Summary. We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric GauB-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigrid V-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.

On page 398 two different norms are denoted in the same way making Theorem 4.1 to a tautology. Formula (4.9) should read as follows: Illulll2=UouH2 +lul 2, u~nH(Q).(4.9)With this notation the correct version of Theorem 4.1 is:

Multigrid methods axe the fastest known methods for the solution of the large systems of equations arising from the discretization of partial differential equations. For self-adjoint and coercive linear elliptic boundary value problems (with Laplace's equation and the equations of linear elasticity as two typical examples), the convergence theory reached a mature, if not its final state. The present article reviews old and new developments for this type of equation and describes the recent advances.

We consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points. We are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations. We present an iterative method involving two levels of iteration, similar in some respects to the Uzawa algorithm. We relate the rates of convergence of the outer and inner iterations, proving that, under natural hypotheses, the outer iteration achieves the rate of convergence of the inner iteration. The technique is applied to finite element approximations of the Stokes equations.

The paper presents the mathematical concepts underlying the new adaptive finite element code KASKADE, which, in its present form, applies to linear scalar second-order 2-D elliptic problems on general domains. Starting point for the new development is the recent work on hierarchical finite element bases due to Yserentant (1986). It is shown that this approach permits a flexible balance between iterative solver, local error estimator, and local mesh refinement device -which are the main components of an adaptive PDE code. Without use of standard multigrid techniques, the same kind of computational complexity is achieved -independent of any uniformity restrictions on the applied meshes. In addition, the method is extremely simple and all computations are purely local -making the method particularly attractive in view of parallel computing. The algorithmic approach is illustrated by a well-known critical test problem.