Domain decomposition algorithms for the partial differential equations of linear elasticity

The use of the finite element method for elasticity problems results in extremely large, sparse linear systems. Historically these have been solved using direct solvers like Choleski's method. These linear systems are often ill-conditioned and hence require good preconditioners if they are to be solved iteratively. We propose and analyze three new, parallel iterative domain decomposition algorithms for the solution of these linear systems. The algorithms are also useful for other elliptic partial differential equations. Domain decomposition algorithms are designed to take advantage of a new generation of parallel computers. The domain is decomposed into overlapping or nonoverlapping subdomains. The discrete approximation to a partial differential equation is then obtained iteratively by solving problems associated with each subdomain. The algorithms are often accelerated using the conjugate gradient method. The first new algorithm presented here borrows heavily from multi-level type algorithms. It involves a local change of basis on the interfaces between the substructures to accelerate the convergence. It works well only in two dimensions. The second algorithm is optimal in that the condition number of the iteration operator is bounded independently of the number of subdomains and unknowns. It uses non-overlapping subdomains, but overlapping regions of the interfaces between subdomains. This is an additive Schwarz algorithm, which works equally well in two or three dimensions. The third algorithm is designed for problems in three dimensions. It includes a coarse problem associated with the unknowns on the wirebaskets of the subdomains. The new method offers more potential parallelism than previous algorithms proposed for three dimensional problems since it allows for the simultaneous solution of the coarse problem and the local problems.