Uniform grid approximation of nonsmooth solutions to the mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a rectangle

Pointwise approximation of corner singularities for a singularly perturbed reaction-diffusion equation in an $L$-shaped domain

“…Note that Assumption A is made only to simplify the presentation. Combining the analysis that we present under Assumption A with the analyses [2,3] for the rectangular domain, yields our main result, Theorem 3.1, without Assumption A.…”

“…Clavero et al [5] argue under the assumption that the compatibility conditions of up to second order are satisfied at the corners of the domain, which, combined with an analogue of our assumption (1.3), yields u ∈ C 4,λ (Ω); it is proved then that the error on a Shishkin mesh is O(N −2 ln 2 N ) uniformly in ε. Andreev [2,3] drops the unrealistic compatibility conditions assumption and proves the same error estimate for the same numerical method assuming only an analogue of condition (1.3), i.e., when the exact solution u is only in C 1,λ (Ω). Note that [3] also addressed the case of Dirichlet-Neumann boundary conditions and thus u being only in C λ (Ω).…”

“…|i|+|j| >C for some sufficiently largeC. Using Taylor series expansions, we observe that (3,0) see Figure 1. Table 1 presents numerical results for our method with the parameter α := (1.1) −1 , which is used in (1.4) to compute the mesh transition parameter σ.…”

“…. , 6, of angle π/2 are typical phenomena in the case of a rectangular domain; both their analytical and numerical analyses are presented in [5,2,3]. To simplify our presentation, we shall focus on the corner singularity at the vertex p 1 of interior angle 3π/2 and hence make the following assumption.…”

“…We also refer the reader to papers [2,3,5], which present maximum norm error estimates for finite difference approximations of problem (1.1), (1.2) posed in the unit square. Clavero et al [5] argue under the assumption that the compatibility conditions of up to second order are satisfied at the corners of the domain, which, combined with an analogue of our assumption (1.3), yields u ∈ C 4,λ (Ω); it is proved then that the error on a Shishkin mesh is O(N −2 ln 2 N ) uniformly in ε. Andreev [2,3] drops the unrealistic compatibility conditions assumption and proves the same error estimate for the same numerical method assuming only an analogue of condition (1.3), i.e., when the exact solution u is only in C 1,λ (Ω).…”