On the accuracy of grid approximations to nonsmooth solutions of a singularly perturbed reaction-diffusion equation in the square

Andreev, V. B.

doi:10.1134/s0012266106070044

Cited by 20 publications

(30 citation statements)

References 4 publications

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“…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.…”

Section: Assumption a Let The Boundary Condition Functionmentioning

confidence: 90%

“…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).…”

Section: Introductionmentioning

confidence: 90%

“…(2.2b) Finally, consider the ε-neighbourhood of the vertex p 1 of interior angle 3π/2, which can be covered by a domain Ω ε ⊂ (−Cε, Cε) 2 ∩ Ω such that ∂Ω ε \p 1 is smooth. Then, by Remark 2.3, we have u ∈ C 4,λ (∂Ω ε \p 1 ).…”

Section: Assumption a Let The Boundary Condition Functionmentioning

confidence: 99%

“…. , 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.…”

Section: Introductionmentioning

confidence: 99%

“…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).…”

Section: Introductionmentioning

confidence: 99%

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

Andreev

Kopteva

2008

Math. Comp.

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Abstract. A singularly perturbed reaction-diffusion equation is posed in a two-dimensional L-shaped domain Ω subject to a continuous Dirchlet boundary condition. Its solutions are in the Hölder space C 2/3 (Ω) and typically exhibit boundary layers and corner singularities. The problem is discretized on a tensor-product Shishkin mesh that is further refined in a neighboorhood of the vertex of angle 3π/2. We establish almost second-order convergence of our numerical method in the discrete maximum norm, uniformly in the small diffusion parameter. Numerical results are presented that support our theoretical error estimate.

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Section: Assumption a Let The Boundary Condition Functionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 90%

Section: Assumption a Let The Boundary Condition Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Andreev

Kopteva

2008

Math. Comp.

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A parameter‐robust numerical method for a system of reaction–diffusion equations in two dimensions

Kellogg

Madden

Stynes

2007

Numerical Methods Partial

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A system of M(≥ 2) coupled singularly perturbed linear reaction-diffusion equations is considered on the unit square. Under certain hypotheses on the coupling, a maximum principle is established for the differential operator. The relationship between compatibility conditions at the corners of the square and the smoothness of the solution on the closed domain is fully described. A decomposition of the solution of the system is constructed. A finite-difference method for the solution of the system on a Shishkin mesh is presented, and it is proved that the computed solution is second-order accurate (up to a logarithmic factor). Numerical results are given to support this result and to investigate the effect of weaker compatibility assumptions on the data.

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Reaction-Diffusion Equations

Ai¹,

Chen²,

Hastings³

2013

Encyclopedia of Systems Biology

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A system of M (≥ 2) coupled singularly perturbed linear reaction-diffusion equations is considered on the unit square. Under certain hypotheses on the coupling, a maximum principle is established for the differential operator. The relationship between compatibility conditions at the corners of the square and the smoothness of the solution on the closed domain is fully described. A decomposition of the solution of the system is constructed. A finite difference method for the solution of the system on a Shishkin mesh is presented and it is proved that the computed solution is second-order accurate (up to a logarithmic factor). Numerical results are given to support this result and to investigate the effect of weaker compatibility assumptions on the data.

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On the accuracy of grid approximations to nonsmooth solutions of a singularly perturbed reaction-diffusion equation in the square

Cited by 20 publications

References 4 publications

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

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

A parameter‐robust numerical method for a system of reaction–diffusion equations in two dimensions

Reaction-Diffusion Equations

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