Graded Meshes in Optimal Control for Elliptic Partial Differential Equations: An Overview

Chapter

Abstract

It is well known that singularities in the solution of boundary value problems due to corners and edges of the domain lead to a reduction of the convergence order of the standard finite element method when quasi-uniform meshes are used. It is also well known that locally graded meshes are suited to recover the optimal convergence order. Less well known are the critical angles when mesh grading becomes necessary; it is not always the same but depends on the norm in which the error is estimated. In this paper, an overview of the results is given and lacking estimates are pointed out. Since the error estimates for optimal control problems are based on those for pure boundary value problems both cases are always considered.

Keywords

Elliptic partial differential equation Finite elements A priori error estimates Mesh grading Optimal control 

Mathematics Subject Classification (2010)

49M25 65N15 65N30 65N50 

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Institut für Mathematik und BauinformatikUniversität der Bundeswehr MünchenMünchenGermany
  2. 2.Fakultät für MathematikUniversität Duisburg–EssenEssenGermany

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