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- Title
Energy-norm and balanced-norm supercloseness error analysis of a finite volume method on Shishkin meshes for singularly perturbed reaction–diffusion problems.
- Authors
Meng, Xiangyun; Stynes, Martin
- Abstract
A singularly perturbed reaction–diffusion problem posed on the unit square in R 2 is considered. To solve this problem numerically, a finite volume method (FVM) whose primal mesh is Shishkin is constructed; the FVM solution is piecewise bilinear on this mesh. Working in the standard energy norm, a superclose result (for the difference between the FVM solution and the Lagrange interpolant of the exact solution) is derived. This result yields an improved bound for the L 2 error of the FVM solution, and implies that a simple postprocessing of the FVM solution produces (in the energy norm) a higher-order approximation of the true solution. Next, we analyse errors in a balanced norm that is stronger than the energy norm; using a more complicated approximant of the exact solution from our piecewise bilinear space, we prove an optimal-order error bound and an associated supercloseness result showing that the difference between the FVM solution and our approximant is of higher order than the error itself. Finally, numerical experiments demonstrate the sharpness of our error bounds.
- Subjects
FINITE volume method; BILINEAR forms; FINITE difference method
- Publication
Calcolo, 2023, Vol 60, Issue 3, p1
- ISSN
0008-0624
- Publication type
Article
- DOI
10.1007/s10092-023-00535-3