Error estimates for finite-element solutions of elliptic boundary value problems

This thesis presents a comprehensive analysis of the Duality Method in error analysis. This method is applied to problems of solid mechanics. In this analysis a procedure for computing a statically admissible stress-field is described. The statically admissible stress-fields are computed by using the data of the problem (prescribed traction and body-force) and the kinematically admissible solutions. The error-indicator is defined by the difference between the kinematically and statically admissible stress-fields, measured in the strain energy norm, over the elements. The relative value of the error-indicator with respect to the strain energy of the deformed body defines the local error-measure. The error-bound for the approximate solution of the problem obtained by the finite-element method is computed by the summation of the local error-measures over all elements in the mesh. The Duality Method is determined to be a robust method of error analysis.; In addition to the analysis of the statically admissible stress-field, this thesis innovates several approximations to statically admissible stress-fields. The approximate stress-fields are compared with statically admissible stress-fields by their features. These stress-fields are used to define a number of error-measures which are shown to be reliable tools to control the accuracy of the finite-element solutions. Some of these error-measures are implemented in a computer program and tested for some problems*.; To compare the results obtained with the results computed by using methods in the literature, the error-estimates presented by Zienkiewicz and Zhu are implemented in the computer program and the same test problems are solved. Then the errors in the solutions are estimated using this method. In addition, the exact errors in the solutions are computed for problems for which exact solutions exist. The computed results are presented and compared with each other. The accuracy and reliability of the error-measures presented in this work supports the viability of the methods of error analysis.; Also, this thesis presents a summary of the Residual Method in error analysis. Correction indicator, Error-indicators, and error-estimators corresponding to this method are quoted. The overall analysis of this method indicates that the error-measures computed by this method are less reliable than those from the duality method.; In the last part, the presented error-bounds and error-estimates are summarized. Their advantages and limitations from the perspectives of accuracy, complexity, and computational cost are stated. For comparison purposes, the method of error analysis by Zienkiewicz and Zhu, and the Residual Method are included in this argument. ftn*Four methods are implemented in a computer program.