The numerical solution of the convection-diffusion-reaction problem is considered in two and three dimensions. A stabilized finite element method of Galerkin/Least squares type accomodates diffusion-dominated as well as convection- and/or reaction- dominated situations. The resolution of boundary layers occuring in the singularly perturbed case is accomplished using anisotropic mesh refinement in boundary layer regions. In this paper, the standard analysis of the stabilized Galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement. Simplicial Lagrangian elements of arbitrary order are used.

Systems of grid equations that approximate elliptic boundary value problems on locally modified grids are considered. The triangulation, which approximates the boundary with second order of accuracy, is generated from an initial uniform triangulation by shifting nodes near the boundary according to special rules. This "locally modified" grid possesses several significant features: this triangulation has a regular structure, the generation of the triangulation is rather fast, this construction allows to use multilevel preconditioning (BPX-like) methods. The proposed iterative methods for solving elliptic boundary value problems approximately are based on two approaches: The fictitious space method, i.e. the reduction of the original problem to a problem in an auxiliary (fictitious) space, and the multilevel decomposition method, i.e. the construction of preconditioners by decomposing functions on hierarchical grids. The convergence rate of the corresponding iterative process with the preconditioner obtained is independent of the mesh size. The construction of the grid and the preconditioning operator for the three dimensional problem can be done in the same way.

No description available.

The numerical solution of the convection-diffusion-reaction problem is considered in two and three dimensions. A stabilized finite element method of Galerkin/Least squares type accomodates diffusion-dominated as well as convection- and/or reaction- dominated situations. The resolution of boundary layers occuring in the singularly perturbed case is accomplished using anisotropic mesh refinement in boundary layer regions. In this paper, the standard analysis of the stabilized Galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement. Simplicial Lagrangian elements of arbitrary order are used.

The paper is concerned with the finite element resolution of layers appearing in singularly perturbed problems. A special anisotropic grid of Shishkin type is constructed for reaction diffusion problems. Estimates of the finite element error in the energy norm are derived for two methods, namely the standard Galerkin method and a stabilized Galerkin method. The estimates are uniformly valid with respect to the (small) diffusion parameter. One ingredient is a pointwise description of derivatives of the continuous solution. A numerical example supports the result. Another key ingredient for the error analysis is a refined estimate for (higher) derivatives of the interpolation error. The assumptions on admissible anisotropic finite elements are formulated in terms of geometrical conditions for triangles and tetrahedra. The application of these estimates is not restricted to the special problem considered in this paper.