An Anisotropic hp-mesh Adaptation Method for Time-Dependent Problems Based on Interpolation Error Control

We propose an efficient mesh adaptive method for the numerical solution of time-dependent partial differential equations considered in the fixed space-time cylinder Ω × ( 0 , T ) . We employ the space-time discontinuous Galerkin method which enables us to use different meshes at different time levels in a natural way. The mesh adaptive algorithm is based on control of the interpolation error in the L ∞ ( 0 , T ; L q ( Ω ) ) -norm. The goal is to construct a sequence of conforming triangular meshes in such a way that the interpolation error bound is under a given tolerance and the number of degrees of freedom is minimal. The resulting grids consist of anisotropic mesh elements with varying polynomial approximation degrees with respect to space. We present a theoretical framework of this approach as well as several numerical examples demonstrating the accuracy, efficiency, and applicability of the method.