A basic convergence result for conforming adaptive finite element methods

PEDRO MORIN; KUNIBERT G. SIEBERT; ANDREAS VEESER

Oberwolfach, Alemania

Workshop; Workshop on Adaptive Numerical Methods for Partial Differential Equations; 2007

Mathematisches Institut Oberwolfach

Abstract:We consider the approximate solution with adaptive finite elements ofa large class of linear boundary value problems, which includesproblems of `saddle point´ type. For the adaptive algorithm we supposethe following framework: refinement relies on unique quasi-regularelement subdivisions and generates locally quasi-uniform grids, thefinite element spaces are conforming, nested, and satisfy the inf-supconditions, the error estimator is reliable as well as locally anddiscretely efficient. Under the assumption that all marked elementsare refined at least once, we give a sufficient and essentiallynecessary condition on the marking strategy for the convergence of thefinite element solutions to the exact one. This condition is satisfiedby all popular marking strategies, including maximum,equidistribution, and Dörfler´s marking strategy. We emphasize thatthe convergence holds under no additional marking due to dataoscillation, and with a minimal refinement on the marked elements,i.e., without enforcing the so-called "interior node property".