Block preconditioning and domain decomposition methods. II. (English) Zbl 0658.65094
Domain decomposition methods for the solution of partial differential equations are attractive on parallel processors, because each processor can work independently on a large subtask. The corresponding stiffness matrix takes a sparse block structure, for which preconditioned iterative methods can be used when solving linear systems with the stiffness matrix. For domains decomposed in strips we get a blocktridiagonal structure for which a new block LU preconditioner was presented in part I: Report 8735, Dept. Math., Univ. Nijmegen (1987).
An alternative method, and also the one more commonly used for substructuring methods is based on approximation of the Schur complement matrix i.e., the reduced system for the dividing line. This approximation is frequently done by various difference methods. In the present paper we examine methods on algebraic approximation methods, for the Schur complement and for the normal form of it. It is found that a certain pentadiagonal matrix is spectrally equivalent to the normal form and can be readily computed just using the action of the matrix on certain vectors. Hence the explicit form of the Schur complement is not required.
An alternative method, and also the one more commonly used for substructuring methods is based on approximation of the Schur complement matrix i.e., the reduced system for the dividing line. This approximation is frequently done by various difference methods. In the present paper we examine methods on algebraic approximation methods, for the Schur complement and for the normal form of it. It is found that a certain pentadiagonal matrix is spectrally equivalent to the normal form and can be readily computed just using the action of the matrix on certain vectors. Hence the explicit form of the Schur complement is not required.
Reviewer: O.Axelsson
MSC:
| 65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
| 65F10 | Iterative numerical methods for linear systems |
| 35J25 | Boundary value problems for second-order elliptic equations |
Keywords:
Domain decomposition methods; preconditioned iterative methods; stiffness matrix; block LU preconditioner; Schur complement matrix; normal formReferences:
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