Robust Estimates in Balanced Norms for Singularly Perturbed Reaction Diffusion Equations Using Graded Meshes

Robust Estimates in Balanced Norms for Singularly Perturbed Reaction Diffusion Equations Using Graded Meshes

The goal of this paper is to provide almost robust approximations of singularly perturbed reaction-diffusion equations in two dimensions by using finite elements on graded meshes. When the mesh grading parameter is appropriately chosen, we obtain quasioptimal error estimations in a balanced norm for...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 96; no. 1; p. 18
Main Authors: Armentano, María Gabriela, Lombardi, Ariel L., Penessi, Cecilia
Format: Journal Article
Language:English
Published: New York Springer US 01-07-2023
Springer Nature B.V
Subjects:
Algorithms
Approximation
Computational Mathematics and Numerical Analysis
Energy
Estimates
Exact solutions
Finite element analysis
Interpolation
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Norms
Reaction-diffusion equations
Robustness (mathematics)
Theoretical
Singularly perturbed problems
65N15
Graded meshes
Reaction diffusion problems
Balanced norms
65N30
Supercloseness
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Summary:The goal of this paper is to provide almost robust approximations of singularly perturbed reaction-diffusion equations in two dimensions by using finite elements on graded meshes. When the mesh grading parameter is appropriately chosen, we obtain quasioptimal error estimations in a balanced norm for piecewise bilinear elements, by using a weighted variational formulation of the problem introduced by N. Madden and M. Stynes, Calcolo 58(2) 2021. We also prove a supercloseness result, namely, that the difference between the finite element solution and the Lagrange interpolation of the exact solution, in the weighted balanced norm, is of higher order than the error itself. We finish the work with numerical examples which show the good performance of our approach.
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-023-02245-y