Spectral Properties of Discontinuous Galerkin Space Operators on Curved Meshes

Abstract

Grids with curved elements are necessary to fully benefit from the high order of accuracy provided by the Discontinuous Galerkin (DG) method, when dealing with complex geometries. We study the relation between the quadratic shape of simplex elements and the spectral properties of the semi-discrete space operators, with emphasis on consequences for the maximum allowable timestep for stability in Runge–Kutta DG methods. A strong influence of element curvature on the eigenvalue spectrum is put in evidence, but no explicit relation could be found to describe the evolution of the spectral radius in function of geometric properties of elements. Furthermore, we show that a correct estimation of stability bounds cannot be obtained by considerations on the norm of integration matrices involved in the DG Method.