Schur complement method

The Schur complement method is the basic and the earliest version of non-overlapping domain decomposition method, also called iterative substructuring. A finite element problem is split into non-overlapping subdomains, and the unknowns in the interiors of the subdomains are eliminated. The remaining Schur complement system on the unknowns associated with subdomain interfaces is solved by the conjugate gradient method.

The Schur complement is usually not stored, but the multiplication of a vector by the Schur complement is implemented by solving the Dirichlet problem on each subdomain. The multiplication of a vector by the Schur complement is a discrete version of the Poincaré-Steklov operator, also called the Dirichlet to Neumann mapping.

There are two benefits of this method. First, the elimination of the interior unknowns on the subdomains, that is the solution of the Dirichlet problems, can be done in parallel. Second, passing to the Schur complement reduces condition number and thus tends to decrease the number of iterations. For second-order problems, such as the Laplace equation or linear elasticity, the matrix of the system has condition number of the order $1/h^2$, where $h$ is the characteristic element size. The Schur complement, however, has condition number only of the order $1/h$.

For performance, the Schur complement method is combined with preconditioning, at least a diagonal preconditioner. The Neumann-Neumann method and the Neumann-Dirichlet method are the Schur complement method with particular kinds of preconditioners.