Convergence of finite elements adapted for weaker norms

PEDRO MORIN; SIEBERT, K. G.; VEESER, A.

Baia Samuele (RG) Italia

Congreso; SIMAI Conference 2006; 2006

We consider finite elements that are adapted to a (semi)norm that isweaker than the one of the trial space.  We establish convergence ofthe finite element solutions to the exact one under the followingconditions: refinement relies on unique quasi-regular elementsubdivisions and generates locally quasi-uniform grids; the finiteelement spaces are conforming, nested, and satisfy the inf-supcondition; the error estimator is reliable and appropriately locallyefficient; the indicator of a non-marked element is bounded by theestimator contribution associated with the marked elements, and eachmarked element is subdivided at least once.  This abstract convergenceresult is illustrated by two examples.