Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods
Discontinuous finite element methods (DGFEM) are a promising approach for discretizing advection equations. The paper analyzes the dispersion and the dissipation error of \(N\)-th order DGFEM on partitions of \(\mathbb R^d\), \(d\in\{2,3\}\), consisting of tensor product mesh cells of size \(h^d\). In the case that the wave number \(k\) is small with respect to the mesh size, it is shown that the order of convergence of the DGFEM is higher than of the standard Galerkin FEM. Consideration of the leading order term of the relative errors leads to the suggestion that it is more efficient to increase \(N\) than to decrease \(h\). The case of fixed \(h\) and varying \(N\) is analyzed in detail. Choosing \(N\) such that \(2N+1 \approx \kappa hk\) with \(\kappa > 1\) is proposed to be the most efficient approach and an exponential order of convergence is proven in this case. The relative errors are related to certain Padé approximants of the exponential. The remainder in these Padé approximants are analyzed and they are linked to a certain eigenvalue problem.
scientific article; zbMATH DE number 2091353
scientific article