A multiscale sparse grid finite element method for a two-dimensional singularly perturbed reaction-diffusion problem

2014 ◽  
Vol 41 (6) ◽  
pp. 987-1014 ◽  
Author(s):  
Niall Madden ◽  
Stephen Russell
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Sparse Grid ◽  
Singularly Perturbed ◽  
Two Dimensional ◽  
Element Method

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.

A fast finite difference/finite element method for the two-dimensional distributed-order time-space fractional reaction–diffusion equation

2020 ◽  
Vol 11 (02) ◽  
pp. 2050016
Author(s):  
Yaping Zhang ◽  
Jiliang Cao ◽  
Weiping Bu ◽  
Aiguo Xiao
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Two Dimensional ◽  
Time Space ◽  
Distributed Order ◽  
Fractional Reaction ◽  
Element Method

In this work, we develop a finite difference/finite element method for the two-dimensional distributed-order time-space fractional reaction–diffusion equation (2D-DOTSFRDE) with low regularity solution at the initial time. A fast evaluation of the distributed-order time fractional derivative based on graded time mesh is obtained by substituting the weak singular kernel for the sum-of-exponentials. The stability and convergence of the developed semi-discrete scheme to 2D-DOTSFRDE are discussed. For the spatial approximation, the finite element method is employed. The convergence of the corresponding fully discrete scheme is investigated. Finally, some numerical tests are given to verify the obtained theoretical results and to demonstrate the effectiveness of the method.

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