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

不连续伽辽金法是近年来备受关注的一种高阶数值方法。在DG方法中，可以在元素级别上局部选择近似顺序。由于计算成本随着逼近阶数的增加而增加，应考虑采用基于局部误差控制精度的自适应方法，以便以最小的自由度达到给定的误差容限。由于不连续伽辽金法是一种有限元法，局部自适应有两种选择，即自适应局部网格步长和自适应局部逼近阶数。在这个项目的前两年（已经由DFG资助），新的自主网格控制以及新的误差估计器已经开发和比较。在第二阶段，我们希望将该方法扩展到一个时空方案。这将允许自适应地控制误差也沿着时间维度导致方案，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.