Variational Multiscale Method with Flexible Fine-Scale Basis for Diffusion-Reaction Equation: Built-In a Posteriori Error Estimate and Heterogenous Coefficients

Abstract

The diffusion-reaction equation develops sharp boundary and/or internal layers for the reaction-dominated case (i.e. singularly perturbed case). In this regime, spurious oscillations pollute the solution obtained with the Galerkin finite element method (FEM). To address this issue, we employ a stabilized Variational Multiscale (VMS) method that relies on an improved expression for the fine-scale stabilization parameter that is derived via the fine-scale variational formulation facilitated by the VMS framework. The flexible fine scale basis consists of enrichment functions which may be nonzero at element edges. The stabilization parameter thus derived has spatial variation over element interiors and along inter-element boundaries that helps model the rapid variation of the residual of the Euler-Lagrange equations over the domain. This feature facilitates consistent stabilization across boundary and internal layers as well as capturing anisotropic refinement effects. In addition, VMS methods come equipped with useful a posteriori error estimators. New numerical results are presented that show the performance of this VMS method with a flexible fine-scale basis for singularly perturbed diffusion-reaction equation. These include an evaluation of the built-in error estimate for homogenous domain, and an optional modification of the method for heterogeneous domains that may result in savings in the computational cost.