Analysis of a discontinuous Galerkin method for heterogeneous diffusion problems with low-regularity solutions (Q2910806)

The authors consider a diffusion problem with a diffusion coefficient which is piecewise constant in open \(d\)-dimensional polyhedrons forming a partition of the \(d\)-dimensional polygonal solution domain (\(d\geq 2\)) under the low regularity assumption that the exact solution is piecewise in \(W^{2,p}\) for \(p\in(2d/(d+2),2]\). They propose, for its numerical solution, a symmetric weighted interior penalty discontinuous Galerkin method that differs from the classical version of \textit{D. N. Arnold} [SIAM J. Numer. Anal. 19, 742--760 (1982; Zbl 0482.65060)], by the use of harmonic averages for the traces and when multiplying the penalty parameter. Concerning the grid they assume that it is not only shape regular but also aligned with the discontinuities of the diffusion coefficient (but may possess hanging nodes). They prove boundedness of the corresponding bilinear form, consistency, stability and convergence with optimal order for \(p\in(2d/(d+2),2]\). In the case that the exact solution is only in the energy space, they show convergence, but without giving estimates of the speed of convergence.