Multilevel Methods for Nonconforming FEM Systems
April 25, 2011
4:00 p.m.
Svetozar Margenov
Abstract
The talk is devoted to multilevel preconditioning methods of AMLI type based on block matrix approximate factorization of the stiffness matrix. It is focused on the case of nonconforming finite elements.
The theory of the robust AMLI methods for anisotropic Crouzeix-Raviart problems will be presented at the first part of the talk. The main result is that a) the condition number is uniformly bounded for any ratio of coefficient jumps and anisotropy, and b) the total computational complexity is of optimal order, i.e. it is proportional to the number of the degrees of freedom. The analysis is based on uniform estimated of the constant in the strengthened CBS inequality and the locally introduced preconditioner of the pivot block in the heirarchical two-by-two splitting. The robustness with respect to the coefficient jumps is shaped automatically, while as we shall see, the robustness with respect to anisotropy needs special constructions.
The sound part will adress several selected applications of AMLI methods for nonconforming FEM systems including: a)Rannacher-Turek rotated tri-linear approximation of high-frequency-high-contrast coefficients: b) Crouzeix-Raviart FEM discretization of time dependent Navier-Stokes problems; c) A composite 3D algorithm for almost incompressible linear elasticity equations.
The theory to be presented is in the spirit of the book: J. Kraus, S. Margenov, Robust Algebraic Multilevel Methods and Algorithms, Radon Series on Computational and Applied Mathematics, 5, de Gruyter, 2009. Some more recent results will also be included ending with several open problems.