Estimation and control of the discretization error in thehp finite element method (Spanish text).

摘要

The more efficient way to control the discretization error of a finite element solution, from the point of view of the number of degrees of freedom needed to reach the desired accuracy, is an adaptive hp-refinement. This type of refinement strategy combines the capacity of isolation of singular points in the h-method, with the greater convergence rate of the error that presents the p-method in domains where the solution is smooth. The result is an exponential convergence of the error for any problem if the mesh is conveniently optimized.; However, few commercial codes offer capacity for adaptive p-refinements and practically none for adaptive hp-refinements. The causes can be the great difficulty to estimate a reliable discretization error in the p-method, especially at local level, and the complexity associated with any hp-refinement procedure, mainly due to the data structure that is needed.; The existent techniques are revised in this Thesis with the purpose of improving the estimate and control of the error in the hp-version of the FEM. An error estimate and an adaptive hp-refinement procedure for linear elastostatics problems in bi-dimensional domains are proposed. The proposed methods have a low computational cost and they do not require several analyses to use extrapolation techniques, neither estimates of the singularity intensity or the convergence rate.; The proposed error estimator is an extension of that of Zienkiewicz-Zhu with a local correction, which depends on the polynomial degree of the elements. The improved stress field is obtained by solving complementary problem in each element.; The proposed hp-refinement procedure uses the a priori convergence law of the error to optimize the discretization by distributing the error uniformly.; The numerical verifications performed with different examples show that the error estimator presents good reliability at global level. It is also sufficiently reliable at local level if it is combined with the proposed refinement procedure. The hp-refinement strategy considerably reduces the number of degrees of freedom required to control the error in comparison with adaptive h- and p-refinements.