Keywords: finite elements, adaptive work, preconditioned conjugate gradient.

We look for efficient implementations of adaptive finite element simulations for problems in mechanics and physics. Here, efficient means both with respect to memory requirements and with respect to time.
The first wish "low memory" requires data structures which allow quick execution of the main tasks

• mesh handling (refinement/coarsening)

• forming (element) matrix / right hand side

• solving the linear systems

• post processing / error estimation [1,4] / error control for mesh refinement

by storing as less information as possible. The second wish "short running times" require such data structures with effort for all the main tasks sketched above, when is the number of nodes (actually in one mesh of the adaptive series).

In our experiments we have defined three data structures:

"nodes":	containing coordinates and solution values of each node
"edges":	containing 3 nodes and refinement information to son-edges
	or geometry/boundary conditions of finest edges, resp.
"elements":	containing its edges and nodes, material information
	and the element matrix and right hand side

For completing a finite element calculation on these data structures, we have to incorporate special projection procedures for handling "hanging nodes", (a necessary tool for most simple refinement ideas) and for all kinds of restrictions from boundary conditions.

We give a special variant of the well-known Preconditioned Conjugate Gradient Method running in a subspace of , which is defined from a special projection . Then we describe the structure of the projector for solving these problems for linear and qudratic elements:

• In case of hanging nodes, calculate a solution that belongs to the conformal subspace (continuous function).
• Satisfy boundary restrictions for some boundary nodes.

We give an example of a Signorini contact simulation [3,2], where all these projections are included. Here, the solve up to nodes requires near the same running time as one linear elastic problem with prescribed displacements.

1

M. Ainsworth, J. Oden, "A Posteriori Error Estimation in Finite Element Analysis", Comp.Meth.Appl.Mech.Eng. 142(1997)1-2, 1-88. doi:10.1016/S0045-7825(96)01107-3

2

J.Schöberl, "Solving the Signorini problem on the basis of domain decomposition techniques", Computiong 60 (1993), 323-344. doi:10.1007/BF02684379

3

A. Signorini, "Sopra akune questioni di elastistatica", Attil della Societa Italiana per il Progresso delle Scienzie (1993).

4

R. Verfürth, "A Review of a posteriori Error Estimation and Adaptive Mesh Refinement Techniques", Wiley-Teubner, Chichester, Stuttgart (1996).

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