The Discretely-Discontinuous Galerkin Coarse Grid for Domain Decomposition

TL;DR

This paper introduces an algebraic coarse grid method for domain decomposition that uses a low-dimensional polynomial space and a Discontinuous Galerkin basis, achieving optimal convergence and scalability.

Contribution

It presents a mesh-independent algebraic coarse grid construction based on input vectors and a Discontinuous Galerkin approach, with proven high-order convergence.

Findings

Converges in a constant number of iterations regardless of problem size.

Achieves optimal scaling in a two-level Schwarz preconditioner.

Effective for various scalar and vector PDEs.

Abstract

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.