Efficient solvers for saddle-point problems arising from domain decompositions with Lagrange multipliers

摘要

In this paper, we are concerned with the domain decomposition method with Lagrange multipliers for solving three-dimensional elliptic problems with variable coefficients. We shall first introduce a weighted saddle-point problem resulting from this domain decomposition, which can be solved by existing iterative methods. Then we will construct two simple preconditioners, one for the system associated with the displacement variable and the other for the Schur complement system associated with the multiplier variable, that are applicable to various discretization schemes. The new preconditioners possess such local properties that they can be implemented cheaply. We will show that the condition number of the global preconditioned system grows only as the logarithm of the dimension of the local problem associated with an individual substructure and is independent of the large variations of the coefficients across the local interfaces.