International Journal of Differential Equations

Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in L1

Abdeluaab Lidouh and Rachid Messaoudi

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We consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in LΩ and the right-hand side belongs to L1Ω; we extend the results where the case of linear finite elements approximation is considered. We prove that the unique solution of the discrete problem converges in W01,qΩ for every q with 1q<d/d-1 (d=2 or d=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 in W01,qΩ when the right-hand side f belongs to LrΩ verifying TkfH1Ω for every k>0, for some r>1.

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Int. J. Differ. Equ., Volume 2018 (2018), Article ID 4650512, 15 pages.

Received: 4 February 2018
Accepted: 21 May 2018
First available in Project Euclid: 19 September 2018

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Lidouh, Abdeluaab; Messaoudi, Rachid. Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in ${L}^{1}$. Int. J. Differ. Equ. 2018 (2018), Article ID 4650512, 15 pages. doi:10.1155/2018/4650512.

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