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