Iterative Solution Methods

The Galerkin discretization of variational problems as described in Chapter 8 leads to large linear systems of algebraic equations. In the case of an elliptic and self adjoint partial differential operator the system matrix is symmetric and positive definite. Therefore we may use the method of conjugate gradients to solve the resulting system iteratively. Instead, the Galerkin discretization of a saddle point problem, e.g. when considering a mixed finite element scheme or the symmetric formulation of boundary integral equations, leads to a linear system where the system matrix is positive definite but block skew symmetric. By applying an appropriate transformation this system can be solved again by using a conjugate gradient method. Since we are interested in iterative solution algorithms where the convergence behavior is independent of the problem size, i.e. which is robust with respect to the mesh size, we need to use appropriate preconditioning strategies. For this we describe and analyze first a quite general approach which is based on the use of operators of the opposite order, and give later two examples for both finite and boundary element methods. For a more detailed theory of general iterative methods we refer to [4, 11, 70, 143].