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A feedback finite element method with a posteriori error estimation. I: The finite element method and some basic properties of the a posteriori error estimator. (English) Zbl 0593.65064
This paper is the first in a series of two in which we discuss some theoretical and practical aspects of a feedback finite element method for solving systems of linear second-order elliptic partial differential equations (with particular interest in classical linear elasticity). In this first part we introduce some nonstandard finite element spaces, which though based on the usual square bilinear elements, permit local mesh refinement. The algebraic structure of these spaces and their approximation properties are analyzed. An ”equivalent estimator” for the \(H^ 1\) finite element error is developed. In the second paper we shall discuss the asymptotic properties of the estimator and computational experience.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
35J25 Boundary value problems for second-order elliptic equations
65N15 Error bounds for boundary value problems involving PDEs
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