A parameter-uniform numerical method for a singularly perturbed two parameter elliptic problem

Abstract

In this paper, a class of singularly perturbed elliptic partial differential equations posed on a rectangular domain is studied. The differential equation contains two singular perturbation parameters. The solutions of these singularly perturbed problems are decomposed into a sum of regular, boundary layer and corner layer components. Parameter-explicit bounds on the derivatives of each of these components are derived. A numerical algorithm based on an upwind finite difference operator and a tensor product of piecewise-uniform Shishkin meshes is analysed. Parameter-uniform asymptotic error bounds for the numerical approximations are established.

Superconvergence analysis of a finite element method for a two-parameter singularly perturbed problem

“…The fundamental character of the solutions of the problem (2.1) is significantly different to the character of the solutions of problems considered in [16,17,[23][24][25]. Shishkin considers elliptic problem posed on bounded [25] and unbounded [23,24] domain for which the totally reduced equation (when ε 1 = ε 2 = 0) is of the first order and the presence of various kinds of boundary layers (exponential, parabolic and hyperbolic) is possible.…”

“…Two-parameter elliptic equation whose solution exhibits only exponential layers is considered in O'Riordan at al. [16,17]. We emphasize the importance of studying problems whose solutions exhibit parabolic layers since they also have the layer quality of fast decay in a narrow region along the characteristic boundaries but have much more complicated structure.…”

“…In contrast to [16,17,[23][24][25], where finite differences are used to obtain a numerical solution, to the best of our knowledge [29] is the only paper considering a finite element method for a two-parameter singularly perturbed problem in two dimensions.…”

“…some typical cases only, since the analysis of the other terms relies on analogous arguments.Technique similar to[32] can be applied for deriving (A 1619)…”

A robust layer adapted difference method for singularly perturbed two-parameter parabolic problems

“…Fitted operator methods have also been developed for the two-parameter problem in [14,17,20]. The two-dimensional counterpart of the steady-state two-parameter problem was considered in [9,13], where the authors studied the two-parameter problem for all the values of the parameters in the parameter space {( , μ) | 0 < ≤ 1, 0 < μ ≤ 1} on the Shishkin mesh for an upwind difference scheme which gave an almost first-order accuracy. In [18,19], Teofanov and Roos gave a solution decomposition and numerical solution using the finite-element method for steady-state elliptic two-parameter problem with parabolic layers in addition to exponential and corner layers.…”