Numerical Approximation of PDEs and Clément’s Interpolation

Abstract

In this short paper, we present a formalism which specifies the notions of consistency and stability of finite element methods for the numerical approximation of nonlinear partial differential equations of elliptic and parabolic type. This formalism can be found in [4], [7], [10], and allows to establish a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods. In concrete cases, the Cléement’s interpolation technique [6] is very useful in order to establish local a posteriori error estimates. This paper uses some ideas of [10] and its main goal is to show in a very simple setting, the mathematical arguments which lead to the stability and convergence of Galerkin methods. The bibliography concerning this subject is very large and the references of this paper are no exhaustive character. In order to obtain a large bibliography on the a posteriori error estimates, we report the lecturer to Verfürth’s book and its bibliography [12].KeywordsNumerical ApproximationPosteriori ErrorPosteriori Error EstimateNonsmooth FunctionLocal RegularizationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.