Keywords: quadrilateral, finite elements, approximation, serendipity, mixed finite elements.

We review in a unified approach results concerning the approximation properties of quadrilateral finite elements on general regular (possibly nonaffine) meshes. These results have been presented recently to the mathematical community; the main references are [1] for the scalar case, [2] for the Stokes problem, [3] for several numerical tests, [4] for the vector case, and [5] for possible consequences for the Reissner-Mindlin plate. We believe that these results are of great importance for the design of quadrilateral finite elements and the aim of this contribution is to provide the engineering community with a unified analysis of this topic.

We shall distinguish between the cases of scalar and vector functions. Indeed, in the former case, we consider the standard mapping (based on the pull-back) which preserves the function continuity. For the latter case, we shall handle two different situations. On one side, if we are interested in the continuity of the normal component (like in problems arising from elasticity or acoustics, when the underlying functional space is ), we shell consider the Piola (variant) transform. On the other side, when the continuity of the tangential component is of interest (as it is the case for problems arising from electromagnetism which involve the space ), we shall consider the covariant transform.

In the the scalar case, we are interested in optimal approximation estimates for the function and its gradient in the norm, which read

where refers, as usual, to the mesh parameter and the finite element space contains polynomials of degree . The main result is that estimates (49), (50) hold on general regular quadrilateral meshes if and only if the reference finite element space contains , the space of polynomials of degree at most in each variable, separately.

The extension to the vector case is not a straightforward application to the results in the scalar case. We are interested in optimal approximation estimates of the vectorfields and their divergences in the norm; namely

Also in this case, we shall show necessary and sufficient conditions for estimates (51), (52) to hold, which characterize the finite element space on the reference element. Such conditions are a little more complicated to describe than the previous one and we do not anticipate them in this abstract.

We then present several numerical tests coming from different applications. The first application concerns the Laplace/Poisson problem. We show that commonly used serendipity elements on general quadrilateral meshes present a lack of convergence. This is the case for the 8-node element which is asymptotically second order accurate in as compared to the full 9-node element which presents a third order behavior.

Then we present the application to the popular Stokes element. We recall that two approaches are possible in order to define the pressure space. A local approach and a global approach The classical inf-sup stability condition is satisfied for both approaches, but the local approach cannot provide optimal order of convergence, since the reference pressure space does not contain all of . Numerical tests confirm the bad behavior of the local approach; it is interesting to notice that, as it should be expected when dealing with mixed methods, the suboptimal convergence of the pressures implies a bad convergence of the velocities as well.

We then show that the popular Raviart-Thomas element does not achieve optimal convergence on general quadrilateral meshes. A new mixed finite element family introduced in [4] proves to be optimal order convergent also on highly distorted meshes.

Finally, we consider two eigenvalue problems. Numerical experiments confirm the theory and show that the lowest order Raviart-Thomas element provide stable, but not convergent solutions on distorted meshes. Some new numerical results show that higher order Raviart-Thomas elements provide suboptimally convergent solutions. On the other hand, the new family of finite elements introduced in [4] presents optimal convergence behavior.

1

D.N. Arnold, D. Boffi, R.S. Falk, "Approximation by quadrilateral finite elements", Mathematics of Computation, 71(239), 909-922, 2002. doi:10.1090/S0025-5718-02-01439-4

2

D. Boffi, L. Gastaldi, "On the quadrilateral element for the Stokes problem", International Journal for Numerical Methods in Fluids, 39(11), 1001-1011, 2002. doi:10.1002/fld.358

3

D.N. Arnold, D. Boffi, R.S. Falk, L. Gastaldi, "Finite element approximation on quadrilateral meshes", Communication in Numerical Methods in Engineering, 17(11), 805-812, 2001. doi:10.1002/cnm.450

4

D.N. Arnold, D. Boffi, R.S. Falk, "Quadrilateral H(div) finite elements", to appear in SIAM Journal on Numerical Analysis. doi:10.1137/S0036142903431924

5

D.N. Arnold, D. Boffi, R.S. Falk, "Remarks on quadrilateral Reissner-Mindlin plate elements", WCCM V, Fifth World Congress on Computational Mechanics. Eds. H.A. Mang, F.G. Rammerstorfer, J. Eberhardsteiner.

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