Collective marking for adaptive least-squares finite element methods with optimal rates

Abstract

All previously known optimal adaptive least-squares finite element methods (LSFEMs) combine two marking strategies with a separate $L^2$ data approximation as a consequence of the natural norms equivalent to the least-squares functional. The algorithm and its analysis in this paper circumvent the natural norms in a div-LSFEM model problem with lowest-order conforming and mixed finite element functions and allow for a simple collective Dörfler marking for the first time. A refined analysis provides discrete reliability and quasi-orthogonality in the weaker norms $L^2\times H^1$ rather than $H(\operatorname {div})\times H^1$ and replaces data approximation terms by data oscillations. The optimal convergence rates then follow for the lowest-order version from the axioms of adaptivity for the newest-vertex bisection without restrictions on the initial mesh-size in any space dimension.