Postprocessing Techniques and Adaptivity in the Finite Element Method
The Finite Element (FE) method is an approximate method and as such the accuracy of its solutions must be controlled to allow for a reliable application. This thesis deals with error estimation and improvements of the FE solutions based on superconvergent properties of the FE solution in certain points.
Two new postprocessing procedures are developed for elliptic problems. The procedures are based on local patches of elements and developed in such a way that the methods function at local level. Being local methods, the cost involved in implementing the procedures is small. The derivatives of the FE solutions are discontinuous across element boundaries. The primary purpose of the proposed postprocessing procedures is to smooth the discontinuous derivatives of finite element solutions like stresses and fluxes. The smoothed solution is used both as a new improved solution and in an error estimation.
The presented postprocessing methods do not only smooth the stress discontinuities but also improve the satisfaction of the governing differential equation. This is in contrast to the fact that the FE solutions fulfill equilibrium conditions only in a weak and global sense. Similarly, the FE solutions do not generally satisfy the prescribed natural boundary conditions satisfactorily. On the other hand, the solutions obtained by the proposed postprocessing procedure satisfy the prescribed boundary conditions. Satisfaction of both the equilibrium and boundary conditions is achieved in the sense of the least squares.
The proposed methods yield superconvergent solutions meaning that the accuracy of the postprocessed solutions are at least one order higher than the finite element solutions. This in its turn means that use of these postprocessed solutions in error estimations gurantees that the error estimate is asymptotically exact or that the estimated discretization error converges towards the true error in the limit.
Several numerical test examples are given to validate the accuracy and effectiveness of these new methods. The presented methods are also implemented in an adaptive finite element procedure of the h-version or mesh enrichment. The postprocessing procedures proposed in this thesis can be regarded as a step forward towards an accurate and automated adaptive analysis in which the ultimate goal is to provide finite element solutions of high quality in an economical manner. Starting from some arbitrary mesh, such an automated adaptive procedure should provide an improved finite element model that meets a user specified level of accuracy. To accomplish this goal, every FE program package should be equipped with efficient postprocessing procedures.