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The KASKADE toolbox defines an interface to a set of C subroutines which can be used to implement adaptive multilevel Finite Element Methods solving systems of elliptic equations in two and three space dimensions. The manual contains the description of the data structures and subroutines. The main modules of the toolbox are a runtime environment, triangulation and node handling, assembling, direct and iterative solvers for the linear systems, error estimators, refinement strategies, and graphic utilities. Additionally, we included appendices on the basic command language interface, on file formats, and on the definition of the partial differential equations which can be solved. The software is available on the ZIB ftp--server {\tt elib} in the directory {\tt pub/kaskade}. TR 93--5 supersedes TR 89--4 and TR 89--05.
We consider the discretization of obstacle problems for second order elliptic differential operators in three space dimensions by piecewise linear finite elements. Linearizing the discrete problems by suitable active set strategies, the resulting linear sub--problems are solved iteratively by preconditioned cg--iterations. We propose a variant of the BPX preconditioner and prove an $O(j)$ estimate for the resulting condition number. To allow for local mesh refinement we derive semi--local and local a posteriori error estimates. The theoretical results are illustrated by numerical computations.
{\def\enorm {\mathop{\mbox{\boldmath{$|\!|$}}}\nolimits} Let $u \in H$ be the exact solution of a given self--adjoint elliptic boundary value problem, which is approximated by some $\tilde{u} \in {\cal S}$, $\cal S$ being a suitable finite element space. Efficient and reliable a posteriori estimates of the error $\enorm u - \tilde{u}\enorm $, measuring the (local) quality of $\tilde{u}$, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A well--known class of error estimates can be derived systematically by localizing the discretized defect problem using domain decomposition techniques. In the present paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations.}