Superconvergence Analysis of Discontinuous Galerkin Methods for Systems of Second-Order Boundary Value Problems

Abstract

In this paper, we present an innovative approach to solve a system of boundary value problems (BVPs), using the newly developed discontinuous Galerkin (DG) method, which eliminates the need for auxiliary variables. This work is the first in a series of papers on DG methods applied to partial differential equations (PDEs). By consecutively applying the DG method to each space variable of the PDE using the method of lines, we transform the problem into a system of ordinary differential equations (ODEs). We investigate the convergence criteria of the DG method on systems of ODEs and generalize the error analysis to PDEs. Our analysis demonstrates that the DG error’s leading term is determined by a combination of specific Jacobi polynomials in each element. Thus, we prove that DG solutions are superconvergent at the roots of these polynomials, with an order of convergence of O(hp+2).

Global error analysis of discontinuous Galerkin methods for systems of boundary value problems

“…Furthermore, compared to other DG methods, our approach guarantees optimal convergence and superconvergence without using internal penalty terms. Another advantage is that the current scheme achieves superconvergence results [25] that can be used to construct a posteriori error estimates by solving a local problem on each element. Moreover, the careful choice of the numerical fluxes ensures the robust stability of the DG scheme.…”