2015/04/03 by Essex Edwards, Edwards, Essex, Robert Bridson +1
Engineering · Physics and Astronomy · #65N55 #Advanced Numerical Analysis Techniques #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #G.1.8 #Model Reduction and Neural Networks #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1504.00907
openalex publication_date 2015/04/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We present an algebraic method for constructing a highly effective coarse grid correction to accelerate domain decomposition. The coarse problem is constructed from the original matrix and a small set of input vectors that span a low-degree polynomial space, but no further knowledge of meshes or continuous functionals is used. We construct a coarse basis by partitioning the problem into subdomains and using the restriction of each input vector to each subdomain as its own basis function. This basis resembles a Discontinuous Galerkin basis on subdomain-sized elements. Constructing the coarse problem by Galerkin projection, we prove a high-order convergent error bound for the coarse solutions. Used in a two-level symmetric multiplicative overlapping Schwarz preconditioner, the resulting conjugate gradient solver shows optimal scaling. Convergence requires a constant number of iterations, independent of fine problem size, on a range of scalar and vector-valued second-order and fourth-order PDEs.