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

Stephen Russell and Martin Stynes


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.

JEL Classification: 65N12; 65N15; 65N30

  1. Funding: The research of the second author is supported in part by the National Natural Science Foundation of China under grants 91430216 and NSAF-U1530401.


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Received: 2017-06-21
Revised: 2017-10-28
Accepted: 2017-10-30
Published Online: 2018-02-01
Published in Print: 2019-03-26

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