scholarly journals Singularly Perturbed Reaction–Diffusion Problems as First order systems

2021 ◽  
Vol 89 (2) ◽  
Author(s):  
Sebastian Franz
Keyword(s):  
Convergence Analysis ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Second Order ◽  
Singularly Perturbed ◽  
Convergence Order ◽  
First Order ◽  
Optimal Convergence ◽  
Balanced Norm ◽  
Diffusion Problems

AbstractWe consider a singularly perturbed reaction diffusion problem as a first order two-by-two system. Using piecewise discontinuous polynomials for the first component and $$H_{{{\,\mathrm{{div}}\,}}}$$ H div -conforming elements for the second component we provide a convergence analysis on layer adapted meshes and an optimal convergence order in a balanced norm that is comparable with a balanced $$H^2$$ H 2 -norm for the second order formulation.

Balanced-norm error estimates for sparse grid finite element methods applied to singularly perturbed reaction–diffusion problems

2019 ◽  
Vol 27 (1) ◽  
pp. 37-55 ◽  
Author(s):  
Stephen Russell ◽  
Martin Stynes
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Two Dimensions ◽  
Sparse Grid ◽  
Singularly Perturbed ◽  
Balanced Norm ◽  
Norm Error ◽  
Diffusion Problems

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.

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

Symmetry ◽  
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.

scholarly journals Reduced basis approximations of the solutions to spectral fractional diffusion problems

2020 ◽  
Vol 28 (3) ◽  
pp. 147-160
Author(s):  
Andrea Bonito ◽  
Diane Guignard ◽  
Ashley R. Zhang
Keyword(s):  
Computational Cost ◽  
Fractional Power ◽  
Reaction Diffusion ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Quadrature Point ◽  
Reduced Basis ◽  
Fast Evaluation ◽  
Bottle Neck ◽  
Diffusion Problems

AbstractWe consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction–diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction–diffusion problems. The reduced basis does not depend on the fractional power s for 0 < smin ⩽ s ⩽ smax < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.

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