Robust Preconditioners for H(grad), H(curl) and H(div) Systems with Strongly Discontinuous Coefficients.

This dissertation is devoted to practical design and theoretical analysis of efficient and robust preconditioners for solving algebraic systems arising from the approximation of partial differential equations, with special emphasis on the problems with strongly discontinuous coefficients. The problems considered here include the standard second order elliptic equations (H( grad) or H1 equations), as well as the second order elliptic systems given in terms of curl and divergence operators (H(curl) and H(div) systems). In regard to the H1 equations with jump coefficients, we study both the multilevel and domain decomposition preconditioners. We analyze the eigenvalue distribution of the preconditioned systems, and prove that only a small number of eigenvalues may deteriorate with respect to coefficients and mesh size. We show that the other eigenvalues are bounded uniformly with respect to the coefficients of the PDE and bounded polylogarithmically with respect to mesh size. As a result, the asymptotic convergence rate of the PCG algorithms is uniform with respect to the coefficients, and nearly uniform (up to a logarithmic factor) with respect to the mesh size. Various numerical experiments justify the theoretical results. For H(curl) and H(div) systems, we give a comprehensive analysis of auxiliary space preconditioners, which rely on regular decompositions. Through such constructions, these preconditioners reduce H(curl) and H(div) systems to the solution of several H(grad) equations, which are amenable by standard algebraic multigrid (AMG) techniques. We also develop a preconditioner for H(curl) systems with jump coefficients. We show that the condition number of the preconditioned system is uniformly bounded with respect to coefficients and mesh size. Another class of preconditioners that we propose are the compatible AMG preconditioners for H( curl) and H(div) systems. This approach makes use of a compatible discretization framework. We reformulate the discrete systems into equivalent 2-by-2 block systems based on discrete Hodge decompositions on co-chains, and then construct the AMG preconditioners for the 2-by-2 block systems. As an important application, we present the augmented Lagrangian method for solving mixed formulations of elliptic boundary value problems, which reduces a saddle point problem to a nearly singular H(div) system.