AbstractBlending function interpolants on rectangular elements are used to construct an overall interpolant which exactly matches function and normal derivative on the perimeter of a rectangular region. This overall interpolant is incorporated into the Ritz‐Galerkin version of the finite element method and a test problem involving the biharmonic equation is solved. The numerical results obtained demonstrate the considerable increase in accuracy of the exact boundary methods as compared with the usual method using interpolated boundary conditions. A similar investigation of second order elliptic problems with Dirichlet boundary conditions where the rectangular region is divided up into triangular elements yields the perhaps surprising result that blended interpolants on triangular elements do not necessarily improve the accuracy of the Ritz‐Galekin version of the Finite Element Met