ABSTRACT
In search of a higher order scheme, the obvious combination Q n ~ Q n -i offers balanced approximation orders but is again unstable. The stability may be recovered by reducing the pressure space and using the combination Q n ~ Q n - 2 • This combination is stable, but the inf-sup constant degrades as fijsr = 0 ( N ~ 1^ 2) as shown in [14]. In addition, the element gives unbalanced approximation orders for a fixed order TV. However, for a spectral element or p-version finite element method [5, 6], whereby the mesh is fixed and convergence is achieved by increasing the polynomial order TV, the approximation orders in TV are balanced. This element is often used in a spectral method for incompressible flow, despite the fact that the inf-sup constant degrades. Another alternative is to reduce the pressure space to V n - i rather than Q n - 2 • The element Q n - V n - i represents the best possible state of affairs, offering balanced approximation orders in both the mesh-size h and the degree TV of the elements, along with a uniformly bounded inf-sup constant as shown in [7]. These properties make the element attractive for hp-finite element approximation.
