Autonomous Control of Mesh Adaptations for the Discontinuous Galerkin Method Applied to Problems in Electrodynamics

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The Discontinuous Galerkin Method (DG) is a high order numerical Method, which attracted considerable attention in recent years. Within the DG method the approximation order can be chosen locally on the element level. As computational costs rise alongside with the approximation order the application of adaptive methods allowing for controlling accuracy based on the local error should be considered in order to reach a given error tolerance with the minimal number of degrees of freedom. As the Discontinuous Galerkin Method is a Finite Element Method two options for local adaptation are available, i.e., the adaptation of the local mesh step size and the local approximation order. During the first two years of this project (already funded by the DFG) new, autonomous mesh controls as well as new error estimators have been developed and compared. In the second phase we wish to complete the extension of the method to a space-time scheme. This will allow for adaptively controlling errors also along the time dimension leading to a scheme, which can reach a prescribed error tolerance in space and time at minimal computational costs.