Institute for Parallel Processing
Bulgarian Academy of Sciences
The iterative methods play an important role in solving linear equations that arise in real-world applications. Numerous properties of the problem may affect the efficiency of the solution. This talk deals with algorithms for the solution of linear systems of algebraic equations with large-scale sparse matrices, with a focus on problems that are obtained after discretization of partial differential equations using nonconforming finite element methods. The Crouzeix-Raviart and Rannacher-Turek elements are considered as stable approximation tools for 3D elliptic problems in strongly heterogeneous media. The robustness issues for problems with coefficient jumps of high frequency and high contrast are currently of a strongly growing interest.
Algebraic MultiLevel Iteration (AMLI) methods of optimal computational complexity are presented. In the case of nonconforming elements, the finite element spaces associated with two consecutive (nested) mesh refinements are not nested. To enable the use of the general multilevel scheme, several hierarchical two-level decompositions of the stiffness matrices are proposed. Uniform estimates of the constant in the strengthened Cauchy-Bunyakowski-Schwarz inequality are obtained, ensuring the optimality of the related AMLI methods. The presented numerical tests illustrate both, the robust convergence and the time scalability.
The second part of the talk demonstrates how several multilevel techniques can be integrated and extended in the construction of new AMLI solvers for more complex real life cases. A two phase linear elasticity problem is considered. Crouzeix-Raviart finite elements are used to get a locking-free discretization. The method is applied for numerical homogenization of bone microstructures. The solid phase of trabecular bone tissues is obtained by a computer tomography (CT) image at a micron scale. The complex geometry is described in terms of voxels the size of which corresponds to the resolution of the CT. Such problems are also referred to as finite element analysis of voxel structures. The presented numerical results illustrate the contribution of the fluid phase to the homogenized elasticity properties. The nonlinear influence of the porosity, i.e. of the level of osteoporosis, is also shown.
J. Kraus, S. Margenov, Robust Algebraic Multilevel Methods and Algorithms, Radon Series on Computational and Applied Mathematics, 5, de Gruyter, 2009