A p-Strategy with a Local Time-Stepping Method in a Discontinuous Galerkin Approach to Solve Electromagnetic Problems

We present a local spatial approximation or p-strategy Discontinuous Galerkin method to solve the time-domain Maxwell equations. First, the Discontinuous Galerkin method with a local time-stepping strategy is recalled. Next, in order to increase the efficiency of the method, a local spatial approximation strategy is introduced and studied. While preserving accuracy and by using different spatial approximation orders for each cell, this strategy is very efficient to reduce the computational time and the required memory in numerical simulations using very distorted meshes. Several numerical examples are given to show the interest and the capacity of such method. 1. Introduction In early works, we have proposed an efficient new Discontinuous Galerkin (DG) method to solve the Maxwell equations in time domain [1]. However, in order to treat industrial problems, we have generally to cope with a very distorted meshes with a large difference between the sizes of the cells. In the DG simulations, the spatial approximation order taken into account is obtained by considering the size of the largest cell and the largest frequency in the spectrum of the excitation source. Consequently, our method suffers from a sampling that is too high for the smallest cells in the mesh and from a time step that is very small to ensure stability. To improve the choice of the time step, we have proposed a local time-stepping strategy in the method [2] which allows reduction of the computational time. Nevertheless, the spatial order of approximation remains the same for all the cells, and for a given source, some cells are too sampled and lead to a local time step on these cells still too small. To avoid this problem, a method with local order on the cells can be used [3]. Some works have been already realized for Maxwell's equation on this subject, with classical upwind DG formulation and ADER strategy [4]. In this paper, we present, for a particular DG method well adapted to the Maxwell equations, a strategy to mix spatial orders of approximation in each cell on the whole mesh. This strategy allows decreasing the order of the small cells and then increasing the CFL condition for these cells and consequently their local time-step. Indeed, it has been shown, for the DG method used, that the CFL condition decreases when the spatial order of approximation increases [5]. In the first part of this paper, we recall the mathematical formulation of the DG schema studied and the local time-stepping strategy used. In the second part, the formulation of a local spatial approximation order or