Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source. (English) Zbl 1514.65135
Summary: The analyses of interior penalty discontinuous Galerkin methods of any order \(k\) 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 a priori error estimates in the \(L^2\) 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 \(L^2\) norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the \(L^2\) 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 \(L^2\) in time and in space. Numerical results for the elliptic problem are added to support the theoretical results.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 35J75 | Singular elliptic equations |
| 35K10 | Second-order parabolic equations |
| 35R06 | PDEs with measure |
Keywords:
interior penalty dG; convergence; local \(L^2\) estimates; local energy estimates; singular solutionsReferences:
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