A nonsymmetric discontinuous Galerkin finite element method with interior penalties is considered for two-dimensional convection-diffusion problems with regular and parabolic layers. On an anisotropic Shishkintype mesh with bilinear elements we prove error estimates (uniformly in the perturbation parameter) in an integral norm associated with this method. On different types of interelement edges we derive the values of discontinuitypenalization parameters. Numerical experiments complement the theoretical results.Mathematics Subject Classification (1991): 65N30, 65N12, 65N15
MSC: 65L11 65L20 65L60 65L70
Keywords:Singularly perturbed problem Two small parameters Galerkin finite element method Bakhvalov-type mesh Clément quasi-interpolant a b s t r a c t A singularly perturbed problem with two small parameters is considered. On a Bakhvalovtype mesh we prove uniform convergence of a Galerkin finite element method with piecewise linear functions. Arguments in the error analysis include interpolation error bounds for a Clément quasi-interpolant as well as discretization error estimates in an energy norm. Numerical experiments support theoretical findings.
We consider a convection-diffusion problem with Dirichlet boundary conditions posed on a unit square. The problem is discretized using a combination of the standard Galerkin FEM and an h-version of the nonsymmetric discontinuous Galerkin FEM with interior penalties on a layer-adapted mesh with linear/bilinear elements. With specially chosen penalty parameters for edges from the coarse part of the mesh, we prove uniform convergence (in the perturbation parameter) in an associated norm. In the same norm we also establish a supercloseness result. Numerical tests support our theoretical estimates.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.