A second-order overlapping Schwarz method for a 2D singularly perturbed semilinear reaction-diffusion problem

Natalia Kopteva, Maria Pickett

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

An overlapping Schwarz domain decomposition is applied to a semilinear reaction-diffusion equation posed in a smooth two-dimensional domain. The problem may exhibit multiple solutions; its diffusion parameter ε2 is arbitrarily small, which induces boundary layers. The Schwarz method invokes a boundary-layer subdomain and an interior subdomain, the narrow subdomain overlap being of width O(ε| ln h|), where h is the maximum side length of mesh elements, and the global number of mesh nodes does not exceed O(h−2). We employ finite differences on layer-adapted meshes of Bakhvalov and Shishkin types in the boundary-layer subdomain, and lumped-mass linear finite elements on a quasiuniform Delaunay triangulation in the interior subdomain.
For this iterative method, we present maximum norm error estimates for ε ∈ (0, 1]. It is shown, in particular, that when ε ≤ C| ln h|−1, one iteration is sufficient to get second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the maximum norm uniformly in ε. Numerical results are presented to support our theoretical conclusions.
LanguageEnglish
Pages81-105
Number of pages25
JournalMathematics of Computation
Volume81
Issue number277
Early online date18 Jul 2011
DOIs
Publication statusPublished - Jan 2012

Fingerprint

Schwarz Methods
Reaction-diffusion Problems
Singularly Perturbed
Semilinear
Overlapping
Boundary Layer
Boundary layers
Maximum Norm
Interior
Layer-adapted Mesh
Mesh
Shishkin Mesh
Iteration
Delaunay triangulation
Semilinear Equations
Multiple Solutions
Triangulation
Domain Decomposition
Iterative methods
Reaction-diffusion Equations

Keywords

  • semilinear reaction-diffusion
  • singular perturbation
  • domain decomposition
  • overlapping Schwarz
  • Bakhvalov mesh
  • Shishkin mesh
  • supra-convergence
  • lumped-mass finite elements

Cite this

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title = "A second-order overlapping Schwarz method for a 2D singularly perturbed semilinear reaction-diffusion problem",
abstract = "An overlapping Schwarz domain decomposition is applied to a semilinear reaction-diffusion equation posed in a smooth two-dimensional domain. The problem may exhibit multiple solutions; its diffusion parameter ε2 is arbitrarily small, which induces boundary layers. The Schwarz method invokes a boundary-layer subdomain and an interior subdomain, the narrow subdomain overlap being of width O(ε| ln h|), where h is the maximum side length of mesh elements, and the global number of mesh nodes does not exceed O(h−2). We employ finite differences on layer-adapted meshes of Bakhvalov and Shishkin types in the boundary-layer subdomain, and lumped-mass linear finite elements on a quasiuniform Delaunay triangulation in the interior subdomain.For this iterative method, we present maximum norm error estimates for ε ∈ (0, 1]. It is shown, in particular, that when ε ≤ C| ln h|−1, one iteration is sufficient to get second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the maximum norm uniformly in ε. Numerical results are presented to support our theoretical conclusions.",
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A second-order overlapping Schwarz method for a 2D singularly perturbed semilinear reaction-diffusion problem. / Kopteva, Natalia; Pickett, Maria.

In: Mathematics of Computation, Vol. 81, No. 277, 01.2012, p. 81-105.

Research output: Contribution to journalArticle

TY - JOUR

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AU - Pickett, Maria

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N2 - An overlapping Schwarz domain decomposition is applied to a semilinear reaction-diffusion equation posed in a smooth two-dimensional domain. The problem may exhibit multiple solutions; its diffusion parameter ε2 is arbitrarily small, which induces boundary layers. The Schwarz method invokes a boundary-layer subdomain and an interior subdomain, the narrow subdomain overlap being of width O(ε| ln h|), where h is the maximum side length of mesh elements, and the global number of mesh nodes does not exceed O(h−2). We employ finite differences on layer-adapted meshes of Bakhvalov and Shishkin types in the boundary-layer subdomain, and lumped-mass linear finite elements on a quasiuniform Delaunay triangulation in the interior subdomain.For this iterative method, we present maximum norm error estimates for ε ∈ (0, 1]. It is shown, in particular, that when ε ≤ C| ln h|−1, one iteration is sufficient to get second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the maximum norm uniformly in ε. Numerical results are presented to support our theoretical conclusions.

AB - An overlapping Schwarz domain decomposition is applied to a semilinear reaction-diffusion equation posed in a smooth two-dimensional domain. The problem may exhibit multiple solutions; its diffusion parameter ε2 is arbitrarily small, which induces boundary layers. The Schwarz method invokes a boundary-layer subdomain and an interior subdomain, the narrow subdomain overlap being of width O(ε| ln h|), where h is the maximum side length of mesh elements, and the global number of mesh nodes does not exceed O(h−2). We employ finite differences on layer-adapted meshes of Bakhvalov and Shishkin types in the boundary-layer subdomain, and lumped-mass linear finite elements on a quasiuniform Delaunay triangulation in the interior subdomain.For this iterative method, we present maximum norm error estimates for ε ∈ (0, 1]. It is shown, in particular, that when ε ≤ C| ln h|−1, one iteration is sufficient to get second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the maximum norm uniformly in ε. Numerical results are presented to support our theoretical conclusions.

KW - semilinear reaction-diffusion

KW - singular perturbation

KW - domain decomposition

KW - overlapping Schwarz

KW - Bakhvalov mesh

KW - Shishkin mesh

KW - supra-convergence

KW - lumped-mass finite elements

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