We consider a mixed finite element method for a model second anisotropic elliptic equation on the unit square \Omega of the following form
- ( p_xx + \eps p_yy ) = f in \Omega, p = 0 in \Gamma,where \Gamma is the boundary of \Omega and \eps is a small constant. We study a multilevel preconditioner for this problem on uniform rectangular and triangular meshes.
We use a two-level result by Bramble, Pasciak, and Zhang [East-West J. Numer. Math., 4 (1996), pp.99-120], in which no approximation or regularity conditions are required. Our "coarse" level problem will be the finite element problem on the same mesh as in the mixed finite element problem.
In the rectangular case, we use the Schur complement of the mixed problem at the "fine" level. A mesh dependent form is used in the analysis and implementation. In the triangular case, we take the equivalent nonconforming finite element problem to the mixed finite element problem as the "fine" level problem. In both cases, line Jacobi smoothers are used to effectively reduce the error in the "fine" level.