A new hp method for the −grad(div) operator in non-Cartesian geometries

This paper describes a new hp method for the numerical solution of the two-dimensional −grad(div) eigenvalue problem in primal variational formulation. In standard methods with triangular or quadrangular finite elements, or with spectral elements, the spectrum contains spurious modes if the mesh is non-Cartesian. With the new hp method described in this paper we show that, although it delivers a Pp–Pp approximation on the same grid for both velocity components, the spectrum is represented without any spurious mode whether a Cartesian or a non-Cartesian quadrangular grid is chosen. Spectra computed with standard high-order methods and with the new element are presented and compared with the analytic solutions.