Helena Zarin

Interior penalty discontinuous approximations of convection–diffusion problems with parabolic layers

SummaryA 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 Shishkin–type 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 discontinuity–penalization parameters. Numerical experiments complement the theoretical results.

A second-order scheme for singularly perturbed differential equations with discontinuous source term

Abstract A Galerkin finite element method that uses piecewise linear functions on Shishkin- and Bakhvalov–Shishkin-type of meshes is applied to a linear reaction-diffusion equation with discontinuous source term. The method is shown to be convergent, uniformly in the perturbation parameter, of order N –2 ln2 N for the Shishkin-type mesh and N –2 for the Bakhvalov–Shishkin-type mesh, where N is the mesh size number. Numerical experiments support our theoretical results.

The streamline-diffusion method for a convection-diffusion problem with a point source

A singularly perturbed convection-diffusion problem with a point source is considered. The problem is solved using the streamline-diffusion finite element method on a class of Shishkin-type meshes. We prove that the method is almost optimal with second order of convergence in the maximum norm, independently of the perturbation parameter. We also prove the existence of superconvergent points for the first derivative. Numerical experiments support these theoretical results.

On discontinuous Galerkin finite element method for singularly perturbed delay differential equations

Abstract A nonsymmetric discontinuous Galerkin FEM with interior penalties has been applied to one-dimensional singularly perturbed problem with a constant negative shift. Using higher order polynomials on Shishkin-type layer-adapted meshes, a robust convergence has been proved in the corresponding energy norm. Numerical experiments support theoretical findings.

Continuous-discontinuous finite element method for convection-diffusion problems with characteristic layers

We study convergence properties of a numerical method for convection-diffusion problems with characteristic layers on a layer-adapted mesh. The method couples standard Galerkin with an h-version of the nonsymmetric discontinuous Galerkin finite element method with bilinear elements. In an associated norm, we derive the error estimate as well as the supercloseness result that are uniform in the perturbation parameter. Applying a post-processing operator for the discontinuous Galerkin method, we construct a new numerical solution with enhanced convergence properties.

Approximation of singularly perturbed reaction-diffusion problems by quadratic C 1 -splines

Collocation with quadratic C1-splines for a singularly perturbed reaction-diffusion problem in one dimension is studied. A modified Shishkin mesh is used to resolve the layers. The resulting method is shown to be almost second order accurate in the maximum norm, uniformly in the perturbation parameter. Furthermore, a posteriori error bounds are derived for the collocation method on arbitrary meshes. These bounds are used to drive an adaptive mesh moving algorithm. Numerical results are presented.

A singularly perturbed problem with two parameters on a Bakhvalov-type mesh

A singularly perturbed problem with two small parameters is considered. On a Bakhvalov-type 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 Clement quasi-interpolant as well as discretization error estimates in an energy norm. Numerical experiments support theoretical findings.

The Discontinuous Galerkin Finite Element Method for Singularly Perturbed Problems

A nonsymmetric discontinuous Galerkin finite element method with interior penalties is considered for two—dimensional singularly perturbed problems. On an anisotropic Shishkin mesh with bilinear elements we prove error estimates (uniformly in the perturbation parameter) in an integral norm associated with this method. We perform separate analyses for the cases of reaction—diffusion and convection—diffusion problems. On different types of interelement edges we derive the values of discontinuity—penalization parameters. Numerical experiments support the theoretical results.

On graded meshes for a two-parameter singularly perturbed problem

A one-dimensional reaction-diffusion-convection problem is numerically solved by a finite element method on two layer-adapted meshes, Duran-type mesh and Duran-Shishkin-type mesh, both defined by recursive formulae. Robust error estimates in the energy norm are proved. Numerical results are given to illustrate the parameter-uniform convergence of numerical approximations.

A singularly perturbed problem with two parameters in two dimensions on graded meshes

Abstract A numerical approximation of a convection–reaction–diffusion problem by standard bilinear finite elements is considered. Using Duran–Lombardi and Duran–Shishkin type meshes we prove first order error estimates in an energy norm. Numerical examples confirm our theoretical results and show smaller errors compared to the well-known Shishkin mesh.