On using the cell discretization algorithm for mixed-boundary value problems and domain decomposition

The cell discretization algorithm is used to approximate solutions to self-adjoint elliptic equations with general nonhomogeneous Dirichlet or Neumann or mixed-boundary values. Error estimates are obtained showing general convergence. This provides the framework for a nonoverlapping iterative algorithm for domain decomposition. The domain of a Dirichlet problem is partitioned into two subdomains. Solutions on the two subdomains can be patched together to form a solution to the original problem provided the solutions agree across the common interface of the subdomains and the normal derivatives there have equal absolute values and are opposite in sign. We generate such a solution by alternating between imposing Neumann and Dirichlet conditions on the interface, with boundary data adapted from the results of the previous approximation. A posteriori error estimates show that we have global convergence provided computed interface errors are sufficiently small.