Discontinuous Galerkin Discretization of the Heat Equation in Any Dimension: the Spectral Symbol∗

The multidimensional heat equation, along with its more general version involving variable diffusion coefficients, is discretized by a discontinuous Galerkin (DG) method in time and a finite element (FE) method of arbitrary regularity in space. We show that the resulting space-time discretization matrices enjoy an asymptotic spectral distribution as the mesh fineness increases, and we determine the associated spectral symbol, i.e., the function that carefully describes the spectral distribution. The analysis of this paper is carried out in a stepwise fashion, without omitting details, and it is supported by several numerical experiments. It is preparatory to the development of specialized solvers for linear systems arising from the DG/FE approximation of the heat equation in the case of both constant and variable diffusion coefficients.