Modeling of steep layers in singularly perturbed diffusion–reaction equation via flexible fine-scale basis

Abstract

We present a new stabilized method for the diffusion–reaction equation which develops sharp boundary and/or internal layers for the reaction-dominated case (i.e. singularly perturbed case). The method relies on an improved expression for the stabilization parameter that is derived via the fine-scale variational formulation facilitated by the variational multiscale (VMS) framework. The proposed fine-scale basis consists of enrichment functions which may be nonzero at element edges. The derived stabilization parameter enjoys 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 also facilitates consistent stabilization across boundary and internal layers as well as capturing anisotropic refinement effects. The method is able to better satisfy the maximum principle as compared to other existing methods. We present a priori mathematical analysis of the stability and convergence of the method for the diffusion–reaction equation. Optimal convergence on meshes comprised of linear triangles and bilinear quadrilateral elements are presented for smooth problems, as well as for problems with steep boundary layers. Stability and accuracy features of the method for problems with discontinuous forcing function, internal layers, and boundary layers are shown and its performance on unstructured and distorted meshes comprised of quadrilateral and triangular elements is highlighted.