Schur complement domain decomposition algorithms for spectral methods

Spectral methods have been using a domain decomposition approach for handling irregular domains. The main focus has been on appropriate matching conditions for the solutions across the subdomain boundaries. In this paper, we propose an efficient method for solving the discrete equations based on solving the Schur complement system for the interface variables. We consider both the Funaro-Maday-Patera weak C¹ matching of the solutions on the interfaces and Orszag's exact C¹ matching. Numerical results for the model problem show that the condition number of the Schur complement system is of order O(n²). We show how this can be improved to nearly O(1) by a boundary probe preconditioner. We also compare briefly our method and the alternating Neumann-Dirichlet method of Funaro-Quarteroni-Zanolli.