Superconvergence of Direct Discontinuous Galerkin Methods: Eigen-structure Analysis Based on Fourier Approach

Abstract

This paper investigates superconvergence properties of the direct discontinuous Galerkin (DDG) method with interface corrections and the symmetric DDG method for diffusion equations. We apply the Fourier analysis technique to symbolically compute eigenvalues and eigenvectors of the amplification matrices for both DDG methods with different coefficient settings in the numerical fluxes. Based on the eigen-structure analysis, we carry out error estimates of the DDG solutions, which can be decomposed into three parts: (i) dissipation errors of the physically relevant eigenvalue, which grow linearly with the time and are of order 2k for $P^k(k=2,3)$ approximations; (ii) projection error from a special projection of the exact solution, which is decreasing over the time and is related to the eigenvector corresponding to the physically relevant eigenvalue; (iii) dissipative errors of non-physically relevant eigenvalues, which decay exponentially with respect to the spatial mesh size $\mathrm{\Delta }x$. We observe that the errors are sensitive to the choice of the numerical flux coefficient for even degree $P^2$ approximations, but are not for odd degree $P^3$ approximations. Numerical experiments are provided to verify the theoretical results.

Funding

National Natural Science Foundation of China(11871428); Division of Mathematical Sciences(1620335); Natural Science Foundation for Colleges and Universities of Jiangsu Province(20KJB110011); Simons Foundation(637716)