Global maximum norm parameter-uniform numerical method for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient

Abstract

A singularly perturbed convection-diffusion problem, with a discontinuous convection coefficient and a singular perturbation parameter ε, is examined. Due to the discontinuity an interior layer appears in the solution. A finite difference method is constructed for solving this problem, which generates ε-uniformly convergent numerical approximations to the solution. The method uses a piecewise uniform mesh, which is fitted to the interior layer, and the standard upwind finite difference operator on this mesh. The main theoretical result is the ε-uniform convergence in the global maximum norm of the approximations generated by this finite difference method. Numerical results are presented, which are in agreement with the theoretical results.

Parameter-Uniform Improved Hybrid Numerical Scheme for Singularly Perturbed Problems With Interior Layers

“…Here, the convection coefficient a, the reaction term b and the source term f satisfy the following assumptions The BVP (1.1)-(1.2) admits a unique solution u ∈ C 1 (Ω) ∩ C 2 (Ω − ∪ Ω + ) (see [2]). Throughout the paper, we assume that a, b, f ∈ C 3 (Ω − ∪ Ω + ) so that these functions can be extended into Ω − and Ω + in C 3 .…”

“…Over the last few years, several researchers developed the fitted mesh methods for solving singularly perturbed problems with non-smooth data, one can refer the articles [1,2,4,10,12] for the stationary case and [7,8,9] for the nonstationary case. However, it is worth mentioning that the hybrid numerical scheme proposed by Cen in [1], is an well-known fitted mesh method for solving singularly perturbed BVPs of the form (1.1)-(1.2) with discontinuous convection coefficient and the method is almost second-order accurate throughout the domain [0, 1] provided the perturbation parameter ε satisfies ε N −1 , otherwise the method is at worst first-order uniformly convergent with respect to ε in the discrete supremum norm (see the detailed discussion in Section 6).…”

Fitted Mesh Method for Singularly Perturbed Robin Type Boundary Value Problem with Discontinuous Source Term

“…Our goal is to construct an ε uniform numerical method for solving this problem, that is a numerical method which generates ε uniformly convergent numerical approximations to the solution and its derivatives. Note that problems with discontinuous data were treated theoretically, in the case of the solution of the convection diffusion with Dirichlet case problem [3,4]. In [5][6][7][8] the authors discussed a self-adjoint Dirichlet type problem with discontinuous source term.…”

“…Motivated by the works of [3][4][5][6] we, in the present paper, develop a computational method to solve SPBVPs for second order equations of the type:…”

A Petrov-Galerkin method for a singularly perturbed ordinary differential equation with non-smooth data

“…The classical finite difference method and finite element method are not fitted to solve this problem on the uniform mesh [4,1]. Since the classical Shishkin scheme successfully solved this singularly perturbed problem, the accuracy could not reach first order (see [3]). In this paper, we use a Petrov-Galerkin finite element method with the piecewiseexponential test function and the piecewise-linear trial function.…”