Finite element decomposition and minimal extension for flow equations

Altmann, R.; Heiland, J.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1489-1509.

In the simulation of flows, the correct treatment of the pressure variable is the key to stable time-integration schemes. This paper contributes a new approach based on the theory of differential-algebraic equations. Motivated by the index reduction technique of minimal extension, a remodelling of the flow equations is proposed. It is shown how this reformulation can be realized for standard finite elements via a decomposition of the discrete spaces and that it ensures stable and accurate approximations. The presented decomposition preserves sparsity and does not call on variable transformations which might change the meaning of the variables. Since the method is eventually an index reduction, high index effects leading to instabilities are eliminated.

Classification : 76M10, 65L80, 65J10
Mots clés : Navier−Stokes equations, time integration schemes, finite element method, index reduction, operator DAE

R. Altmann, Index reduction for operator differential-algebraic equations in elastodynamics. Z. Angew. Math. Mech. (ZAMM) 93 (2013) 648–664. | DOI | Zbl

R. Altmann and C. Carstensen, P${}_1$-nonconforming finite elements on triangulations into triangles and quadrilaterals. SIAM J. Numer. Anal. 50 (2012) 418–438. | DOI | Zbl

M. Arnold, Half-explicit Runge-Kutta methods with explicit stages for differential-algebraic systems of index 2. BIT 38 (1998) 415–438. | DOI | Zbl

M. Arnold, K. Strehmel and R. Weiner, Half-explicit Runge-Kutta methods for semi-explicit differential-algebraic equations of index 1. Numer. Math. 64 (1993) 409–431. | DOI | Zbl

U. M. Ascher, H. Chin, L.R. Petzold and S. Reich, Stabilization of constrained mechanical systems with DAEs and invariant manifolds. Mech. Struct. Mach. 23 (1995) 135–157. | DOI

R. Becker and S. Mao, Quasi-optimality of adaptive nonconforming finite element methods for the Stokes equations. SIAM J. Numer. Anal. 49 (2011) 970–991. | DOI | Zbl

C. Bernardi and G. Raugel, Analysis of some finite elements for the Stokes problem. Math. Comput. 44 (1985) 71–79. | DOI | Zbl

D. Braess, Finite Elements − Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, New York, 3rd edition (2007). | Zbl

S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edition. Springer-Verlag, New York (2008). | Zbl

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). | Zbl

S.L. Campbell and C.W. Gear, The index of general nonlinear DAEs. Numer. Math. 72 (1995) 173–196. | DOI | Zbl

P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | Zbl

M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Franc. Automat. Inform. Rech. Operat. 7 (1973) 33–75. | Numdam | Zbl

A. Gaul, Krypy. Public Git Repository, Commit: 110a1fb756fb. Iterative Solvers for Linear Systems. Available at https://github.com/andrenarchy/krypy.

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag. Berlin (1986). | MR | Zbl

P.M. Gresho and R.L. Sani, Incompressible Flow and the Finite Element Method. Isothermal Laminar Flow, vol. 2. Wiley, Chichester (2000). | Zbl

E. Hairer, C. Lubich and M. Roche, The numerical solution of differential-algebraic systems by Runge-Kutta methods, vol. 1409 of Lect. Notes Math. Springer-Verlag, Berlin (1989). | MR | Zbl

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edition. Springer-Verlag, Berlin (1996). | MR

J. Heiland, TayHoodMinExtForFlowEqns. Public Git Repository, Commit: 8eb641f21d. Solution of time-dependent 2D nonviscous flow with nonconforming minimal extension. Available at https://github.com/highlando/TayHoodMinExtForFlowEqns.

J. Heinrich and C. Vionnet, The penalty method for the Navier-Stokes equations. Arch. Comput. Method E. 2 (1995) 51–65. | DOI | MR

M. Hinze, Optimal and instantaneous control of the instationary Navier-Stokes equations. Habilitationsschrift, Technische Universität Berlin, Institut für Mathematik (2000).

R. Kouhia and R. Stenberg, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Engrg. 124 (1995) 195–212. | DOI | MR | Zbl

P. Kunkel and V. Mehrmann, Index reduction for differential-algebraic equations by minimal extension. Z. Angew. Math. Mech. 84 (2004) 579–597. | DOI | MR | Zbl

P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution. European Mathematical Society (EMS), Zürich (2006). | MR | Zbl

S. Le Borne and D. Cook II, Construction of a discrete divergence-free basis through orthogonal factorization in $\mathscr{H}$-arithmetic. Computing 81 (2007) 215–238. | MR | Zbl

P. Lin, A sequential regularization method for time-dependent incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 34 (1997) 1051–1071. | DOI | MR | Zbl

V.H. Linh and V. Mehrmann, Efficient integration of matrix-valued non-stiff DAEs by half-explicit methods, Technische Universität Berlin, Germany (2011) Preprint 2011–16.

A. Logg, K. Ølgaard, M. Rognes and G. Wells, Ffc: the fenics form compiler. In Automated Solution of Differential Equations by the Finite Element Method. Springer-Verlag, Berlin (2012) 227–238. | Zbl

G. Matthies and F. Schieweck, A multigrid method for incompressible flow problems using quasi divergence free functions. SIAM J. Sci. Comput. 28 (2006) 141–171. | DOI | MR | Zbl

G.-P. Ostermeyer, On Baumgarte stabilization for differential-algebraic equations. In Real-Time Integration Methods for Mechanical System Simulation. In vol. 69 of NATO ASI Series, edited by E. Haug and R. Deyo. Springer-Verlag, Berlin (1991) 193–207. | Zbl

C. Park and D. Sheen, $P_1$-nonconforming quadrilateral finite element methods for second-order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 624–640. | DOI | MR | Zbl

A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin (1994). | MR | Zbl

R. Rannacher, On the numerical solution of the incompressible Navier-Stokes equations. Z. Angew. Math. Mech. 73 (1993) 203–216. | DOI | MR | Zbl

R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Eqs. 8 (1992) 97–111. | DOI | MR | Zbl

J. Shen, On error estimates of the penalty method for unsteady Navier-Stokes equations. SIAM J. Numer. Anal. 32 (1995) 386–403. | DOI | MR | Zbl

L. Tartar,An Introduction to Navier-Stokes Equation and Oceanography. Springer-Verlag, Berlin (2006). | MR | Zbl

C. Taylor and P. Hood, A numerical solution of the Navier-Stokes equations using the finite element technique. Int. J. Comput. Fluids 1 (1973) 73–100. | DOI | MR | Zbl

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam (1977). | MR | Zbl

S. Turek, Efficient Solvers for Incompressible Flow Problems. An Algorithmic and Computational Approach. Springer-Verlag, Berlin (1999). | MR | Zbl

R. Verfürth, Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO Anal. Numér. 18 (1984) 175–182. | DOI | Numdam | MR | Zbl

R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Stuttgart (1996). | Zbl

J. Weickert, Applications of the Theory of Differential-Algebraic Equations to Partial Differential Equations of Fluid Dynamics. Ph.D. thesis, TU Chemnitz, Fakultät Mathematik, Chemnitz (1997). | Zbl