Finite element approximation on quadrilateral meshes. (English) Zbl 0999.76073

Summary: Quadrilateral finite elements are generally constructed by starting from a given finite-dimensional space of polynomials \(\widehat V\) on the unit reference square \(\widehat K\). The elements of \(\widehat V\) are then transformed by using the bilinear isomorphisms \(F_K\), which map \(\widehat K\) to each convex quadrilateral element \(K\). It has been recently proven that a necessary and sufficient condition for approximation of order \(r+1\) in \(L^2\) and \(r\) in \(H^1\), is that \(\widehat V\) contains the space \(Q_r\), of all polynomial functions of degree \(r\) separately in each variable. In this paper several numerical experiments are presented which confirm the theory. The tests are taken from various examples of applications: Laplace operator, Stokes problem, and an eigenvalue problem arising in fluid-structure interaction modelling.


76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
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