On the Use of Lagrange Multipliers in Domain Decomposition for Solving Elliptic Problems

The primal hybrid method for solving second-order elliptic equations is extended from finite element approximations to general bases. Variational techniques are used to show convergence of approximations to the solution of the homogeneous Dirichlet problem for selfadjoint equations. Error estimates are obtained and examples are given. Introduction Lagrange multipliers have been employed to define classes of functions used to approximate solutions of elliptic partial differential equations in a number of ways. Greenstadt has described the cell discretization method, where the domain of a problem is partitioned into cells; approximations are made on each cell, and the approximations are forced to be weakly continuous across the boundaries of each cell by using Lagrange multipliers in a method called moment collocation [5, 6, 13-16]. These results are discussed in §4. Babuska has shown how Lagrange multipliers can be used to make finite element approximations match the boundary data in elliptic problems [1] (see also [4]). Dorr [9] has applied the methods of Babuska to force continuity across an internal interface formed by dividing a domain in R2 into two parts, using a finite element basis. The primal hybrid finite element method of Raviart and Thomas [18] shows how Lagrange multipliers can be used to ensure that nonconforming finite element approximations converge to solutions as the size of the mesh of the finite element grid becomes small. We show here that convergence of the Greenstadt method occurs in quite general situations. The cells do not diminish in size. The only requirement for convergence is that the basis functions on each cell constitute a Schauder basis in an appropriate space and that the weight functions defined on the boundary segments of each cell that are used to enforce moment collocation also be a Schauder basis. The algorithm is naturally suited for parallel computational methods. Received by the editor November 26, 1990 and, in revised form, October 3, 1991. 1991 Mathematics Subject Classification. Primary 65N30; Secondary 65N35, 65N15.