Data-Sparse Schur Complement Domain Decomposition

A class of hierarchical matrices (H-matrices) allows the data-sparse approximation to integral and more general nonlocal operators (say, the Poincaré-Steklov operators) with almost linear cost. We consider the H-matrix-based approximation to the Schur complement on the interface [2] corresponding to the FEM discretisation of an elliptic operator L with jumping coefficients in Rd. As with the standard Schur complement domain decomposition methods, we split the elliptic inverse L−1 as a sum of local inverses associated with subdomains (this can be implemented in parallel), and the corresponding Poincaré-Steklov operator on the interface. Using the hierarchical formats based on either standard or weakened admissibility criteria (cf. [1]) we elaborate the approximate Schur complement inverse in an explicit form that is proved to have a linear-logarithmic cost O(NΓ log q NΓ), where NΓ is the number of degrees of freedom on the interface. The H-matrix-based preconditioner can be also applied