This paper presents an optimum technique based on the least squares methodfor the derivation of the bubble functions to enrich the standard linear finiteelements employed in the formulation of Galerkin weighted-residual statements.The element-level linear shape functions are enhanced with supplementarypolynomial bubble functions with undetermined coefficients. The best leastsquares minimization of the residual functional obtained from the insertion ofthese trial functions into model equations results in an algebraic system ofequations whose solution provides the unknown coefficients in terms ofelement-level nodal values. The normal finite element procedures for theconstruction of stiffness matrices may then be followed with no extra degree offreedom incurred as a result of such enrichment. The performance of theproposed method has been tested on a number of benchmark linear transportequations with the results compared against the exact and standard linearelement solutions. It has been observed that low order bubble enriched elementsproduce more accurate approximations than the standard linear elements with noextra computational cost despite employing relatively crude mesh. However, forthe solution of strongly convection or reaction dominated problemssignificantly higher order enrichments as well as extra mesh refinements willbe required.
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