A Domain Decomposition Preconditioner Based on a Change to a Multilevel Nodal Basis

A domain decomposition method based on a simple change of basis on the interfaces and vertices is presented. It is shown that this leads to an effective preconditioner compared to the ones previously considered, such as the preconditioner by Bramble, Pasciak, and Schatz (BPS) [Math. Comp., 47 (1986), pp. 103-134], and the hierarchical basis domain decomposition (HBDD) preconditioner by Smith and Widlund [SIAM J. Sci. Statist. Comput., 11 (1990), pp. 1212-1226]. This domain-decomposed preconditioner is based on Bramble, Pasciak, and Xu’s multilevel nodal basis preconditioner [Math. Comp., to appear]. It is shown that analytically this method and the HBDD method give the same order of condition number, namely, O(log2(H/h)) for problems with smooth coefficients. Numerically this method appears to be more effective with little additional cost and for the model Poisson problem, the condition numbers appear to be O(1). Key words, domain decomposition, hierarchical basis, multilevel nodal basis, preconditioned conjugate gradient methods, Schur complement AMS(MOS) subject classifications. 65F10, 65N30