A two‐dimensional mixed finite‐element pair on rectangles

Abstract

AbstractMixed finite‐element discretizations provide a promising method for extending highly desirable C‐grid properties to high order on quasi‐uniform horizontal grids. For the degree 1 Raviart–Thomas–quadrilateral (RT1–Q1) finite‐element discretization of the one‐dimensional linear gravity‐wave equations, it has recently been shown that the group velocity goes to zero for waves with wavelength close to or equal to twice the element width, but that this problem can be addressed by partially lumping mass matrices. As an essential stepping stone to generalization to the two‐dimensional linear shallow‐water equations, recent analysis is extended herein to the one‐dimensional linear shallow‐water equations. It is found that the previous two‐parameter partial mass lumping no longer, in general, addresses the dispersion problem; in particular, and surprisingly, there is an unanticipated spectral gap for the asymptotic limit of pure inertial oscillations. Fortunately, the ensuing dispersion problems can be addressed by deducing an additional constraint on parameter values. This is most easily obtained by ensuring removal of the spectral gap for pure inertial oscillations and leads to the original (generally) asymmetric two‐parameter partial mass lumping reducing to a symmetric single‐parameter one.The resulting modified RT1–Q1 discretization is then extended to two dimensions by taking a tensor product on rectangles. Using numerical simulations of the linear Rossby adjustment problem, it is confirmed that problematic group velocity modes also exist for the unmodified scheme in two dimensions but that they can again be removed by symmetric single‐parameter partial mass lumping. Crucially, not only does this modification lead to accurate Rossby adjustment but it also preserves the desirable properties of the unmodified RT1–Q1 scheme.