The focus of this three-year research effort is the design, analysis and software implementation of a class of parallel nonlinear iterative methods for the numerical solution of some highly nonlinear partial differential equations (PDEs) arising from important applications in computational fluid dynamics and computational biology. The nonlinear PDEs to be considered in the project are usually not strongly elliptic, and they often contain non-elliptic components causing the solution to be nonsmooth and have local singularities, such as boundary layers or sharp fronts. For smooth nonlinear problems, traditional nonlinear methods, such as Newton's methods, are capable of reducing the global nonlinearities at a nearly quadratic convergence rate but become very slow once the local singularities appear somewhere in the computational domain, even if this happens to only a few components of a largenonlinear system. The proposed algorithm is motivated by the class of nonlinear preconditioning algorithms introduced by Cai and Keyes in 2001 for solving algebraic nonlinear equations that have unbalanced nonlinearities. In nonlinear preconditioning, the global problem is partitioned into subproblems, and subspace nonlinear eliminations are performed on all subproblems. The subproblem are then 'glued' together by a Schwarz type domain decomposition method. Due to the subspace nonlinear elimination, the local singularities are removed, and the global system therefore has more uniform nonlinearity. A family of such algorithms will be studied using a combination of multilevel/multigrid, domain decomposition, nonlinear preconditioning, and nonlinear elimination methods. Roughly speaking, in these algorithms, domain decomposition provides the parallelism, multilevel provides the scalability with respect to the problem size and to the number of processors on parallel computers, and nonlinear elimination removes the sensitivity to the local singularities. Several important application problems will be considered, including the steady state incompressible Navier-Stokes equations with high Reynolds number and the optimization of a steady state biofluid problem. To study the parallel performance of the algorithms on high performance computers, such as a cluster of workstations and supercomputers, a library will be developed as a plug-in package that is fully interoperable with PETSc of Argonne National Laboratory. The proposed algorithm and software development will have a great impact on the application areas, and will also have substantial influence on other areas of computational sciences where large nonlinear equations need to be solved.
