Nonlinear Domain Decomposition Methods and Software for Multicomponent Problems

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The focus of the research is the design and software implementation of a class of parallel nonlinear domain decomposition methods for the numerical solution of highly nonlinear multicomponent partial differential equations arising from important applications in computational fluid dynamics and computational biology. The algorithms and software development will have a great impact on several important application areas, and will also have substantial influence on other areas of computational sciences and engineering where large nonlinear equations need to be solved. To broaden the impact of the research, the software will be made fully compatible with the widely used PETSc package. Current users of PETSc, as wellas users of other packages with interfaces to PETSc, will be able to access to the algorithms and software.These multicomponent problems are usually not elliptic, and these non-elliptic components often cause the solution to have local singularities, such as boundary layers or sharp fronts. The new algorithms are motivated by the class of nonlinear elimination algorithms and nonlinear preconditioning algorithms for solvingalgebraic nonlinear equations with unbalanced nonlinearities. In the new algorithms the misscaled nonlinear components are first identified, then using the implicit function theorem of calculus these components are replaced by a function of the remaining more uniformly scaled components. The original system is hence reducedimplicitly to contain only the smooth components. Several important classes of applications are considered including problems in traditional CFD and problems in the emerging field of computational biology. To study the parallel performance of the algorithms on high performance computers a library will be developed as a plug-inpackage that is fully interoperable with PETSc of Argonne National Laboratory.