Interior penalty discontinuous Galerkin method for Maxwell's equations: energy norm error estimates
The development of new more sophisticated algorithms for the numerical solution of Maxwell's equations is dictated by increasingly complex applications in electromagnetics. Edge elements are probably the most satisfactory from a theoretical point of view, in particular because they correctly represent singular behaviour at reentrant corners. However, they are less attractive for time-dependent computations, because the solution of a linear system is required at every time iteration. Discontinuous Galerkin (DG) FEM offer an attractive alternative to edge elements for the numerical solution of Maxwell's equations, in particular for time-dependent problems. They lead to (block-) diagonal mass matrices and therefore yield fully explicit, inherently parallel methods when coupled with explicit time stepping. Because continuity across element interfaces is weakly enforced merely by adding suitable bilinear forms (so-called numerical fluxes) to the standard variational formulation, the implementation of DG-FE methods is straightforward within existing FE software libraries. For first-order hyperbolic systems, various DGFEM are available. Here, authors propose and analyze the symmetric interior penalty DG method for the spatial discretization of Maxwell's equations in second order form. After stating the model problem, they describe the interior penalty DG variational formulation and obtain optimal a-priori error bounds in the energy norm. In the case of solutions with smoothness beyond \(H^1\) , the error bound holds for arbitrary DG-FE discretizations, whereas in the case of lower regularity, the error bound only holds for conforming meshes. The article ends with all the proofs and technical approximation results, and with some concluding remarks.
scientific article; zbMATH DE number 5161211
scientific article