Abstract: Domain decomposition preconditioners for high-order Galerkin methods in two dimensions are often built from modules associated with the decomposition of the discrete space into subspaces of functions related to the interior of elements, individual edges, and vertices. The restriction of the original bilinear form to a particular subspace gives rise to a diagonal block of the preconditioner, and the action of its inverse on a vector has to be evaluated in each iteration. Each block can be replaced by a preconditioner in order to decrease the cost. Knowledge of the quality of this local preconditioner can be used directly in a study of the convergence rate of the overall iterative process.

The Schur complement of an edge with respect to the variables interior to two adjacent elements is considered. The assembly and factorization of this block matrix are potentially expensive, especially for polynomials of high degree. It is demonstrated that the diagonal preconditioner of one such block has a condition number that increases approximately linearly with the degree of the polynomials. Numerical results demonstrate that the actual condition numbers are relatively small.