On the coupling of hyperbolic and parabolic systems: analytical and numerical approach
Applied Numerical Mathematics - Spectral multi-domain methods
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
SIAM Journal on Numerical Analysis
Superconvergence of the Local Discontinuous Galerkin Method for Elliptic Problems on Cartesian Grids
SIAM Journal on Numerical Analysis
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems
SIAM Journal on Numerical Analysis
Robust A Posteriori Error Estimates for Stationary Convection-Diffusion Equations
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Journal of Scientific Computing
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We develop and analyze a Discontinuous Galerkin (DG) method based on weighted interior penalties (WIP) applied to second order (elliptic) PDEs and in particular to advection-diffusion-reaction equations featuring non-smooth and possibly vanishing diffusivity. First of all, looking at the derivation of a DG scheme with a bias to domain decomposition methods, we carefully discuss the set up of the discretization scheme in a general framework putting into evidence the helpful role of the weights and the connection with the well known Local Discontinuous Galerkin schemes (LDG). Then, we address the a-priori error analysis of the method, recovering optimal error estimates in suitable norms. By virtue of the introduction of the weighted penalties, these results turn out to be robust with respect to the diffusion parameter. Furthermore, we discuss the derivation of an a-posteriori local error indicator suitable for advection-diffusion-reaction problems with highly variable, locally small diffusivity. All the theoretical results are illustrated and discussed by means of numerical experiments.