  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.
