Past

We propose a new parallel domain decomposition algorithm to solve symmetric linear systems of equations derived from the discretization of PDEs on general unstructured grids of triangles or tetrahedra. The algorithm is based on a single-level Schwarz alternating procedure and a modified conjugate gradient solver. A single layer of overlap has been adopted in order to simplify the data-structure and minimize the overhead. This approach makes the global convergence rate vary slightly with the number of domains and the algorithm becomes highly scalable. The algorithm has been implemented in Fortran 90 using MPI and hence portable to different architectures. Numerical experiments have been carried out on a SunFire 15K parallel computer and have been shown superlinear performance in some cases.

In S18

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Non-normal matrices can be tiresome; some eigenvalues may be phlegmatic while others may be volatile. Computable error bounds are rarely used in such computations. We offer a way to proceed. Let (e,q,p') be an approximate eigentriple for non-normal B. Form column and row residuals r = Bq - qe and s' = p'B - ep'. We establish the relation between the smallest perturbation E, in both spectral and Frobenius norms, that makes the approximations correct and the norms of r and s'. Our results extend to the case when q and p are tall thin matrices and e is a small square matrix. Now regard B as a perturbation of B-E to obtain a (first order) bound on the error in e as a product of ||E|| and the condition number of e, namely (||q|| ||p'||)/|p'q|.