Multilevel Sparse Approximate Inverse Preconditioners for Adaptive Mesh Refinement∗

We present an efficient and effective preconditioning method for time-dependent simulations with dynamic, adaptive mesh refinement and implicit time integration. Adaptive mesh refinement greatly improves the efficiency of simulations where the solution develops steep gradients in small regions of the computational domain that change over time. Unfortunately, adaptive mesh refinement also introduces a number of problems for preconditioning in (parallel) iterative linear solvers, as the changes in the mesh lead to structural changes in the linear systems we must solve. Hence, we may need to compute a new preconditioner at every time-step. Since this would be expensive, we propose preconditioners that are cheap to adapt for dynamic changes to the mesh; more specifically, we propose preconditioners that require only localized changes to the preconditioner for localized changes in the mesh. Our preconditioners combine sparse approximate inverses and multilevel techniques. We demonstrate significant improvements in convergence rates of Krylov subspace methods and significant reductions of the overall runtime. Furthermore, we demonstrate experimentally level-independent convergence rates for various problems.