Ljiljana Teofanov

The discrete minimum principle for quadratic spline discretization of a singularly perturbed problem

We consider a singularly perturbed boundary value problem with two small parameters. The problem is numerically treated by a quadratic spline collocation method. The suitable choice of collocation points provides the discrete minimum principle. Error bounds for the numerical approximations are established. Numerical results give justification of the parameter-uniform convergence of the numerical approximations.

Family of quadratic spline difference schemes for a convection-diffusion problem

We consider the finite difference approximation of a singularly perturbed one-dimensional convection–diffusion two-point boundary value problem. It is discretized using quadratic splines as approximation functions, equations with various piecewise constant coefficients as collocation equations and a piecewise uniform mesh of Shishkin type. The family of schemes is derived using the collocation method. The numerical methods developed here are non-monotone and therefore apart from the consistency error we use Greens grid function analysis to prove uniform convergence. We prove the almost first order of convergence and furthermore show that some of the schemes have almost second-order convergence. Numerical experiments presented in the paper confirm our theoretical results.

A modification of the Shishkin discretization mesh for one-dimensional reaction-diffusion problems

In this paper we consider a modification of the Shishkin discretization mesh designed for the numerical solution of one-dimensional singularly perturbed reaction-diffusion problems. The modification consists of a slightly different choice of the transition points between the fine and coarse parts of the mesh. We prove that this change does not affect the order of convergence of the numerical solution obtained by using the central finite-difference scheme. However, due to a better layer-resolving mesh, numerical results show an improvement in the accuracy of the computed solution when compared to the results on the standard Shishkin mesh.

A robust layer-resolving spline collocation method for a convection–diffusion problem

Abstract We consider finite difference approximation of a singularly perturbed one-dimensional convection–diffusion two-point boundary value problem. The problem is numerically treated by a quadratic spline collocation method on a piecewise uniform slightly modified Shishkin mesh. The position of collocation points is chosen so that the obtained scheme satisfies the discrete minimum principle. We prove pointwise convergence of order O ( N - 2 ln 2 N ) inside the boundary layer and second order convergence elsewhere. The uniform convergence of the approximate continual solution is also given. Further, we approximate normalized flux and give estimates of the error at the mesh points and between them. The numerical experiments presented in the paper confirm our theoretical results.

On the quasilinear boundary-layer problem and its numerical solution

We obtain improved derivative estimates for the solution of the quasilinear singularly perturbed boundary-value problem. This enables us to modify the transition point between the fine and coarse parts of the Shishkin discretization mesh. The resulting mesh may be denser in the layer than the standard Shishkin mesh. When this is the case, numerical experiments show an improvement in the accuracy of the computed solution.

A modified Bakhvalov mesh

Abstract We consider a few numerical methods for solving a one-dimensional convection–diffusion singularly perturbed problem. More precisely, we introduce a modified Bakvalov mesh generated using some implicitly defined functions. Properties of this mesh and convergence results for several methods on it are given. Numerical results are presented in support of the theoretical considerations.

A uniform numerical method for semilinear reaction-diffusion problems with a boundary turning point

Motivated by problems arising in semiconductor-device modeling, this paper is concerned with a singularly perturbed semilinear reaction-diffusion problem with a boundary turning point. It is proved that the problem has a unique solution with two boundary layers. Based on the estimates of the derivatives of the solution, a numerical method is proposed which uses the classical finite-difference discretization on a Bakhvalov-type mesh. Second-order accuracy, uniform with respect to the perturbation parameter, is proved in the maximum norm. Numerical results are presented in support of the theoretical ones.

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.