| We study numerical methods and iterative solution techniques for a model second-order anisotropic elliptic partial differential equation on a unit square. In particular, we consider a mixed finite element approximation for this problem on uniform rectangular and triangular meshes and derive error estimates explicitly giving the behavior of the anisotropy parameter. To efficiently solve the resulting linear system from the mixed finite element problem, a two-level preconditioner is constructed and analyzed. Here, the fine and coarse level problems correspond to the mixed finite element and the standard finite element problems, respectively, on the same mesh. Utilizing the multigrid/multilevel preconditioners for the finite element problem, a multilevel preconditioner for the mixed system is obtained. To relate the mixed finite element problem with the standard finite element problem, we take the Schur complement of the mixed system in the rectangular case and an equivalent nonconforming problem in the triangular case. As smoothers in the multilevel preconditioners, the line Jacobi and line Gauss-Seidel smoothers are used. It is shown that this approach gives a preconditioner for the mixed system which is uniform both in the anisotropy parameter and the mesh size. Uniform multigrid preconditioners for the standard finite element method for the anisotropic problem are also discussed and numerical results for the two-level preconditioning are presented. |
