Isogeometric Analysis: new stable elements for the Stokes equation

SUMMARYIn this paper we discuss the application of IsoGeometric Analysis to incompressible viscous flow problems, forwhich preliminary results were presented in [1, 2, 3]. Here we consider, as a prototype problem, the Stokes systemand we propose various choices of compatible Spline spaces for the approximations to the velocity and pressurefields. The proposed choices can be viewed as extensions of the Taylor-Hood, N´ed elec and Raviart-Thomas pairs´of finite element spaces, respectively. We study the stability and convergence properties of each method and discussthe conservation properties of the discrete velocity field in each case. Copyright c 2000 John Wiley & Sons, Ltd. KEY WORDS : Isogeometric Analysis; NURBS; stability; incompressibility; Stokes flow 1. INTRODUCTIONThe concept of Isogeometric Analysis (IGA) was introduced in [4] with the aim of bridging the gapbetween computer aided design (CAD) and the finite element method. This aim is pursued by adoptingthe same spline or Non Uniform Rational B-spline (NURBS) basis functions used to design domaingeometries in CAD to construct both trial and test spaces in the discrete variational formulation ofdifferential problems. As an additional benefit with respect to standard finite elements, the use of thesefunctions allows to construct finite dimensional spaces of higher regularity. In this paper we discussthe application of IGA to incompressible viscous flow problems, for which preliminary results werepresented in [1, 2, 3]. Here we consider, as a prototype problem, the Stokes system modeling the flowof a viscous, incompressible fluid with constant viscosity :ˆ r

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