire diameters are available in increments of 0.01 in, the coils can be fabricated in increments of 1/16th of an inch, and the number of coils must be an integer. Assume functions of the problem to be differentiable. Compare the continuous and discrete solutions.

15.11

Consider the spring design problem formulated in Section 2.9 and solved in Section 13.5. Assume that the wire diameters are available in increments of 0.01 in, the coils can be fabricated in increments of 1/16th of an inch, and the number of coils must be an integer. Assume the functions of the problem to be nondifferentiable. Compare the continuous and discrete solutions.

15.12

Consider the spring design problem formulated in Section 2.9 and solved in Section 13.5. Assume that the wire diameters are available in increments of 0.015 in, the

coils can be fabricated in increments of 1/8th of an inch, and the number of coils must be an integer. Assume functions of the problem to be differentiable. Compare the continuous and discrete solutions.

15.13

Consider the spring design problem formulated in Section 2.9 and solved in Section 13.5. Assume that the wire diameters are available in increments of 0.015 in, the coils can be fabricated in increments of 1/8th of an inch, and the number of coils must be an integer. Assume the functions of the problem to be nondifferentiable. Compare the continuous and discrete solutions.

15.14

Solve problems of Exercises 15.8 and 15.10. Compare the two solutions, commenting on the effect of the size of the discreteness of variables on the optimum solution. Also, compare the continuous and discrete solutions.

15.15

Formulate the problem of optimum design of prestressed concrete transmission poles described in Kocer and Arora (1996a). Use a mixed variable optimization procedure to solve the problem. Compare the solution to that given in the reference.

15.16

Formulate the problem of optimum design of steel transmission poles described in Kocer and Arora (1996b). Solve the problem as a continuous variable optimization problem.

15.17

Formulate the problem of optimum design of steel transmission poles described in Kocer and Arora (1996b). Assume that the diameters can vary in increments of 0.5 in and the thicknesses can vary in increments of 0.05 in. Solve the problem as a discrete variable optimization problem.

15.18

Formulate the problem of optimum design of steel transmission poles using standard sections described in Kocer and Arora (1997). Compare your solution to the solution given there.

15.19

Solve the following mixed variable optimization problem (Hock and Schittkowski, 1981):

minimize

$\begin{array}{l}f={\left(x_1-10\right)}^2+5{\left(x_2-12\right)}^2+x_3^4+3{\left(x_4-11\right)}^2+10x_5^6\\ +7x_6^2+x_7^4-4x_6{x}_7-10x_6-8x_7\end{array}$

subject to

$\begin{array}{l}{g}_1=2x_1^2+3x_2^4+x_3+4x_4^2+5x_5\le 127\\ g_2=7x_1+3x_2+10x_3^2+x_4-x_5\le 282\\ g_3=23x_1+x_2^2+6x_6^2-8x_7\le 196\\ g_4=4x_1^2+x_2^2-3x_1{x}_2+2x_3^2+5x_6-11x_7\le 0\end{array}$

The first three design variables must be selected from the following sets

x ₁ ∈ {1, 2, 3, 4, 5}; x ₂, x ₃ ∈ {1, 2, 3, 4, 5}

15.20

Formulate and solve the three-bar truss of Exercise 3.50 as a discrete variable problem where the cross-sectional areas must be selected from the following discrete set:

A _i ∈ {50, 100, 150, …, 4950, 5000} mm²

Check your solution using the graphical method of Chapter 3. Compare continuous and discrete solutions.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B978012064155050015X

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