Masri, RamiShen, BoqianRiviere, Beatrice2023-04-252023-04-252023Masri, Rami, Shen, Boqian and Riviere, Beatrice. "Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source." <i>ESAIM: Mathematical Modelling and Numerical Analysis,</i> 57, no. 2 (2023) EDP Sciences: 585-620. https://doi.org/10.1051/m2an/2022095.https://hdl.handle.net/1911/114844The analyses of interior penalty discontinuous Galerkin methods of any order <i>k<i/> for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving <i>a priori<i/> error estimates in the <i>L<i/><sup>2<sup/> norm and in weighted energy norms. In addition, we prove almost optimal local error estimates in the energy norm for any approximation order. Further, almost optimal local error estimates in the <i>L<i/><sup>2<sup/> norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the <i>L<i/><sup>2<sup/> norm are shown for any polynomial degree. For the time-dependent case, convergence of semi-discrete and of backward Euler fully discrete scheme is established by proving error estimates in <i>L<i/><sup>2<sup/> in time and in space. Numerical results for the elliptic problem are added to support the theoretical results.engThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Journal articlem2an220085https://doi.org/10.1051/m2an/2022095