The iterative solution of a sequence of linear systems arising from nonlinear finite element analysis

Research on failure mechanisms of engineering components often focuses on modeling complex, nonlinear response. The analysis by finite element methods requires large-scale, very refined 3D solid models. Domain decomposition methods are frequently employed. Finite element methods for quasi-static and transient responses over longer time scales generally adopt an implicit formulation. Together with a Newton scheme for the nonlinear equations, such implicit formulations require the solution of large linear systems, thousands of times, to accomplish a realistic analysis. This represents an enormous computational burden. Robust, domain based iterative solvers are essential to increase scalability in parallel simulation codes.; We consider improvements to solver technology to reduce the overall solution time for a sequence of linear systems. Linear solvers that retain a subspace determined while solving previous systems can use that subspace to reduce the cost of solving the next system in the sequence. We call this process "Krylov subspace recycling". We develop two recycling solvers, and demonstrate on several model problems that we can reduce the iteration count required to solve a linear system by up to factor of two.; We analyze the convergence of one of the new solvers, which recycles nearly invariant subspaces, and establish residual bounds that suggest a convergence rate similar to one obtained by removing select eigenvector components from the initial residual vector. Experimental and theoretical results show that while recycling invariant subspaces can be beneficial, better choices exist.; Improved preconditioners form one avenue to bolster the performance of iterative solvers for this problem class. Domain decomposition preconditioners based on substructuring have been applied successfully to many engineering problems. For a domain decomposition method to exhibit satisfactory scalability, it must employ a "coarse-space" preconditioner. We consider the finite element tearing and interconnecting (FETI) method, as it is a popular domain decomposition method showing both numerical and parallel scalability. Application of the one-level FETI method produces a KKT (Karush-Kuhn-Tucker) linear system. We develop new connections between KKT and FETI solvers and preconditioners and show potential improvements to the FETI method, including the use of a less expensive approximate Schur complement in the FETI coarse problem.