It is well known that the solution of second order elliptic problems with interfaces may feature internal layers and/or singularities. We present an adaptive discontinuous Galerkin (DG) method to suitably approximate such problems. First, we introduce the weighted interior penalty method, which generalizes the classical interior penalty DG schemes by replacing the arithmetic means with suitably weighted averages where the weights depend on the coefficients of the problem. Then, we discuss the construction of a family of residual based local error indicators for the energy norm, applied to advection-diffusion-reaction equations featuring a diffusivity parameter that may be discontinuous along an interface. In particular, we demonstrate how the weights can incorporate into the scheme some a-priori knowledge of the exact solution that improves the efficacy of the estimator and of the corresponding adapted mesh. The theoretical results are confirmed by means of numerical experiments.
Energy norm a-posteriori error estimates for a discontinuous Galerkin scheme applied to elliptic problems with an interface
ZUNINO, PAOLO
2009-01-01
Abstract
It is well known that the solution of second order elliptic problems with interfaces may feature internal layers and/or singularities. We present an adaptive discontinuous Galerkin (DG) method to suitably approximate such problems. First, we introduce the weighted interior penalty method, which generalizes the classical interior penalty DG schemes by replacing the arithmetic means with suitably weighted averages where the weights depend on the coefficients of the problem. Then, we discuss the construction of a family of residual based local error indicators for the energy norm, applied to advection-diffusion-reaction equations featuring a diffusivity parameter that may be discontinuous along an interface. In particular, we demonstrate how the weights can incorporate into the scheme some a-priori knowledge of the exact solution that improves the efficacy of the estimator and of the corresponding adapted mesh. The theoretical results are confirmed by means of numerical experiments.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.