## International Journal of Differential Equations

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

#### Abstract

We consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in ${L}^{\mathrm{\infty }}(\mathrm{\Omega })$ and the right-hand side belongs to ${L}^{\mathrm{1}}(\mathrm{\Omega })$; 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 ${W}_{\mathrm{0}}^{\mathrm{1},q}(\mathrm{\Omega })$ for every $q$ with $\mathrm{1}\le q ($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 ${W}_{\mathrm{0}}^{\mathrm{1},q}(\mathrm{\Omega })$ when the right-hand side $f$ belongs to ${L}^{r}(\mathrm{\Omega })$ verifying ${T}_{k}(f)\in {H}^{\mathrm{1}}(\mathrm{\Omega })$ 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
Accepted: 21 May 2018
First available in Project Euclid: 19 September 2018

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

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