Hierarchical Mixed Hybridized Methods for Elliptic Problems

In this article, we propose a computational procedure for the efficient implementation of Dual Mixed Hybridized methods of arbitrary degree. The procedure relies on the decomposition of the finite element spaces into a “vertical” p-type hierarchy, consisting of a lower order part and a defect correction, coupled with an additional “horizontal” decomposition of the defect correction space for the vector variable based on the Helmholtz principle. An appropriate definition of the basis function set allows us to obtain a systematic substructuring of the block matrix system. This property, in turn, naturally gives rise to an efficient implementation of the procedure through an approximate fixed-point block iteration. Exploiting the equivalence between the principle of defect correction and the Variational Multiscale Modeling Theory, we also devise and numerically validate a hierarchical a posteriori error estimator for Dual Mixed methods in hybridized form.