Mixed finite element methods for elliptic problems

Douglas N. Arnold

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101 Scopus citations

Abstract

This paper treats the basic ideas of mixed finite element methods at an introductory level. Although the viewpoint presented is that of a mathematician, the paper is aimed at practitioners and the mathematical prerequisites are kept to a minimum. A classification of variational principles and of the corresponding weak formulations and Galerkin methods-displacement, equilibrium and mixed-is given and illustrated through four significant examples. The advantages and disadvantages of mixed methods are discussed. The concepts of convergence, approximability and stability and their interrelations are developed, and a résumé is given of the stability theory which governs the performance of mixed methods. The paper concludes with a survey of techniques that have been developed for the construction of stable mixed methods and numerous examples of such methods.

Original languageEnglish (US)
Pages (from-to)281-300
Number of pages20
JournalComputer Methods in Applied Mechanics and Engineering
Volume82
Issue number1-3
DOIs
StatePublished - Sep 1990

Bibliographical note

Funding Information:
Here /-2 denotes the region in three-dimensional space, R 3, occupied by the elastic body, u : 1-2 ~ R 3 denotes the displacement field, ~(u) denotes the corresponding infinitesimal strain tensor, (i.e., the symmetric part of the gradient of u, eq(u) = (ui,j + ui,i)/2), f denotes the imposed volume load, and 5e: O---, ~1 ~3×3 (the space of symmetric 3 x 3 tensors) denotes the stress field. The divergence of b", div re, is applied to each row of ~, so that (div S¢)i = E i sij.j. The material properties are determined by the compliance tensor A which is a positive definite symmetric operator from ~,y I1~3x3 to itself 1, possibly depending on the point x E/2. The * This work was supported by NSF grant DMS-89-02433. 1 This means that the action of A can be written as (Are)q = r.kt aqktSk~ with the components aiikt satisfying the usual major symmetries aqk t = aklq, minor symmetries ai\]kl = aj~kt, and positivity condition ~ijkl aqktSqSkl >~ Y ~q Sq, 2 for all S¢, where y > 0.

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