Comparisons of piecewise Ritz and weighted-residual methods

To date, finite element techniques using the Ritz Method have found wide use in solving differential equations in the applied sciences, particularly in structural analysis. Finite Element Techniques using weighted-residual methods, however, have not been tried. This thesis develops Finite Element Techniques using the Collocation, Subdomain, Galerkin, and Least-Squares weighted-residual methods, and compares them to the Ritz Method. Numerical results are presented for a one-dimensional problem using all weighted-residual methods and the Ritz Method. From a standpoint of accuracy and ease of formulation and programming, the Subdomain Method compared favorably with the Ritz Method. The solution of a two-dimensional problem was attempted. The weighted-residual techniques led to a singular matrix. This was due, probably, to the poor choice of an approximation family.