Superconvergence Study of the Direct Discontinuous Galerkin Method and Its Variations for Diffusion Equations

Abstract

In this paper, we apply the Fourier analysis technique to investigate superconvergence properties of the direct disontinuous Galerkin (DDG) method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009), the DDG method with the interface correction (DDGIC) (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010), the symmetric DDG method (Vidden and Yan in Comput Math 31(6):638–662, 2013), and the nonsymmetric DDG method (Yan in J Sci Comput 54(2):663–683, 2013). We also include the study of the interior penalty DG (IPDG) method, due to its close relation to DDG methods. Error estimates are carried out for both $P^2$ and $P^3$ polynomial approximations. By investigating the quantitative errors at the Lobatto points, we show that the DDGIC and symmetric DDG methods are superior, in the sense of obtaining $(k+2)$th superconvergence orders for both $P^2$ and $P^3$ approximations. Superconvergence order of $(k+2)$ is also observed for the IPDG method with $P^3$ polynomial approximations. 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 carried out at the same time and the numerical errors match well with the analytically estimated errors.

Funding

National Natural Science Foundation of China(11871428)

Innovative Research Group Project of the National Natural Science Foundation of China(11621101)

National Science Foundation (US)(DMS-1620335)

Simons Foundation(637716)