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 conditions. 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 r72-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+aregularity, 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 0(m'a/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 i/2-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 refined triangulations suitable for these problems are not included in the theory so far. The aim of the present paper is to fill this gap. Received August 31, 1982; revised November 16, 1983. 1980 Mathematics Subject Classification. Primary 65N30, 65F10. ©1986 American Mathematical Society 0025-5718/86 $1.00 + $.25 per page 399 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 400 HARRY YSERENTANT We take the boundary value problem -Au + u = f in ß, (1.1) M = 0 on rD, du/dn = 0 on TN, as a prototype of the problems under consideration, ß is an open bounded polygonal subset of R2, the boundary of which is subdivided in two parts, rD and TN, each consisting of finitely many pieces of straight lines. Let //¿(ß) be the space of all functions u e Hl(ti) with u = 0 on TD in the sense of traces. We understand the model problem (1.1) in the usual weak sense: Given a linear continuous functional