Convergence of finite elements adapted for weaker norms

Abstract

We consider finite elements that are adapted to a (semi)norm that is weaker than the one of the trial space. We establish convergence of the finite element solutions to the exact one under the following conditions: refinement relies on unique quasi-regular element subdivisions and generates locally quasi-uniform grids; the finite element spaces are conforming, nested, and satisfy the inf-sup condition; the error estimator is reliable and appropriately locally efficient; the indicator of a non-marked element is bounded by the estimator contribution associated with the marked elements, and each marked element is subdivided at least once. This abstract convergence result is illustrated by two examples.

Keywords: Adaptivity, conforming finite elements, convergence, weaker (semi)norms, mesh-dependent norms


Published: Applied and Industrial Matematics in Italy II, Selected Contributions from the 8th SIMAI Conference, Vincenzo Cutello, Giorgio Fotia, Luigia Puccio, eds. Series on Advances in Mathematics for Applied Sciences - Vol. 75. World Scientific, 2007.