Biorthogonal basis functions in hphp-adaptive FEM for elliptic obstacle problems

In this paper, the discretization of a non-symmetric elliptic obstacle problem with hphp-adaptive H1(Ω)H1(Ω)-conforming finite elements is discussed. For this purpose, a higher-order mixed finite element discretization is introduced where the dual space is discretized via biorthogonal basis functions. The hphp-adaptivity is realized via automatic adaptive mesh refinement based on a residual a posteriori error estimation which is also derived in this paper. The use of biorthogonal basis functions leads to unilateral box constraints and componentwise complementarity conditions enabling the highly efficient application of a quadratically converging semi-smooth Newton scheme, which can be modified to ensure global convergence. hphp-adaptivity usually implies meshes with hanging nodes and varying polynomial degrees which have to be handled appropriately within the H1(Ω)H1(Ω)-conforming finite element discretization. This is typically done by using so-called connectivity matrices. In this paper, a procedure is proposed which efficiently computes these matrices for biorthogonal basis functions. Finally, the applicability of the theoretical findings is demonstrated with several numerical experiments.