An optimal streamline diffusion finite element method for a singularly perturbed problem

Abstract

Abstract. The stability and accuracy of a streamline diffusion finite element method (SDFEM) on arbitrary grids applied to a linear 1-d singularly perturbed problem are studied in this paper. With a special choice of the stabilization quadratic bubble function, the SDFEM is shown to have an optimal second order in the sense thatwhere u h is the SDFEM approximation of the exact solution u and V h is the linear finite element space. With the quasi-optimal interpolation error estimate, quasi-optimal convergence results for the SDFEM are obtained. As a consequence, an open question about the optimal choice of the monitor function for a second order scheme in the moving mesh method is answered.

The Capriciousness of Numerical Methods for Singular Perturbations

“…But a careful study of the pointwise error [5] shows that the convergence rate can be N −δ if the coarse mesh sizes…”

“…In fact, for streamline diffusion the convergence rate is insensitive to the perturbation of the grid; see [5].…”

Convergence analysis of the LDG method for singularly perturbed two-point boundary value problems

“…Another effective way for globally solving this problem is to construct a numerical scheme on a layer adapted mesh, such as Shishkin type meshes and Bakhvalov type meshes. There are plenty of theoretical results about FEMs and stabilized FEMs on layer adapted meshes [6,17,23]. Recently, the LDG method was considered [20,22] for numerically solving singularly perturbed problems on layer adapted meshes.…”

Error Estimates of the Finite Element Method with Weighted Basis Functions for a Singularly Perturbed Convection-Diffusion Equation

“…To avoid non-physical numerical solutions, many special finite element techniques have been developed, including upwind finite element [1,4], Petrov-Galerkin finite element [7], streamline diffusion finite element methods [2,8,9], and exponentially fitted finite elements [18,[21][22][23]. However, these methods do not always give accurate results, especially when a diffusion coefficient has the same magnitude as that of mesh size.…”

“…Although the method proposed in [12] is promising from its numerical performance, except for a simple error bound of order O(h 1/2 | ln ε|) in [14] the mathematical understanding of the method is very limited. Regarding about the convergent results on layer-adapted meshes, streamline diffusion finite element or standard finite element methods can give uniformly optimal convergent rate, the reader is referred to [2,3,11,16,[24][25][26][27][28]. Moreover, spectral methods have been proposed to resolve the bounding layers, which are shown very effective, see, e.g., [29,30].…”