Abstract
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.
| Language | English |
|---|---|
| Pages | 81-105 |
| Number of pages | 25 |
| Journal | Mathematics of Computation |
| Volume | 81 |
| Issue number | 277 |
| Early online date | 18 Jul 2011 |
| DOIs | |
| Publication status | Published - Jan 2012 |
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Keywords
- semilinear reaction-diffusion
- singular perturbation
- domain decomposition
- overlapping Schwarz
- Bakhvalov mesh
- Shishkin mesh
- supra-convergence
- lumped-mass finite elements
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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 journal › Article
TY - JOUR
T1 - A second-order overlapping Schwarz method for a 2D singularly perturbed semilinear reaction-diffusion problem
AU - Kopteva, Natalia
AU - Pickett, Maria
PY - 2012/1
Y1 - 2012/1
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
U2 - 10.1090/S0025-5718-2011-02521-4
DO - 10.1090/S0025-5718-2011-02521-4
M3 - Article
VL - 81
SP - 81
EP - 105
JO - Mathematics of Computation
T2 - Mathematics of Computation
JF - Mathematics of Computation
SN - 0025-5718
IS - 277
ER -