Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source

The analyses of interior penalty discontinuous Galerkin methods any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence method by deriving a priori error estimates in L 2 norm weighted energy norms. In addition, almost optimal local approximation order. Further, obtained case piecewise linear approximations whereas suboptimal bounds shown polynomial degree. time-dependent semi-discrete backward Euler fully discrete scheme is established proving time space. Numerical results problem added to support theoretical results.