Ritz approximations to two-dimensional boundary value problems

Abstract

Various Ritz solutions to the plane strain elasticity and the steady state heat flow boundary value problems for a polygonal domain Ω are considered. Historically, two basic approaches have been used in partitioning Ω into "finite elements", (i) complete triangulation and (ii) rectangles with boundary triangles. In each case, the Ritz solution is the unique function (or vector of functions) which minimizes an energy functional over a finite dimensional vector spaceS. We consider as choices forS, piecewise linear and cubic functions for complete triangulations; and piecewise bilinear and bicubic functions for the case in which Ω is a union of rectangles and boundary triangles. For the elasticity problem,L 2 convergence of the components of stress and strain is established for each choice of the spaceS. L 2 convergence of the displacement vector is also shown for a wide class of boundary conditions. Convergence of the temperature is proven for the heat flow problem also. Numerical comparisons are made of the Ritz solutions based upon each spaceS of trial functions.