Finite difference and finite element discretization procedures with improved continuity of interpolation functions

In the paper, procedures oriented to improve the continuity of interpolation functions in both finite difference and finite element methods are presented. The mathematical formulation of a new method, the least square finite difference method, is given and the features of the method are discussed. Analogously, in the area of finite element method a new procedure is presented allowing to obtain complete polynomial interpolation functions by minimization of quadratic continuity errors on the lower order derivatives not constrained to exact continuity. In both cases, the first results are very encouraging; in the paper, some of them are outlined and discussed.