Stability and accuracy of adapted finite element methods for singularly perturbed problems

Abstract

The stability and accuracy of a standard finite element method (FEM) and a new streamline diffusion finite element method (SDFEM) are studied in this paper for a one dimensional singularly perturbed connvection-diffusion problem discretized on arbitrary grids. Both schemes are proven to produce stable and accurate approximations provided that the underlying grid is properly adapted to capture the singularity (often in the form of boundary layers) of the solution. Surprisingly the accuracy of the standard FEM is shown to depend crucially on the uniformity of the grid away from the singularity. In other words, the accuracy of the adapted approximation is very sensitive to the perturbation of grid points in the region where the solution is smooth but, in contrast, it is robust with respect to perturbation of properly adapted grid inside the boundary layer. Motivated by this discovery, a new SDFEM is developed based on a special choice of the stabilization bubble function. The new method is shown to have an optimal maximum norm stability and approximation property in the sensewhere u N is the SDFEM approximation in linear finite element space V N of the exact solution u. Finally several optimal convergence results for the standard FEM and the new SDFEM are obtained and an open question about the optimal choice of the monitor function for the moving grid method is answered. Mathematics Subject Classification (2000)65L10 · 65L20 · 65L60 · 76R99

Optimal anisotropic meshes for minimizing interpolation errors in $L^p$-norm

“…We have obtained some preliminary results for a one dimensional convection dominated model problem. In [29,30] we show that for a carefully designed streamline diffusion finite element method the discretization error is controlled by the interpolation error (in the L ∞ norm).…”

Shishkin mesh simulation: A new stabilization technique for convection–diffusion problems

“…It is to be remarked, though, that the irregularity of the grid does not affect the SUPG-based SMS method. Some results explaining the degradation of performance of the Galerkin method on irregular grids can be found in [10] and [44]. Also, further numerical experiments with the SMS method on irregular grids can be found in [19].…”

“…Obviously, in order to have the value of α * we have to solve the whole system (7)(8)(9)(10). In the present paper, we introduce a technique to approximate α * without the need to compute the whole approximation on the Shishkin grid.…”

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

“…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].…”