# Iterative methods for non-conforming finite elements

## Dr. Ivan Georgiev

**April 25, 2007, 2 p.m. HF 136**

Non-conforming rotated multilinear finite elements were introduced by Rannacher and Turek (R. Rannacher, S. Turek 1992) as a class of simple elements for stable discretization of the Stokes problem.

The recent activities in the development of efficient solution methods for non-conforming finite element systems are inspired by their attractive properties as a stable discretization tools for ill-conditioned problems. The model anisotropic elliptic second order boundary value problem is considered. Preconditioned Conjugate Gradient iterative method is used for iterative solution of the resulting linear algebraic system. Two preconditioning algorithms are presented.

Modified Incomplete Cholesky (MIC(0)) preconditioner belongs to the class of incomplete LU factorization methods. Modification of the stiffness matrix is the first step of the algorithm, then MIC(0) factorization is applied. This approach is applied for precondition-ing of separate displacement components of the rotated trilinear FEM elasticity systems.

A real-life benchmark problems are presented. Preconditioners based on various multilevel extensions of two-level finite element methods lead to iterative methods which often have an optimal order computational complexity with respect to the number of degrees of freedom of the system. Such methods were first presented by Axelsson and Vassilevski [O. Axelsson, P. Vassilevski 89&90], and are based on (recursive) two-level splittings of the finite element space. The key role in the derivation of optimal convergence rate estimates plays the constant γ in the so-called strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality, associated with the angle between the two subspaces of the splitting. The proposed variants of hierarchical two-level basis are first introduced in a rather general setting. Then, the involved parameters are studied and optimized. The major contribution is the derived estimates of the constant in the strengthened CBS inequality which is shown to allow the efficient multilevel extension of the related two-level preconditioners. Representative numerical tests well illustrate the optimal complexity of the resulting iterative solver.