Castillo, Paul; Cockburn, Bernardo; Perugia, Ilaria; Schötzau, Dominik An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. (English) Zbl 0987.65111 SIAM J. Numer. Anal. 38, No. 5, 1676-1706 (2000). Authors’ summary: We present the first a priori error analysis for the local discontinuous Galerkin method for a model elliptic problem. For arbitrary meshes with hanging nodes and elements of various shapes, we show that, for stabilization parameters of order one, the \(L^2\)-norm of the gradient and the \(L^2\)-norm of the potential are of order \(k\) and \(k+1/2\), respectively, when polynomials of total degree at least \(k\) are used; if stabilization parameters of order \(h^{-1}\) are taken, the order of convergence of the potential increases to \(k+1\). The optimality of these theoretical results is tested in a series of numerical experiments on two-dimensional domains. Reviewer: H.Marcinkowska (Wrocław) Cited in 3 ReviewsCited in 253 Documents MSC: 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs Keywords:finite elements; discontinuous Galerkin methods; elliptic problems; error analysis; stabilization; convergence; numerical experiments PDFBibTeX XMLCite \textit{P. Castillo} et al., SIAM J. Numer. Anal. 38, No. 5, 1676--1706 (2000; Zbl 0987.65111) Full Text: DOI