Philippe G. Ciarlet

Maximum principle and uniform convergence for the finite element method

Abstract The solution of the Dirichlet boundary value problem over a polyhedral domain Ω ⊂ R n , n ≥ 2, associated with a second-order elliptic operator, is approximated by the simplest finite element method, where the trial functions are piecewise linear. When the discrete problem satisfies a maximum principle, it is shown that the approximate solution u h converges uniformly to the exact solution u if u ϵ W 1, p (Ω), with p > n , and that ∥u−u h ∥ L ∞(Ω) = O(h) if u ϵ W 2 , p (Ω), with 2 p > n . In the case of the model problem − Δu + au = f in Ω, u = u o on δΩ, with a ⩾ 0, a simple geometrical condition is given which insures the validity of the maximum principle for the discrete problem.

Introduction to numerical linear algebra and optimisation

Preface Part I. Numerical Linear Algebra: 1. A summary of results on matrices 2. General results in the numerical analysis of matrices 3. Sources of problems in the numerical analysis of matrices 4. Direct methods for the solution of linear systems 5. Iterative methods for the solution of linear systems 6. Methods for the calculation of eigenvalues and eigenvectors Part II. Optimisation: 7. A review of differential calculus. Some applications 8. General results on optimisation. Some algorithms 9. Introduction to non-linear programming 10. Linear programming Bibliography and comments Main notations used Index.

A Mixed Finite Element Method for the Biharmonic Equation

Publisher Summary This chapter explains a mixed finite element method for the Biharmonic equation. One way to avoid the computational difficulty is to use a non-conforming method, in which the approximate solution lies in a finite-dimensional space. The other way to avoid the computational difficulty is based on a different variational principle called the standard variational formulation of problem. Different variational principle falls into two categories: either the variational principle is explicitly dependent upon a given triangulation of the set Ώ, in which case the method is called hybrid, or otherwise, it is called a mixed method. The chapter also discusses an approximate method related to yet another variational principle, in which the Laplacian Δu of the solution plays a special role.

A justification of a nonlinear model in plate theory

Abstract The asymptotic expansion method is applied to a nonlinear model in three-dimensional elasticity. Without any a priori assumption (either geometrical or mechanical in nature), it is shown that the first term in the expansion is a solution of a known bi-dimensional model in nonlinear plate theory. The existence of the first term is also established.

Discrete maximum principle for finite-difference operators

with XI:R2--.R continuous functions together with their partial derivatives Xij= =coXi/coxj, i , j= 1, 2. The functions ei:]to, + oo[-~R, to>_. o% are such as to guarantee the existence of solutions of any initial problem for (1). The solutions of (1) are said to be convergent if for each pair of solutions of (1), (xl , x2) and (Yl, Y2), which are defined on a neighborhood of the point + o0, we have xi( t ) -y i ( t ) -*O when t ~ + 0% i = 1, 2. We easily see that there exists at the most one periodic (or almost periodic) solution for a system whose solutions are convergent. The main result of this note is the following criterion of convergence:

A justification of the von Kármán equations

The method of asymptotic expansions, with the thickness as the parameter, is applied to the nonlinear, three-dimensional, equations for the equilibrium of a special class of elastic plates under suitable loads. It is shown that the leading term of the expansion is the solution of a system of equations equivalent to those of von Kármán. The existence of solutions of this system is established. It is also shown that the displacement and stress corresponding to the leading term of the expansion have the specific form generally assumed in the usual derivations of the von Kármán equations; in particular, the displacement field is of Kirchhoff-Love type. This approach also clarifies the nature of admissible boundary conditions for both the von Kármán equations and the three-dimensional model from which these equations are obtained. A careful discussion of the limitations of this approach is given in the conclusion.

THE COMBINED EFFECT OF CURVED BOUNDARIES AND NUMERICAL INTEGRATION IN ISOPARAMETRIC FINITE ELEMENT METHODS

Abstract Consider the model problem where Ω is a bounded open subset of R n with a curved boundary Γ. The finite element method with isoparametric finite elements is applied to this problem, with curved finite elements along the boundary, in connection with a numerical quadrature scheme which is used to compute the coefficients of the resulting linear system. However, except in some very special cases, the interior Ω h of the union of the finite elements is not equal to Ω (although the boundary of Ω h is very close to Γ), and thus, taking also numerical integration into account, we obtain an approximate solution u h in the space H 1 0 (Ω h ). For several types of isoparametric finite elements (of simplicial and quadrilateral type) commonly used by the Engineers, we prove asymptotic error estimates of the form where r = 0, 1 or 2 depending upon the type of finite elements which are used, h is the greatest diameter of the finite elements, and either ũ is an extension of the exact solution u to the set Ω h ⊄ Ω, or ũ = u if Ω ⋐ Ω

Asymptotic Analysis of Linearly Elastic Shells. III. Justification of Koiter's Shell Equations

AbstractWe consider as in Part I a family of linearly elastic shells of thickness 2ɛ, all having the same middle surfaceS=ϕ(ϖ)⊂R3, whereω⊂R2 is a bounded and connected open set with a Lipschitz-continuous boundary, andϕ∈l3 (ϖ;R3). The shells are clamped on a portion of their lateral face, whose middle line isϕ(γ0), whereγ0 is any portion of∂ω withlength γ0>0. We make an essential geometrical assumption on the middle surfaceS and on the setγ0, which states that the space of inextensional displacements % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaieWacG% aGC9NvamacaY1gaaWcbGaGCHqaciacaYLFgbaabKaGCbGccGaGCjik% aiadaYfHjpWDcGaGCjykaiadaYLH9aqpieaajqgaadGaa03EaGGadO% Gamaiu8D7aOHGaaiadac1E9aqpcWaGqThkaGIamaiueE7aOnacac1g% aaWcbGaGqjacacLFPbaabKaGqbGccGaGqjykaiadacLHiiIZcGaGqn% isamacacfhaaWcbKaGqfacacLaiaiuigdaaaGccGaGqjikaiadacfH% jpWDcGaGqjykaiadacLHxdaTcGaGqnisamacacfhaaWcbKaGqfacac% LaiaiuigdaaaGccGaGqjikaiadacfHjpWDcGaGqjykaiadacLHxdaT% cGaGqnisamacacfhaaWcbKaGqfacacLaiaiuikdaaaGccGaGqjikai% adacfHjpWDcGaGqjykaiacacLG7aaabaGaa8hiaiaa-bcacaWFGaGa% a8hiaiaa-bcacaWFGaGaa8hiaiaa-bcacaWFGaGaa8hiaiaa-bcaca% WFGaGaa8hiaiadaseH3oaAdGaGeTbaaSqaiaircGaGe5xAaaqajair% aOGaiair-1dacWaGezOaIy7aiairBaaaleacasKaiGgi+zhaaeqcas% eakiad0HdH3oaAdGaGeTbaaSqaiaircGaGeH4maaqajairaOGaiair% -1dacGaGe9hiaiacasuFWaGaiair-bcacGaGe13BaiacasuFUbGaia% ir+bcacWaGer4SdC2aiairBaaaleacasKaiair+bdaaeqcaseakiac% asKGSaGamaireo7aNnacas0gaaWcbGaGejadCHdHXoqycWax4qOSdi% gabKaGebGccGaxejikaiadCreF3oaAcGaxejykaiadCrKH9aqpcGax% eHimaiacCruFGaGaiWfr9LgacGaxe1NBaiacCruFGaGamWfreM8a3L% azaamacGaxakyFaiacCbOGSaaaaaa!D00E!

Existence theorems for two-dimensional linear shell theories

\begin{gathered} V_F (\omega ) = \{ \eta = (\eta _i ) \in H^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \hfill \\ \eta _i = \partial _v \eta _3 = 0 on \gamma _0 ,\gamma _{\alpha \beta } (\eta ) = 0 in \omega \} , \hfill \\ \end{gathered}