The Convergence of Multi-Level Methods for Solving Finite-Element Equations in the Presence of Singularities

The known convergence proofs for multi-level methods assume the quasi-uniformity of the family of domain triangulations used. Such triangulations are not suitable for problems with singularities caused by re-entrant corners and abrupt changes in the boundary condi- tions. In this paper it is shown that families of properly refined grids yield the same convergence behavior of multi-level methods for such singular problems as quasi-uniform subdivisions do for H2-regular problems. 1. The Continuous Problem, Its Discretization and the Multi-Level Method. Using multi-level techniques ((1), (3), (4), (6), (7), (8), (10)), it is possible to solve the large systems of linear equations arising in connection with finite-element methods with an amount of work roughly proportional to the number of unknowns. This property makes multi-level methods at least theoretically superior to all other solution methods, including fast solvers based on FFT-like algorithms which may be directly applied to special problems only. The convergence of multi-level methods was proved, for example, by Nicolaides (10), Bank and Dupont (4) and Hackbusch (8). All these proofs assume a certain amount of elliptic regularity of the continuous problem to be solved approximately, and quasi-uniform subdivisions of the domain in finite elements. Assuming H1+a- regularity, such quasi-uniform triangulations and a Jacobi-like smoothing procedure, Bank and Dupont (4) and Hackbusch (8) showed the following result: The rate of convergence of a full iteration step of the multi-level method behaves like O(m 0/2), uniformly in the number of levels, for a growing number m of smoothing steps per level. In the optimal case a = 1, the problem has to be H2-regular. This means, for example, that the region is not allowed to have re-entrant corners. If this condition is violated, the convergence rate of the multi-level procedure decreases, and, in addition, the approximation properties of the finite-element discretization itself change for the worse because of the presence of singularities in the solution not captured by the quasi-uniform grids. The strongly nonuniform, systematically re- fined triangulations suitable for these problems are not included in the theory so far. The aim of the present paper is to fill this gap.