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In this paper, we construct a finite volume element method (FVEM) on the Shishkin mesh for solving a singularly perturbed reaction–diffusion problem. The stability of the method is established in energy norm. Further, we derive the optimal error estimate in energy norm, under the decomposition of solution. Some numerical experiments are provided to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we construct and analyze an upwind finite volume element method on a Shishkin mesh for singularly perturbed convection–diffusion problems. We prove the stability of the method under the assumption of the convection and reaction term coefficients. The error estimate in the energy norm is presented on the Shishkin mesh, and the optimal error bound O (N − 1 (ln N) 3 / 2) is obtained. This error bound is uniformly valid with respect to the singular perturbation parameter. Numerical examples are provided to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]