An optimal adaptive mixed finite element method

Abstract

Various applications in fluid dynamics and computational continuum mechanics motivate the development of reliable and efficient adaptive algorithms for mixed finite element methods. In order to save degrees of freedom, not all but just some selected set of finite element domains are refined. Hence the fundamental question of as well as the question of optimality require new mathematical arguments. The presented adaptive algorithm for Raviart-Thomas mixed finite element solves the Poisson model problem, with optimal rate. Chen, Holst, and Xu presented convergence and optimality of adaptive mixed finite element methods (2008) following arguments of Rob Stevenson for the conforming finite element method. Their algorithm reduces oscillations separately, before approximating the solution by some adaptive algorithm in the spirit of W. Dorfler (1996). The algorithm proposed here appears more natural in switching to either reduction of the edge-error estimator or of the oscillations.