الفهرس 
SummaryOnly 14 pages are availabe for public view
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Abstract
Differential equations (DEs) are considered one of the most prevalent equations in many engineering, physical and chemical fields. Approximate solutions to these equations are of great importance due to the limited availability of their exact solutions especially on complex domains. There are several methods used to obtain approximate solutions to DEs including finite element methods (FEMs). Researchers grew more interest in FEMs as a result of the progress in computer programs that greatly facilitated complicated mathematical operations required for applying these methods. In this thesis, we explored some techniques that improve the approximate solutions resulting from FEMs and study the potential contribution of these techniques in estimating the error in the goal-oriented problems. The error estimation is an essential step to discover the regions in the finite element mesh where the errors exceeds the allowed limits. This helps in developing refinement algorithms to obtain solutions that are more accurate for goal-oriented problems with less computational cost. We presented a new approach for evaluating a posteriori error estimate for goal-oriented problems using the polynomial preserving recovery (PPR) technique. The PPR is one of the techniques employed to improve the finite element solutions. Also, we deduced the formula for the goal-oriented error for elliptic problems with variable coefficients. Furthermore, we devised a new mesh refinement algorithm using the proposed error estimator in order to be compatible with the goal-oriented problems. In addition, we generalized and modified the suggested refinement algorithm to be more appropriate for other classes of DEs. The proposed technique is employed to obtain an error estimate for one and two-dimensional goal-oriented problems. We studied the behavior of the effectivity index to illustrate the efficiency of the new approach.