| This research work investigates the computational characteristics of linear operators formed using the Galerkin and Least Squares finite element formulations of differential systems modeling boundary value problems. The linear systems generated by these formulations are studied, in particular, for their dependence on smoothness of the approximating bases.; First the differential operators are classified into three broad categories of: (1) linear self adjoint, (2) linear non-self adjoint, and (3) nonlinear. Then the Galerkin and least squares formulations in context with approximation spaces formed using higher orders of smoothness (differentiability) are examined for typical operators in each of the three categories. Consequences of the retained order of smoothness on rates of convergence, capturing of localized high gradients through a numerical process, and conformity to the differential form are illustrated and discussed.; Furthermore, importance of the convex character of least squares functional and conditions under which this convexity can be preserved and hence utilized to ensure convergence to the global minima of the functional during the computational process are presented.; Also, it is discovered that for convection dominated systems, the oscillations that are observed in the approximate solutions are simple consequence of inadequate smoothness in the approximation space. And that with proper meshing and sufficient smoothness the highly localized gradients of such systems can be captured effectively. |
