Fourier Analysis of Local Discontinuous Galerkin Methods for Linear Parabolic Equations on Overlapping Meshes

Abstract

A new local discontinuous Galerkin method for convection–diffusion equations on overlapping mesh was introduced in Du et al. (BIT Numer Math 1–24, 2019). In the new method, the primary variable u and auxiliary variable $$p=u_x$$ are solved on different meshes. The stability and suboptimal error estimates for problems with periodic boundary conditions were derived. Numerical experiments demonstrated that the convergence rates cannot be improved if the dual mesh is constructed by using the midpoint of the primitive mesh. Several alternatives to gain optimal convergence rates were demonstrated in Du et al. (2019). However, the reason for accuracy degeneration is still unclear. In this paper, we will use Fourier analysis to analyze the scheme for linear parabolic equations with periodic boundary conditions in one space dimension. We explicitly write out the error between the numerical and exact solutions, and investigate the reason for the accuracy degeneration. Moreover, we also find out some superconvergence points that may depend on the perturbation constant in the construction of the dual mesh. Since the current work is based on Fourier analysis, we only consider uniform meshes. Numerical experiments will be given to verify the theoretical analysis.