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

间断伽辽金方法(DG)是近年来引起人们极大关注的一种高精度数值方法。在DG方法中，可以在单元水平上局部地选择近似顺序。由于计算成本随着近似顺序的增加而增加，应考虑采用允许根据局部误差控制精度的自适应方法，以便以最少的自由度达到给定的误差容限。由于间断Galerkin方法是一种有限元方法，因此有两种局部自适应方法可供选择，即局部网格步长自适应和局部逼近阶数自适应。在这个项目的头两年(已经由DFG资助)，已经开发了新的自主网格控制以及新的误差估计器并进行了比较。在第二阶段，我们希望完成将该方法扩展到时空方案。这将允许也沿着时间维度自适应地控制误差，从而得到一种方案，该方案可以在最小的计算成本下在空间和时间上达到规定的误差容限。

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.