Singularly perturbed reaction–diffusion problems on a k ‐star graph

Abstract We consider a singularly perturbed linear reaction–diffusion problem posed on the unit square in two dimensions. Standard finite element analyses use an energy norm, but for problems of this type, this norm is too weak to capture adequately the behaviour of the boundary layers that appear in the solution. To address this deficiency, a stronger so-called ‘balanced’ norm has been considered recently by several researchers. In this paper we shall use two-scale and multiscale sparse grid finite element methods on a Shishkin mesh to solve the reaction–diffusion problem, and prove convergence of their computed solutions in the balanced norm.

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Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems

Symmetry ◽

10.3390/sym13122291 ◽

2021 ◽

Vol 13 (12) ◽

pp. 2291

Author(s):

Yanjie Mei ◽

Sulei Wang ◽

Zhijie Xu ◽

Chuanjing Song ◽

Yao Cheng

Keyword(s):

Layer Structure ◽

Reaction Diffusion ◽

Singularly Perturbed ◽

Energy Norm ◽

Approximation Properties ◽

Anisotropic Meshes ◽

Balanced Norm ◽

Ldg Method ◽

Local Projections ◽

Diffusion Problems

We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction–diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and “balanced” norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the “balanced” norm is artificially introduced to capture the boundary layer contribution. We establish a uniform convergence of order k for the LDG method using the balanced norm with the local weighted L2 projection as well as an optimal convergence of order k+1 for the energy norm using the local Gauss–Radau projections. The numerical method, the layer structure as well as the used adaptive meshes are all discussed in a symmetry way. Numerical experiments are presented.

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A Numerical Scheme For Semilinear Singularly Perturbed Reaction-Diffusion Problems

Applied Mathematics and Nonlinear Sciences ◽

10.2478/amns.2020.1.00038 ◽

2020 ◽

Vol 5 (1) ◽

pp. 405-412

Author(s):

Kerem Yamac ◽

Fevzi Erdogan

Keyword(s):

Boundary Value Problems ◽

Numerical Scheme ◽

Boundary Value ◽

Reaction Diffusion ◽

Computational Approach ◽

Singularly Perturbed ◽

Uniform Mesh ◽

Numerical Example ◽

Uniformly Convergence ◽

Diffusion Problems

AbstractIn this study we investigated the singularly perturbed boundary value problems for semilinear reaction-difussion equations. We have introduced a basic and computational approach scheme based on Numerov’s type on uniform mesh. We indicated that the method is uniformly convergence, according to the discrete maximum norm, independently of the parameter of ɛ. The proposed method was supported by numerical example.

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