A Two-Level Preconditioner for Schrödinger-Type Singular Elliptic Operator

Abstract

In this paper a two-level preconditioner is considered for the Schrodinger-type singular operator on a uniform grid on a rectangle. The construction of such preconditioners for a positive definite operator is well defined provided that a restriction of the operator into any subdomain of computational domain is at least positive semidefinite. It is shown that although local matrices approximating the Schrodinger operator on some subdomains are indefinite it is still possible to develop a multilevel preconditioner, which is spectrally equivalent to the operator of the problem. First, explicit estimates of condition number are established for a two-level preconditioner. The main idea of the two-level estimate is to analyze a superelement-based method for a problem with indefinite local superelement matrices, modify the main diagonals of the local matrices so that they are positive definite and still sum up to the same global matrix. This modification is for theoretical purposes only and does not affect the computational method itself. A generalization of the usual two-level method is considered in which different bilinear forms are defined on the coarse and fine levels. Then, the two-level method can be extended to a multilevel one using the standard theory of multilevel methods with inner iterative procedures.