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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.

##### MSC:
 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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