Domain decomposition methods for partial differential equations

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Many problems in science and engineering are posed as partial differential equations (PDEs). For instance, conservations of mass, momentum and energy can be expressed as PDEs. Except for very simple cases, no analytic solution is available and we often rely on a numerical solution of the PDEs. This grant proposal deals with a class of methods called domain decomposition methods. In modern scientific computing, the problems are so large that no single computer can handle the computational and memory demands. The idea is to split up the domain into smaller subdomains and on a parallel computer, assign each processor to a subdomain. We solve the PDE on each subdomain in parallel and then stitch together these solutions to form the global solution. These are iterative methods, meaning that typically, the computed solution converges to the exact solution after infinitely many iterations. Of course in practice, the iteration is stopped when a prescribed error tolerance is satisfied. The goal is a numerical scheme which converges optimally, that is, at a rate which is independent of the discretization parameter. Another important issue is how to define the boundary condition(s) along each interior boundary. There is a subclass of methods, called Optimized Schwarz methods, which cleverly chooses free parameter(s) in the interior boundary conditions to optimize the convergence rate. We shall work on second-order PDEs as well as more difficult fourth-order PDEs. The latter PDEs are more difficult because they are far more ill-conditioned than second-order PDEs - the rate of convergence of traditional iterative methods is far slower for the fourth-order case. This program will implement these new algorithms as well as develop a theory of convergence rate of the methods. If successful, these methods will allow scientists and engineers to solve more complex problems and/or solve problems with a higher resolution.