Concepts of the Finite Element Toolbox Albert

ALBERT is an Adaptive multi-Level nite element toolbox using Bisectioning reenement and Error control by Residual Techniques. Its design is based on appropriate data structures holding geometrical, nite element, and algebraic information. Using such data structures, abstract adaptive methods for stationary and time dependent problems, assembly tools for discrete systems, and dimension dependent tasks like mesh modiications can be provided in a library. This allows dimension{independent development and programming of a general class of applications. In ALBERT, hierarchical 2d and 3d meshes are stored in binary trees. Several sets of nite elements can be used on the same mesh, either using predeened ones, or by adding new sets for special applications. Depending on the currently used nite element spaces, all degrees of freedom are automatically managed during mesh modiications. 1. Introduction and design principles The core part of every nite element program is the problem{dependent assembly and solution of the discretized problem. This holds for programs which solve the discrete problem on a xed mesh as well as for adaptive methods which automatically adjust the underlying mesh to the actual problem and solution. In an adaptive iteration, the solution of a discrete system is necessary after each mesh change. A general nite element toolbox must provide exibility in problems and nite element spaces while on the other hand this core part can be performed eeciently. Data structures are needed which allow an easy and eecient implementation of the problem{dependent parts and also allow to use adaptive methods, mesh modiication algorithms , and solvers for linear and nonlinear discrete problems by calling library routines. Starting point for our considerations is the abstract concept of a nite element space deened (similar to the deenition of a single nite element by Ciarlet 4]) as a triple consisting of a collection of mesh elements; a set of local basis functions on a single element, usually a restriction of global basis functions to a single element; a connection of local and global basis functions giving global degrees of freedom of a nite element function.