Comparisons of piecewise Ritz and weighted-residual methods



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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.