Collection of abstracts 14th GAMM-Seminar Kiel on Concepts of Numerical Software January 23rd to 25th, 1998.

Saturday, January 24th, 1998

ALBERT: An adaptive hierarchical finite element toolbox

ALBERT is an Adaptive multi-Level finite element toolbox using Bisectioning refinement and Error control by Residual Techniques.

The basic idea of this package is to provide a flexible toolbox for adaptive finite element applications in two and three space dimensions with a variety of ansatz functions, even of higher order. It is based on hierarchical data structures.

The goal of adaptive finite element methods is the generation of a mesh and a corresponding discrete solution, such that the error between exact and approximate solutions is below a given tolerance, while the number of degrees of freedom (DOFs) on the mesh is for efficiency reasons as small as possible.
Usually, finite element methods are formulated independent of the space dimension and the actually used ansatz functions. In principle, most parts of a finite element program can be written independently of the space dimension, too. Using general data structures for local basis functions and grid elements, it is possible to remove most dependencies on dimension and ansatz functions.
Multilevel preconditioners and solvers are important tools for the efficient solution of linear or nonlinear (sub-) problems. Finite element methods need hierarchical information about ansatz spaces and meshes to use such routines.

One prerequisite for an adaptive algorithm is a tool for local grid modifications, i.e. refinement and coarsening of mesh elements. This includes the automatical administration of DOFs and data associated to them during mesh changes: Creation of new DOFs during refinement, deletion of obsolete DOFs during coarsening, and interpolation and restriction of data.
In ALBERT, the underlying mesh is a conforming triangulation consisting of simplices (triangles or tetrahedra), with possible DOFs associated to vertices, edges, faces, and element centers. Multiple sets of DOFs can be associated to a mesh. Each set corresponds to a finite element space given by a set of local basis functions on the mesh.
The mesh is stored in hierarchical data structures; this enables a sparse storage of information about the complete hierarchical mesh. Full information is only stored for elements on the coarsest grid, while only few data is stored for each refined element. During hierarchy traversal, all necessary information is generated for each element.
Refinement and coarsening of elements is done by bisection methods which ensure conforming and nested grids in all dimensions. We implement the recursive bisection algorithms of Mitchell (2d) and Kossaczky (3d).
In this toolbox, several adaptive methods for stationary and timedependent problems are available.

ALBERT is implemented in ANSI-C. Most of its standard modules can be replaced by specialized versions, if needed. On the other hand, the toolbox is easily extensible for special applications, for example by adding other basis functions, (non-) linear solvers, etc. Visualization of numerical results can be done via post-processing or interactively using X11, OpenGL, and the procedural Mesh2d/Mesh3d interface of GRAPE.