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The behavior of planetary (Rossby) waves in finite-element numerical models is investigated by using a quasi-geostrophic approximation to the midlatitude, β-plane, two-dimensional linear shallow-water equations. A dispersion relation analysis is employed here to determine the possible occurrence of spurious modes and to ascertain the dispersive/dissipative nature of the finiteelement Galerkin mixed formulation. Three finite-element pairs are considered by using a variety of mixed interpolation schemes. For each pair the frequency or dispersion relation is obtained and analyzed, and the dispersion properties are compared analytically and graphically with the continuous case. It is shown that certain choices of mixed interpolation schemes may lead to significant phase and group velocity errors and spurious solutions in the calculation of slow Rossby waves, due to the coupling between the momentum and continuity equations. Numerical solutions of two test problems to simulate slow Rossby waves are in good agreement with the analytical results.

SUMMARY A posteriori error estimation methods for mesh adaptation often require an accurate computation of the gradient of a Lagrange finite element solution. The precision of the error estimation is directly related to the accuracy of the recovered gradient. We therefore present in this communication a simple method for the evaluation of the gradient of a linear Lagrange finite element solution, and we show that it has significant advantages over existing methods of the same order. The proposed method requires the solution of a global linear system that can be solved by a preconditioned conjugate gradient method. An interesting feature is that it does not require any special treatment for boundary nodes contrarily to classical local patch recovery methods. Copyright © 2012 John Wiley & Sons, Ltd.

SUMMARYThe goal of this study is to evaluate the effect of mass lumping on the dispersion properties of four finiteelement velocity/surface-elevation pairs that are used to approximate the linear shallow-water equations. For each pair, the dispersion relation, obtained using the mass lumping technique, is computed and analysed for both gravity and Rossby waves. The dispersion relations are compared with those obtained for the consistent schemes (without lumping) and the continuous case. The P 0 − P 1 , RT 0 and P NC 1 − P 1 pairs are shown to preserve good dispersive properties when the mass matrix is lumped. Test problems to simulate fast gravity and slow Rossby waves are in good agreement with the analytical results.