Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in L1. (2nd July 2018)

• Record Type: Journal Article
• Authors: Lidouh, Abdeluaab; Messaoudi, Rachid
• Other Names: Wong Patricia J. Y. Academic Editor.
• Is Part Of: International journal of differential equations. Volume 2018(2018)
• Publication Date: 2018-07-02
• Subjects: Differential equations -- Periodicals; Differential equations; Periodicals; 515.35
• DOI: 10.1155/2018/4650512
• Languages: English
• ISSNs: 1687-9643

Abstract

Abstract : We consider the standardaffine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients inL ∞ Ω and the right-hand side belongs toL 1 Ω ; we extend the results where the case oflinear finite elements approximation is considered. We prove that the unique solution of the discrete problem converges inW 0 1, q Ω for everyq with1 ≤ q < d / d - 1 (d = 2 ord = 3 ) to the unique renormalized solution of the problem. Statements and proofs remain valid in our case, which permits obtaining a weaker result when the right-hand side is a bounded Radon measure and, when the coefficients are smooth, an error estimate inW 0 1, q Ω when the right-hand sidef belongs toL r Ω verifyingT k f ∈ H 1 Ω for everyk > 0, for somer > 1 .