Krylov improvements of the Uzawa method for Stokes type operator matrices

The paper is devoted to Krylov type modifications of the Uzawa method on the operator level for the Stokes problem in order to accelerate convergence. First block preconditioners and their effect on convergence are studied. Then it is shown that a Krylov–Uzawa iteration produces superlinear convergence on smooth domains, and estimation is given on its speed.

A Segregated Approach for Modeling the Electrochemistry in the 3-D Microstructure of Li-Ion Batteries and Its Acceleration Using Block Preconditioners

Battery performance is strongly correlated with electrode microstructure. Electrode materials for lithium-ion batteries have complex microstructure geometries that require millions of degrees of freedom to solve the electrochemical system at the microstructure scale. A fast-iterative solver with an appropriate preconditioner is then required to simulate large representative volume in a reasonable time. In this work, a finite element electrochemical model is developed to resolve the concentration and potential within the electrode active materials and the electrolyte domains at the microstructure scale, with an emphasis on numerical stability and scaling performances. The block Gauss-Seidel (BGS) numerical method is implemented because the system of equations within the electrodes is coupled only through the nonlinear Butler–Volmer equation, which governs the electrochemical reaction at the interface between the domains. The best solution strategy found in this work consists of splitting the system into two blocks—one for the concentration and one for the potential field—and then performing block generalized minimal residual preconditioned with algebraic multigrid, using the FEniCS and the Portable, Extensible Toolkit for Scientific Computation libraries. Significant improvements in terms of time to solution (six times faster) and memory usage (halving) are achieved compared with the MUltifrontal Massively Parallel sparse direct Solver. Additionally, BGS experiences decent strong parallel scaling within the electrode domains. Last, the system of equations is modified to specifically address numerical instability induced by electrolyte depletion, which is particularly valuable for simulating fast-charge scenarios relevant for automotive application.

Composable, Scalable Solvers on Staggered Grids

<p>Scalable preconditioners for saddle point problems are essential to the solution of problems in geodynamics and beyond. Recent years have produced a wealth of research into efficient solvers for finite element methods.  These solvers are also effective, however, for orthogonal-grid finite volume discretizations of saddle point problems, also know as "staggered grid" or "marker and cell (MAC)" methods. Perhaps, ironically, due to the highly-structured nature of these discretizations, the use of advanced solvers is stymied due to the lack of a uniform topological abstraction, which is required for most scalable solvers, such as geometric multigrid.  We present new software to allow experimentation with and composition of these advanced solvers.  We focus on variable-viscosity Stokes problems with discontinuous coefficient jumps.  In particular, we attempt to demonstrate how the important know robust preconditioners may be employed, and how new variants may be experimented with.  Important solvers are compositions of block factorizations and multigrid cycles.  We demonstrate as many of these as possible, including triangular block preconditioners with nested multigrid solves, and monolithic multigrid solves with cellwise (Vanka) or field-based (Distributed Gauss-Seidel, Braess-Sarazin) smoothers.  Implementations are provided as part of the PETSc library, using the new DMStag component, and examples from the StagBL library are also shown where appropriate.  These tools are intended to help break down the barrier between cutting-edge research into advanced solvers (which is only becoming more complex, as multi-phase problems are further explored) and practical usage in geophysical research and production codes.</p>