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A new class of iterative methods for nonselfadjoint or indefinite problems. (English) Zbl 0756.65050

A new iterative technique for solving nonsymmetric or indefinite (NSPD) systems is proposed and analyzed. The iterations of the basic algorithm consist of two steps: first, the original NSPD operator is solved exactly in a subspace (coarser grid space) and then the equation with the SPD part of the NSPD operator is solved approximately by a suitable inner iterative method. In both steps the corresponding residual serves as the right hand side.

The algorithm is applied to the solution of finite element systems arising from second-order elliptic boundary value problems with first- order derivatives. It is shown that the convergence factor of the new method is a sum of two items: the first item is given by the convergence factor of the inner iterative method, the second item is given by the approximation properties of the coarser grid space.

For properly choosen coarse grid space the rate of convergence of the new method is close to the rate of convergence of the inner iterative method. For uniformly convergent inner iterations, the new method is also uniformly convergent.

The special choices of the inner iterative method are discussed as e.g. multiplicative domain decomposition or multigrid methods and some modifications of the basic algorithm suitable for these choices are described.

MSC:
65F10Iterative methods for linear systems
65N55Multigrid methods; domain decomposition (BVP of PDE)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
35J25Second order elliptic equations, boundary value problems