The Residual-Free-Bubble Finite Element Method on Anisotropic Partitions

Abstract

Abstract. The subject of this work is the analysis and implementation of stabilized finite element methods on anisotropic meshes. We develop the anisotropic a priori error analysis of the residual-free-bubble (RFB) method applied to elliptic convection-dominated convection-diffusion problems in two dimensions, with finite element spaces of type Q k , k ≥ 1. In the case of P 1 finite elements, relying on the equivalence of the RFB method to classical stabilized finite element methods, we propose a new rule, justified through the analysis of the RFB method, for selecting the stabilization parameter in classical stabilized methods on two-dimensional anisotropic triangulations.Key words. residual-free-bubble finite element method, convection-dominated diffusion problems, stabilized finite element methods AMS subject classifications. 65N12, 65N39, 76M10DOI. 10.1137/0606580111. Introduction. Elliptic convection-diffusion problems arise in a vast number of applications, and their stable, accurate, and efficient solution is of significant theoretical and practical interest. From the computational point of view, problems of this kind become particularly challenging when convection dominates diffusion in the sense that the Péclet number, which measures the magnitude of the convective vector field over the length scale of the computational domain relative to the size of the diffusion coefficient, is large. Convection-dominated diffusion equations exhibit features which resemble those of the reduced, first-order hyperbolic equation arising from the second-order elliptic convection-diffusion equation on neglecting the diffusion term. For example, the solution may contain thin internal layers within the computational domain; also, due to the singular perturbation nature of an elliptic convection-dominated diffusion problem, the solution may exhibit thin boundary layers along sections of the boundary of the computational domain which correspond to the outflow part of the boundary for the reduced problem. As a result of this, on meshes which do not resolve internal and boundary layers, standard Galerkin finite element methods have poor stability and accuracy properties. The difficulties typically manifest themselves as large, maximum-principle-violating, oscillations in the numerical solution which occur predominantly along the characteristics of the reduced problem.The situation may be remedied by using a classical stabilized finite element method (such as a streamline-diffusion method or a Galerkin least-squares method) or a residual-free-bubble (RFB) finite element method; we refer to the monograph [28] for an extensive survey of the literature. Due to the presence of anisotropic numerical dissipation terms in the direction of the characteristics of the reduced equation whose role is to suppress undesirable numerical oscillations, these methods are capable of delivering accurate numerical solutions even on shape-regular computational meshes

On the stabilization parameter in the subgrid scale approximation of scalar convection–diffusion–reaction equations on distorted meshes

“…In particular, the definition of the stabilization parameter using the minimum element length, as the analysis of [23] suggests, is not the most convenient, as will be shown here. This fact was also noted in [10], where anisotropic error estimates for the residual free bubble (RFB) method are developed. Then, based on the equivalence between RFB and classical stabilization techniques, a new definition of the stabilization parameter is proposed on two-dimensional anisotropic triangulations.…”

“…In the case of anisotropic problems we propose to find the direction in which these numbers are bigger, that is, the direction of maximum instability of the problem. This is possible thanks to the definition of the directional dimensionless Pé clet and Damköhler numbers (10). Therefore, as a definition of the direction of k, we propose the direction in which some combination of D k and P k is maximum.…”

“…The situation is similar in the two examples and gives rise to the conclusion that motivated the proposal: k depends on the direction in which the instability of the problem appears. In these examples the way to determine which is the correct definition of the stabiliza- tion parameter was by determining the direction in which the instability appears and this was done by comparing the directional dimensionless numbers (10), which in this case reduce to…”

“…Then, based on the equivalence between RFB and classical stabilization techniques, a new definition of the stabilization parameter is proposed on two-dimensional anisotropic triangulations. However, quoting [10] ''this choice will guarantee stability for any, not too unreasonably designed partition" meaning aspect ratios less than 5.…”

“…In Section 5 some comments on the standard error analysis of the method are made. As usual, convergence estimates depend on interpolation estimates which are difficult to obtain in the anisotropic case (see [10,2] and references therein). However, the analysis clearly shows that the stabilization parameter cannot be defined using the minimum element length.…”