Equivalent operator preconditioning for linear elliptic problems

Finite element or finite difference approximations to linear partial differential equations of elliptic type lead to algebraic systems, normally of very large size. However, since the matrices are sparse, even extremely large-sized systems can be handled. To save computer memory and elapsed time, such equations are normally solved by iteration, most commonly using a preconditioned conjugate gradient (PCG) method. To make this process efficient, it is crucial to choose the preconditioner in such a way that both the rate of convergence of the iterative method is fast and the computational labour, required for each iteration step, is relatively small. The goal of this paper is to survey a general framework to construct such preconditioners. The main idea is as follows: instead of constructing the preconditioner directly for the given finite element (FE) or finite difference (FD) matrix, it can be more efficient to first approximate the given differential operator by some simpler differential operator, and then to use the FE or FD matrix of this operator as preconditioner, hereby using the same discretization mesh as for the original operator. This idea can be described formally as follows. Let