## International Journal of Differential Equations

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

### Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in ${L}^{1}$

Abdeluaab Lidouh and Rachid Messaoudi

#### 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 $r>\mathrm{1}.$

#### Article information

**Source**

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

**Dates**

Received: 4 February 2018

Accepted: 21 May 2018

First available in Project Euclid: 19 September 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.ijde/1537322435

**Digital Object Identifier**

doi:10.1155/2018/4650512

**Mathematical Reviews number (MathSciNet)**

MR3827848

**Zentralblatt MATH identifier**

06915953

#### Citation

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. https://projecteuclid.org/euclid.ijde/1537322435