Petrov - Galerkin Spectral Elements 1 A Petrov - Galerkin Spectral Element Technique for Heterogeneous

A novel spectral element technique is examined in which the test functions are different from the approximating elements. Examples are given for a simple 1D Helmholtz equation, Burger's Equation with a small viscosity, and for Darcy's Equation with a discontinous hydraulic conductivity. 1. Polynomial Approximation. Spectral element approximations have been shown to be an eeective tool for the approximations of certain classes of partial diierential equationss9, 14]. The approach utilized by Patera combines a spectral multi-domain method with the variational formulation commonly used for nite element approximations 14]. The technique proceeds by integrating against \test" functions whose support is limited to either a single subdomain or two adjacent subdomains. Traditionally the test functions are chosen from the same class as the approximating \trial" functions, and an approximation is constructed such that within each subdomain the approximation is a linear combination of orthogonal polynomials. In this paper a new set of trial functions is examined. The new approach is similar in spirit to a mixed nite element method or a Petrov-Galerkin scheme. If the sequence of orthogonal polynomials is given as fp n (x)g, ?1 x 1, then the new trial functions are given by ~ 2 : Integration against these new trial functions is a simpler process than integration against those used in traditional spectral element approaches and the integration can be done with respect to a Jacobi weight function. This new approach is discussed in detail below and is illustrated for two model problems. In addition, a 2-D example for an approximation of Darcy's equation for porous media ow,