Adaptive finite element analysis of elliptic partial differential equations

Abstract

89 6. SUMMARY AND CONCLUSIONS The main aim of the study was to find error indicators for elliptic partial differential equations. During the study, four types of different error estimation algorithms have been adapted to the existing finite element analysis modules. All the algorithms are implemented to the existing finite element analysis modules by using FORTRAN language. It is well known that error estimation is highly problem dependent, e.g. a error estimation algorithm that is derived for parabolic systems may not give satisfactory results for elliptic systems. The resulting adaptive schemes have been examined on various elliptic problems with known and unknown analytical solutions. The convergence rates have been measured by using eL and //e//hoo norms for the problems with known analytical solutions. The convergence rates of classical non-adaptive finite element schemes have been measured for comparison, the values are in the order of one as stated in the corresponding figures. However, if adaptive schemes are applied to the problem the convergence rates do increase up to the order of two. That means, the same quality in the error norms can be achieved with a less number of unknowns, if adaptive schemes are applied to the problem.90 Since the data structure is capable only for linear triangular elements h-type adaptive procedures have been used. However, if the capabilities of the data structure could be improved, p-type or even a combination of h- and p-types of adaptive algorithms could be implemented. Because of core storage requirements, the modules have been designed to run with direct access disk files, but this strategy surely increases the processing time. In a further study, the modules could be assembled to a single finite element program with windowing and interactive graphics capabilities.