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

Abstract

We consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in $L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ and the right-hand side belongs to $L^{\mathrm{1}}\left(\mathrm{\Omega }\right)$; 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 $W_{\mathrm{0}}^{\mathrm{1},q}\left(\mathrm{\Omega }\right)$ for every $q$ with ($d=\mathrm{2}$ or $d=\mathrm{3}$) 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 $W_{\mathrm{0}}^{\mathrm{1},q}\left(\mathrm{\Omega }\right)$ when the right-hand side $f$ belongs to $L^r\left(\mathrm{\Omega }\right)$ verifying $T_k\left(f\right)\in H^{\mathrm{1}}\left(\mathrm{\Omega }\right)$ for every $k>\mathrm{0}$, for some $r>\mathrm{1}.$