ABSTRACT
The Petrov–Galerkin methods use the meshes and piecewise polynomial basis functions for the approximations U as in the finite element methods, that is they use the same trial spaces SEh. Zienkiewicz in O. C. Zienkiewicz et al. was among the first to suggest that the analogy in finite element methods to upwinding the finite difference convection operator was provided by upwinding the test function. This chapter begins by deriving for the one-dimensional model problem error bounds which are comparable to those derived in the last section for conforming Petrov-Galerkin methods. Even the free-space Green’s function has the form of a modified Bessel function; and to apply boundary conditions on the boundary of the support of each global basis function is quite impractical.
