Superconvergent Discontinuous Galerkin Methods for Linear Non-selfadjoint and Indefinite Elliptic Problems

Abstract

Based on Cockburn et al. (Math. Comp. 78:1–24, 2009), superconvergent discontinuous Galerkin methods are identified for linear non-selfadjoint and indefinite elliptic problems. With the help of an auxiliary problem which is the discrete version of a linear non-selfadjoint elliptic problem in divergence form, optimal error estimates of order k+1 in L 2-norm for the potential and the flux are derived, when piecewise polynomials of degree k≥1 are used to approximate both potential and flux variables. Using a suitable post-processing of the discrete potential, it is then shown that the resulting post-processed potential converges with order k+2 in L 2-norm. The article is concluded with a numerical experiment which confirms the theoretical results.