Preconditioning and two-level multigrid methods of arbitrary degree of approximation

Let h be a mesh parameter corresponding to a finite element mesh for an elliptic problem. We describe preconditioning methods for two-level meshes which, for most prob- lems solved in practice, behave as methods of optimal order in both storage and computa- tional complexity. Namely, per mesh point, these numbers are bounded above by relatively small constants for all h - ho, where ho is small enough to cover all but excessively fine meshes. We note that, in practice, multigrid methods are actually solved on a finite, often even a fixed number of grid levels, in which case also these methods are not asymptotically optimal as h - 0. Numerical tests indicate that the new methods are about as fast as the best implementations of multigrid methods applied on general problems (variable coefficients, general domains and boundary conditions) for all but excessively fine meshes. Furthermore, most of the latter methods have been implemented only for difference schemes of second order of accuracy, whereas our methods are applicable to higher order approximations. We claim that our scheme could be added fairly easily to many existing finite element codes. 1. Introduction. Consider the numerical solution of elliptic boundary value prob- lems discretized by finite element methods. We assume that the boundary is polygonal or consists of planes. We note that in practical problems one often has a fine enough grid already after the definition of the boundary and the minimal number of vertices needed for a first (coarse) triangulation. Anyhow, if not so, in most cases one makes only a few steps of mesh refinement. Hence the power of multigrid methods-their optimal order of computational complexity-is most often not achieved fully, because optimality requires a large number of recursively defined meshes (for details see, e.g., (4) and for further references see (7)). Hence one might as well consider other methods, perhaps simpler and more effective on a fixed mesh, but which are not asymptotically optimal. Here we shall describe a method which uses only a fixed mesh, but for which one nevertheless achieves a low order of computational complexity and of seemingly optimal order except for, from a practical viewpoint, excessively small meshes. To be more precise, the computational cost per mesh point is bounded by clog N for N < No, where N is the number of mesh points, No is large enough to cover most applications and c is small enough that the method is competitive with multigrid methods. As is well known, the latter need recursion and the usual smoothing followed by corrections of the solutions on the different mesh levels. We claim that the new method is more suitable for implementation in existing finite element packages. In fact most packages for the multigrid methods are only for second order difference methods.