This paper is concerned with the general question of iterative methods for solving the systems of linear equations that arise from finite element approximations of elliptic problems. The author is concerned in particular with the mortar finite element method: a feature of the approach is that domain decomposition leads to a mixed or saddlepoint problem, the Lagrange multipliers arising from imposition of the constraints at subdomain interfaces.
The developments are carried out in the framework of a standard elliptic partial differential equation. The problem is posed variationally as a mixed problem after subdivision of the domain and imposition of the interface conditions. Grids corresponding to different subdomains are assumed to be nonmatching in general. Some properties of the submatrices appearing in the system of linear equations are explored, for example, the spectral equivalence of certain matrices to \(h^k{\mathbf I}\) for appropriate \(k\), where \(h\) is the mesh parameter.
Block-diagonal preconditioners are constructed in the context of the generalized Lanczos method, which includes as its two main features the method of minimal iterations and the conjugate gradient method. The preconditioners are constructed for the various submatrices in such a way that they have optimal orders of arithmetical complexity, and eventually a spectrally equivalent preconditioner can be defined for the matrix corresponding to the system as a whole.