Parallel Adaptive Numerical Simulation 1 the Finite Element Method

We present the principles of an algorithm which solves partial diierential equations with an adaptive version of the Finite Element Method (FEM), an approach that is very interesting because of the increasing importance of numerical simulations. The calculations are executed in a distributed way on a net of loosely coupled workstations. The algorithm is based on the principle of recur-sive substructuring with static condensation. The advantage of this method is that it requires less CPU-time and storage than the standard FEM. Moreover, the adaptivity of the procedure guarantees that the result is improved to a pre-deened accuracy. The costs for getting this accuracy are much lower than by a non-adaptive procedure. Because of the parallelization of the computations a high speed-up is achieved, and the memory requirement on each workstation is strongly reduced. The theoretical considerations are connrmed by practical experiences with distributed adaptive simulations of problems arising in the eld of structural engineering, like, e.g., a short cantilever under uniform load. The Finite Element Method is a very important tool for the numerical simulation of objects, whose behavior is described by the equations of calculus, e.g. diier-ential, integral, integrodiierential, or variational equations (s. 1]). The goal of the FEM is to calculate the response of these objects under certain loading and boundary conditions. Its advantages are robustness and exibility. But in common implementations there are also some disadvantages, namely a high amount of storage requirement and a high number of necessary oating-point operations. In order to use the given hardware capabilities eeciently, adaptive methods must be applied. The adaptive FEM is an iterative procedure where in several runs of computations the result is improved automatically until a predeened accuracy is reached. In this section, the algorithm and the terms of the FEM will be explained as far as necessary for the comprehension of section 2.