Superconvergent discontinuous Galerkin methods for linear non-selfadjoint and indefinite elliptic problems
The article deals with a superconvergent discontinuous Galerkin method for the following linear non-selfadjoint and indefinite elliptic problem \[ -\nabla \cdot (a(x)\nabla u(x)+ b(x)u)+c(x)u(x)=f(x) \; \text{in} \; \Omega, \] \[ u(x)=0 \; \text{on} \; \partial \Omega. \] The main result of the paper is the estimate \[ \|u-u_h\|+{\| q- q_h\|}+\|\sigma-\sigma_h\| \leq C h^{k+1}, \] where \(q=\nabla u\), \(\sigma=a q \), and the triplet \(u_h, q_h, \sigma_h\) stands for the approximation of \(u\), \(q\), \(\sigma\) by the discontinuous Galerkin method when piecewise polynomials of degree \(k\) are used. With a suitable post-processing \(u^*_{h}\) of \(u_h\) it is shown that \[ \|u-u_h^*\|\leq Ch^{k+2}. \] The article is concluded with numerical examples which confirm the theoretical results.
non-selfadjoint and indefinite linear elliptic problems
scientific article; zbMATH DE number 6134441
scientific article