A Galerkin method with modified piecewise polynomials for solving a second-order boundary value problem

Summary The classical Ritz-Galerkin method is applied to a linear, second-order, self-adjoint boundary value problem. The coefficient functions of the operator exhibit a piecewise smooth behaviour characteristic of some “physical” situations. A trial function is constructed using a modified quintic smooth Hermite space <img src="/fulltext-image.asp?format=htmlnonpaginated&src=_html\211_2005_Article_BF01399081_TeX2GIFIE1.gif" border"0" /> $$\tilde H_0^{(3)} (\pi )$$ , in order to meet some desired regularity conditions for the approximate solution. A collocation technique is used to reduce the amount of computational work. Known convergence properties for the projection method are recalled which, in this particular case, are illustrated by a series of numerical experiments.