Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in L1
We consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in L∞Ω and the right-hand side belongs to L1Ω; we extend the results where the case of linear finite elements approximation is considered. We...
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Wiley
2018-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2018/4650512 |
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| author | Abdeluaab Lidouh Rachid Messaoudi |
| author_facet | Abdeluaab Lidouh Rachid Messaoudi |
| author_sort | Abdeluaab Lidouh |
| collection | DOAJ |
| description | We consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in L∞Ω 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 1≤q<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 Tkf∈H1Ω for every k>0, for some r>1. |
| format | Article |
| id | doaj-art-e6cf3150f510438cb1edd94c6db04bad |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-e6cf3150f510438cb1edd94c6db04bad2025-02-03T05:52:32ZengWileyInternational Journal of Differential Equations1687-96431687-96512018-01-01201810.1155/2018/46505124650512Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in L1Abdeluaab Lidouh0Rachid Messaoudi1Department of Mathematics and Computer Science, Laboratory LACSA, Faculty of Sciences, Mohammed 1st University, BV Mohammed VI, P.O. Box 717, 60000 Oujda, MoroccoDepartment of Mathematics and Computer Science, Laboratory LACSA, Faculty of Sciences, Mohammed 1st University, BV Mohammed VI, P.O. Box 717, 60000 Oujda, MoroccoWe consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in L∞Ω 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 1≤q<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 Tkf∈H1Ω for every k>0, for some r>1.http://dx.doi.org/10.1155/2018/4650512 |
| spellingShingle | Abdeluaab Lidouh Rachid Messaoudi Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in L1 International Journal of Differential Equations |
| title | Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in L1 |
| title_full | Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in L1 |
| title_fullStr | Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in L1 |
| title_full_unstemmed | Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in L1 |
| title_short | Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in L1 |
| title_sort | affine discontinuous galerkin method approximation of second order linear elliptic equations in divergence form with right hand side in l1 |
| url | http://dx.doi.org/10.1155/2018/4650512 |
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