hphp-adaptive IPDG/TDG-FEM for parabolic obstacle problems

For a parabolic obstacle problem two equivalent hphp-FEM discretization methods based on interior penalty discontinuous Galerkin in space and discontinuous Galerkin in time are presented. The first approach is based on a variational inequality (VI) formulation and the second approach on a mixed method in which the non-penetration condition is resolved by a Lagrange multiplier. The discrete Lagrange multiplier is a linear combination of biorthogonal basis functions, allowing to write the discrete VI-constraints as a set of complementarity problems. Employing a penalized Fischer–Burmeister non-linear complementarity function, the discrete mixed problem can be solved by a locally Q-quadratic converging semi-smooth Newton (SSN) method. The hierarchical a posteriori error estimator for the VI-formulation, which under the saturation assumption is both efficient and reliable, allows hphp-adaptivity. The numerical experiments show improved convergence compared to uniform and hh-adaptive meshes. Furthermore, an a priori error estimate is given for the VI-formulation.