Two-level preconditioned conjugate gradient methods with applications to bubbly flow problems

The Preconditioned Conjugate Gradient (PCG) method is one of the most popular iterative methods for solving large linear systems with a symmetric and positive semi-definite coefficient matrix. However, if the preconditioned coefficient matrix is ill-conditioned, the convergence of the PCG method typically deteriorates. Instead, a two-level PCG method can be used. The corresponding two-level preconditioner usually treats unfavorable eigenvalues of the coefficient matrix effectively, so that the two-level PCG method is expected to converge faster than the original PCG method. Many two-level preconditioners are known in the fields of deflation, multigrid and domain decomposition methods. Several of them are discussed in this thesis, where the main focus is on the deflation method. We show some theoretical properties of the deflation method, which give insights into the effectiveness of this method. A crucial component of the deflation preconditioner is the choice of projection vectors. Several choices are discussed and examined. We advocate that subdomain projection vectors, which are based on disjoint and piecewise-constant vectors, are among the best choices for a class of problems. Subsequently, we examine the application of the deflation method to linear systems with singular coefficient matrices. Several mathematically equivalent variants of the original deflation method are proposed to deal with the possible singularity of this coefficient matrix. In addition, two approaches are discussed in order to handle coarse linear systems with a Galerkin matrix, which are involved in each iteration of the deflation method. After the discussion of the implementation and efficiency issues of the deflation method, it is demonstrated that this method is usually faster than the original PCG method. Moreover, we present a comparison between the deflation method and other well-known two-level PCG methods, among them the balancing-Neumann-Neumann, additive coarse-grid correction
Electrical Engineering, Mathematics and Computer Science