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IsoGeometric analysis: Stable elements for the 2D Stokes equation. (English) Zbl 1429.76044

Summary: We discuss the application of IsoGeometric Analysis to incompressible viscous flow problems. We consider, as a prototype problem, the Stokes system and we propose various choices of compatible spline spaces for the approximations to the velocity and the pressure fields. The proposed choices can be viewed as extensions of the Taylor – Hood, Nédélec and Raviart – Thomas pairs of finite element spaces, respectively. We study the stability and convergence properties of each method and discuss the conservation properties of the discrete velocity field in each case.

MSC:

76D07 Stokes and related (Oseen, etc.) flows
76M10 Finite element methods applied to problems in fluid mechanics
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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[1] Hughes, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering 194 (39-41) pp 4135– (2005) · Zbl 1151.74419 · doi:10.1016/j.cma.2004.10.008
[2] Bazilevs, Isogeometric analysis: approximation, stability and error estimates for h-refined meshes, Mathematical Models and Methods in Applied Sciences 16 pp 1031– (2006) · Zbl 1103.65113 · doi:10.1142/S0218202506001455
[3] Bressan A Elementi isogeometrici per il problema di Stokes 2009
[4] Auricchio, A fully ’locking-free’ isogeometric approach for plane linear elasticity problems: a stream function formulation, Computer Methods in Applied Mechanics and Engineering 197 (1-4) pp 160– (2007) · Zbl 1169.74643 · doi:10.1016/j.cma.2007.07.005
[5] Brezzi, Mixed and Hybrid Finite Element Methods (1991) · Zbl 0788.73002 · doi:10.1007/978-1-4612-3172-1
[6] Gerbeau, Spurious velocities in the steady flow of an incompressible fluid subjected to external forces, International Journal for Numerical Methods in Fluids 25 pp 679– (1997) · Zbl 0893.76041 · doi:10.1002/(SICI)1097-0363(19970930)25:6<679::AID-FLD582>3.0.CO;2-Q
[7] Christon, The consistency of pressure-gradient approximations used in multi-dimensional shock hydrodynamics, International Journal for Numerical Methods in Fluids (2009) · Zbl 1375.76081 · doi:10.1002/fld.2143
[8] Carrero, Hybridized globally divergence free LDG methods. part I: the Stokes problem, Mathematics of Computation 75 (254) pp 533– (2005) · Zbl 1087.76061 · doi:10.1090/S0025-5718-05-01804-1
[9] Cockburn, A locally conservative LDG method for the incompressible Navier-Stokes equations, Mathematics of Computation 74 (251) pp 1067– (2004) · Zbl 1069.76029 · doi:10.1090/S0025-5718-04-01718-1
[10] Taylor, A numerical solution of the Navier-Stokes equations using the finite element technique, International Journal of Computers and Fluids 1 (1) pp 73– (1973) · Zbl 0328.76020 · doi:10.1016/0045-7930(73)90027-3
[11] Nédélec, A new family of mixed finite elements in R3, Numerische Mathematik 50 (1) pp 57– (1986) · Zbl 0625.65107 · doi:10.1007/BF01389668
[12] Raviart, Lecture Notes in Mathematics, in: Mathematical Aspects of the Finite Element Method pp 292– (1981)
[13] de Boor, Applied Mathematical Sciences, in: A Practical Guide to Splines (2001) · Zbl 0987.65015
[14] Nédélec, Mixed finite elements in R3, Numerische Mathematik 35 (3) pp 315– (1980) · Zbl 0419.65069 · doi:10.1007/BF01396415
[15] Cockburn, A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations, Journal of Scientific Computing 31 (1-2) pp 61– (2007) · Zbl 1151.76527 · doi:10.1007/s10915-006-9107-7
[16] Cockburn, Local discontinuous Galerkin methods for the Stokes system, SIAM Journal on Numerical Analysis 40 (1) pp 319– (2003) · Zbl 1032.65127 · doi:10.1137/S0036142900380121
[17] Austin, A robust multilevel approach for minimizing H(div)-dominated functionals in an H1-conforming finite element space, Numerical Linear Algebra with Applications 11 (2-3) pp 115– (2004) · Zbl 1164.65516 · doi:10.1002/nla.373
[18] Kanschat G A continuous finite element de Rham complex and its application to the two-dimensional Stokes problem 2007
[19] Bathe, Inf-sup testing of up-wind methods, International Journal for Numerical Methods in Engineering 48 pp 745– (2000) · Zbl 0963.76044 · doi:10.1002/(SICI)1097-0207(20000620)48:5<745::AID-NME904>3.0.CO;2-E
[20] Chapelle, The inf-sup test, Computers and Structures 47 pp 537– (1993) · Zbl 0780.73074 · doi:10.1016/0045-7949(93)90340-J
[21] Buffa, Isogeometric analysis in electromagnetics: B-splines approximation, Computer Methods in Applied Mechanics and Engineering (2010) · Zbl 1227.78026 · doi:10.1016/j.cma.2009.12.002
[22] Sederberg, ACM SIGGRAPH 2003 Papers pp 477– (2003) · doi:10.1145/1201775.882295
[23] Sederberg, T-spline simplification and local refinement, ACM Transactions on Graphics (TOG) 23 (3) pp 276– (2004) · Zbl 05457574 · doi:10.1145/1015706.1015715
[24] Bazilevs, Isogeometric analysis using T-splines, Computer Methods in Applied Mechanics and Engineering 199 pp 229– (2010) · Zbl 1227.74123 · doi:10.1016/j.cma.2009.02.036
[25] Gürcan, Streamline topologies in stokes flow within lid-driven cavities, Theoretical and Computational Fluid Dynamics 17 (1) pp 19– (2003) · Zbl 1068.76510 · doi:10.1007/s00162-003-0095-z
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