A numerical study on the finite element solution of singularly perturbed systems of reaction–diffusion problems

2007 ◽  
Vol 187 (2) ◽  
pp. 1351-1367 ◽  
Author(s):  
C. Xenophontos ◽  
L. Oberbroeckling
Keyword(s):  
Finite Element ◽  
Numerical Study ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Finite Element Solution ◽  
Element Solution ◽  
Singularly Perturbed Systems ◽  
Perturbed Systems ◽  
Diffusion Problems

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 Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction–diffusion problems

2020 ◽  
Vol 54 (6) ◽  
pp. 1951-1973 ◽  
Author(s):  
Iain Smears ◽  
Martin Vohralík
Keyword(s):  
Finite Element ◽  
Upper Bound ◽  
Mesh Size ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
A Posteriori ◽  
Finite Element Approximations ◽  
Arbitrary Space ◽  
Perturbation Parameters ◽  
Diffusion Problems

We consider energy norm a posteriori error analysis of conforming finite element approximations of singularly perturbed reaction–diffusion problems on simplicial meshes in arbitrary space dimension. Using an equilibrated flux reconstruction, the proposed estimator gives a guaranteed global upper bound on the error without unknown constants, and local efficiency robust with respect to the mesh size and singular perturbation parameters. Whereas previous works on equilibrated flux estimators only considered lowest-order finite element approximations and achieved robustness through the use of boundary-layer adapted submeshes or via combination with residual-based estimators, the present methodology applies in a simple way to arbitrary-order approximations and does not request any submesh or estimators combination. The equilibrated flux is obtained via local reaction–diffusion problems with suitable weights (cut-off factors), and the guaranteed upper bound features the same weights. We prove that the inclusion of these weights is not only sufficient but also necessary for robustness of any flux equilibration estimate that does not employ submeshes or estimators combination, which shows that some of the flux equilibrations proposed in the past cannot be robust. To achieve the fully computable upper bound, we derive explicit bounds for some inverse inequality constants on a simplex, which may be of independent interest.

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