Locally optimized preconditioning of Rannacher–Turek FEM systems

In this paper Rannacher–Turek non-conforming rotated bilinear finite elements are applied for the numerical solution of second order elliptic boundary value problems. The preconditioned conjugate gradient method is used for the iterative solution of the arising linear algebraic system A u = f . A locally optimized construction for an M-matrix approximation B of the global stiffness matrix A is the first step of the proposed algorithm. Then, the preconditioner is obtained by modified incomplete Cholesky factorization of the auxiliary M-matrix B. A comparative analysis concerning three different approaches for construction of such matrices B is presented. The related spectral condition number estimates are derived. The most important contributions of the paper is the developed original robust preconditioning scheme for strongly anisotropic problems based on properly skewed meshes. A set of numerical tests is presented to illustrate the theoretical investigations.