Error Estimates and Adaptivity of the Space-Time Discontinuous Galerkin Method for Solving the Richards Equation

Abstract

We present a higher-order space-time adaptive method for the numerical solution of the Richards equation that describes a flow motion through variably saturated media. The discretization is based on the space-time discontinuous Galerkin method, which provides high stability and accuracy and can naturally handle varying meshes. We derive reliable and efficient a posteriori error estimates in the residual-based norm. The estimates use well-balanced spatial and temporal flux reconstructions which are constructed locally over space-time elements or space-time patches. The accuracy of the estimates is verified by numerical experiments. Moreover, we develop the hp-adaptive method and demonstrate its efficiency and usefulness on a practically relevant example.