COMPUTATIONAL COMPARISON BETWEEN THE TAYLOR HOOD AND THE CONFORMING CROUZEIX RAVIART ELEMENT

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1 Proceedings of ALGORITMY 5 pp COMPUTATIONAL COMPARISON BETWEEN E TAYLOR HOOD AND E CONFORMING OUZEIX RAVIART ELEMENT ROLF KRAHL AND EBERHARD BÄNSCH Abstract. Tis paper is concerned wit te computational performance of te P 2 P 1 Taylor Hood element and te conforming P + 2 P 1 Crouzeix Raviart element in te finite element discretization of te incompressible Navier Stokes equations. To tis end various kinds of discretization errors are computed as well as te beavior of two different preconditioners to solve te arising systems are studied. Key words. Taylor Hood element, Crouzeix Raviart element, incompressible fluid flow, preconditioners for Quasi-Stokes AMS subject classifications. 65N, 76M 1. Introduction. In [11] Taylor and Hood proposed te Q (8) 2 Q 1 element on quadrilaterals for solving te Navier Stokes equations numerically. Tis is a variant of te biquadratic-bilinear Q 2 Q 1 element, but wit te central node in te velocity space removed. Since ten, it became common practice to refer to te Q 2 Q 1 element on quadrilaterals as well as to its triangular counterpart, te P 2 P 1 element, bot in 2D and in 3D, as Taylor Hood element. Tis combination of finite element spaces as become one of te most well-known and popular elements for solving te incompressible Navier Stokes equations. In [7] Crouzeix and Raviart analyzed a furter class of finite element spaces on triangular meses for te Stokes equations, of wic at least two became also rater popular in CFD and are referred to as Crouzeix Raviart element nowadays: te non-conforming P 1 P element, were te velocity is continuous at te midpoints of te element faces only, see e.g. [1, 12], and te conforming P + 2 P 1 element, see e.g. [2, 8, ]. Muc effort as been spent to analyze various classes of elements. However, tere are significantly less computational investigations to compare different types of element regarding teir actual discretization errors and computational performance in general. Te aim of tis article is to present some computational results in order to provide some rationale for assessing te beavior and performance of te P 2 P 1 and te conforming P + 2 P 1 elements. Tis paper summarizes results from [13]. Te rest of tis paper is organized as follows: in Section 2 we present te numerical problem we are concerned wit, introduce some notations, and define te preconditioners tested in tis paper. In Section 3 we recall te definition of te Taylor Hood and te conforming Crouzeix Raviart element. Tis is followed by Section 4, were te performance of bot elements is tested in various ways. Preconditioners for te Quasi-Stokes problem are te objective of Section 5. Weierstraß-Institut für Angewandte Analysis und Stocastik, Morenstraße 39, 117 Berlin, Germany, kral@wias-berlin.de Friedric-Alexander-Universität Erlangen-Nürnberg, Lerstul für Angewandte Matematik III, Haberstraße 2, 958 Erlangen, Germany, baensc@mi.uni-erlangen.de 369

2 37 R. KRAHL, E. BÄNSCH 2. Numerical metods. We are concerned wit te numerical solution of te instationary, incompressible Navier Stokes equations in a bounded domain Ω R d, d {2, 3} and some time interval T R + : for a given rigt and side f, Diriclet boundary data u D and initial function u find a pair (u, p) of a velocity and pressure field fulfilling t u + u u 1 u + p = f Re in T Ω, u = in T Ω, u = u D u = u on T Ω, in {} Ω (2.1) wit Re te Reynolds number. In order to determine te pressure uniquely, as usual we impose te condition Ω p =. Te structure of te solver used in tis paper for te computational solution of (2.1) is described in [3]. It uses algoritms proposed in [5]. Te following simplified Quasi-Stokes problem appears as a core subproblem after te time discretization: find (u, p) suc tat µu u + p = f in Ω, u = in Ω, u = u D on Ω (2.2) for certain parameters µ, >. Since te Quasi-Stokes problem is linear, we fix µ = 1 ereafter. Let us remark tat in a more general situation te core problem to be solved in eac time step may be an Oseen equation and tus being nonlinear. In tis case te setting is computationally less convenient, since it proibits te use of CG metods. Oter Krylov space metods migt be used instead Scur complement formulation of te Quasi-Stokes problem. Te weak formulation of (2.2) is written in te usual form: find (u, p) u D + X Y suc tat a(u, ϕ) + b(ϕ, p) = l, ϕ ϕ X, b(u, ψ) = ψ Y (2.3) were X and Y are appropriate function spaces for te velocity and te pressure and a, b, and l are defined as a : X X R, a(u, v) = µ u v + u: v, Ω Ω b : X Y R, b(v, p) = p v, Ω l X, l, v = f v. Using operators A : X X, B : X Y, and B T : Y X defined as Au, v = a(u, v), Bu, q = b(u, q), B T p, v = b(v, p) Ω

3 COMPARISON BETWEEN TAYLOR HOOD AND OUZEIX RAVIART ELEMENT 371 u, v X, p, q Y, problem (2.3) can be written as: find (u, p) X Y suc tat Au + B T p = l, Bu =. (2.4) Note tat in tis paper for ease of presentation we sall use te notation A, B, B T also for matrices representing discretized versions of tese operators. Te meaning will be clear from te context. Now tis problem is equivalent to te Scur complement formulation: find p Y suc tat BA 1 B T p = BA 1 l and u = A 1 (l B T p). (2.5) In tis paper we consider solvers based on tis Scur complement formulation. Preconditioned CG metods are used to solve (2.5). Preconditioning of te Scur complement operator is crucial, since te Scur complement operator degenerates for µ. Te specific coices of preconditioners are addressed in te following section Preconditioners for te Quasi-Stokes problem. In tis section we introduce two different preconditioners for te Scur complement formulation (2.5). Te first one (called Laplace preconditioner ereafter) follows PDE ideas, wile te second one is based on te discretized equations (mass diagonal preconditioner) Laplace preconditioner. In [5] Bristeau, Glowinski, and Periaux proposed a preconditioner for te Quasi-Stokes problem (2.5) based on te solution of a Laplace problem in te pressure space: for a given p Y let q be te solution of q = p in Ω, n q = on Ω. (2.6) Now set S 1 Lp p := µq + p. (2.7) We denote by C Lp = ( Ω ψ ) i ψ j te stiffness matrix in te pressure space. Ten (2.6) can be written as q = C 1 Lp p. Ol sanskii proved in [14] tat in fact tis preconditioner is equal to te inverse of te Scur complement operator of te Quasi-Stokes problem for a particular set of model boundary conditions. Note tat S Lp becomes identity if µ = and = 1. Te Laplace preconditioner as proven to perform very well wit te Taylor Hood element in practice wit more realistic boundary conditions as for instance of Diriclet type, wic is also confirmed by our tests. However, for discontinuous pressure functions, it is not straigtforward ow to formulate and implement tis preconditioner, since te Laplace operator requires H 1 -regularity of te underlying space Mass diagonal preconditioner. As an alternative, we tested a preconditioner based on ideas presented in [15]. To tis end we observe tat te Quasi-Stokes problem does not need any preconditioning if µ is small. Terefore it is sensible to first consider te case µ 1. In tis case we ave 1 µ A = M + µ D M (2.8)

4 372 R. KRAHL, E. BÄNSCH wit M = ( Ω ϕ i ϕ j ) te mass matrix and D = ( Ω ϕ i : ϕ j) te stiffness matrix in te velocity space. Since M 1 is dense, we replace te mass matrix by its spectrally equivalent diagonal part M = (δij M ij ). Te discrete Scur complement operator BA 1 B T is ten approximated by B M 1 B T. To cover te full parameter range of µ and, we set S 1 MD p := µ(b M 1 B T ) 1 p + p. (2.9) Note tat te matrix C MD = B M 1 B T is computable, owever te stencil is larger tan for te usual Laplace s operator. Te incidence matrix involves also neigbors of neigboring nodes of a vertex, see [4, 13] for details. Tis mass diagonal preconditioner sows reasonable performance in practice, altoug not as good as te Laplace preconditioner for te Taylor Hood element, see Section 5. Its main advantage is tat it is based on te matrices of te saddle point problem only and does not introduce additional requirements on te regularity of te finite element spaces. 3. Te Taylor Hood and te Crouzeix Raviart element. In tis section, we recall te definition of te P 2 P 1 Taylor Hood and te P + 2 P 1 conforming Crouzeix Raviart finite element spaces. First we need to fix some notations: assume for simplicity Ω to be polygonally saped. Let T be a triangulation of Ω consisting of simplices. We assume te usual admissibility and sape regularity conditions on T, see e.g. [6, Sec ]. For any k N and any simplex S R d we denote by P k (S) := {p : S R p is a polynomial of degree k}. (3.1) Te P 2 P 1 Taylor Hood element consists of globally continuous, piecewise quadratic functions in te velocity space and of globally continuous, piecewise linear functions in te pressure space: X T H := {u (C (Ω)) d S T : u S (P 2 (S)) d } (H 1 (Ω))d, Y T H := {p C (Ω) S T : p S P 1 (S)} L 2 (Ω). (3.2) According to for instance [9] tis combination of elements is LBB-stable, provided a rater general assumption on te triangulation is fulfilled. In te case of te P + 2 P 1 conforming Crouzeix Raviart element, te pressure space consists of piecewise linear, discontinuous functions. Te additional degrees of freedom must be balanced by enricing te velocity space by e.g. volume bubbles Φ v and face bubbles Φ f for te LBB-condition to old. More precisely, let { d } Φ v (S) := span λ i i= and { d } Φ f (S) := span λ i k {,..., d} i= i k (3.3) were λ i are te barycentric coordinate functions wit respect to S. Define P + 2 (S) := P 2(S) Φ v (S) in 2D, P + 2 (S) := P 2(S) Φ v (S) Φ f (S) in 3D. (3.4)

5 COMPARISON BETWEEN TAYLOR HOOD AND OUZEIX RAVIART ELEMENT 373 Level Elements X T H Y T H X Table 4.1 Uniform refinement of te unit square in 2D. Number of degrees of freedom () for te Taylor Hood and te conforming Crouzeix Raviart element in te velocity and te pressure space. Y Note tat in 2D we ave Φ f (S) P 2 (S). Now define te P + 2 P 1 element by X := {u (C (Ω)) d S T : u S (P + 2 (S))d } (H 1 (Ω)) d, Y := {p L S T : p S P 1 (S)} L 2 (Ω). (3.5) Again, te proof of LBB-stability for tis element can be found in in [9] (actually, tere te proof is given for a variant wit a sligtly smaller velocity space; te LBB-stability for te P + 2 P 1 element as defined ere, is a trivial corollary of tis). One main advantage of te discontinuous pressure functions of conforming Crouzeix Raviart element is tat te elementwise mean values of te divergence of discrete solutions u are zero and tus te discrete solutions fulfill a local mass balance. Tis is stated in te following teorem, wic is readily proved. Teorem 3.1. Let (u, p ) X Y be te solution of te discrete Quasi- Stokes equations. Ten we ave for all S T : u =. (3.6) S 4. Comparison of discretization errors. In tis section we compare te Taylor Hood and te conforming Crouzeix Raviart element wit respect to discretization errors. We start by studying te Quasi-Stokes problem (2.2), for wic an analytic solution is available and tus te error can be computed exactly, see Section 4.1. Next we consider te full Navier Stokes equations. Tere we first consider te dynamic beavior of te discrete solution for an instationary convection, see Section 4.2. Finally, we compare te two elements regarding te error in u, see Section 4.3. We ave run numerical experiments in 2D and in 3D. Since te results in 3D exibit te same picture as te 2D examples, tey are omitted ere for sake of brevity. Let us start by commenting on te triangulation. To keep tings as concise as possible, we consider a uniform subdivision of te square in 2D and te cube in 3D. As for any given triangulation te function spaces X and Y are supersets of X T H and Y T H respectively, it is clear tat te number of degrees of freedom for te conforming Crouzeix Raviart element is iger tan for te Taylor Hood element. Tis increase is muc more pronounced in te pressure space due to te discontinuities tan in te velocity space. Te number of degrees of freedom for a uniform refinement of a square in 2D are sown in Table 4.1. Anoter indicator for te computational effort is te number of non-zero entries of te matrices involved. Table 4.2 sows te maximum number of non-zeros per

6 374 R. KRAHL, E. BÄNSCH Matrix 2D 3D A B B T C MD C Lp 9 27 Table 4.2 Uniform refinement of te unit square in 2D and of te unit cube in 3D. Maximum number of non-zero entries per line in te matrices A, B, B T, C MD, and C Lp for te Taylor Hood and te conforming Crouzeix Raviart element. line for te matrices A, B, and B T from te Quasi-Stokes problem (2.4) and for te matrices C Lp and C MD needed in te preconditioners as defined in Section Quasi-Stokes problem wit known solution. Te functions ( ( cos π ) u(x, y) := 2 (x + y)) cos ( ( π ) π, p(x, y) := sin 2 (x + y)) 2 (x y) (4.1) are solution of te Quasi-Stokes system (2.2) in 2D wit a suitable rigt and side. Wen (2.2) is related to one time step of te time discretized Navier Stokes equations (2.1), te parameter is proportional to t/re, t te time step size, since µ was fixed to µ = 1. Tus in practice, would be rater small. Terefore, in tis section we study te discretization error u u and p p wit respect to variations of ranging from = 6 to =. Tis error is sown wit respect to te number of degrees of freedom () in Figure 4.1. Clearly, for moderate values of te experimental order of convergence (EOC) approaces te expected values of 2 for te velocity and pressure in te H 1 and L 2 -norms, respectively, and 3 for te velocity in te L 2 -norm. Quantitatively, for given te error in te pressure beaves very similarly for te Crouzeix Raviart and te Taylor Hood element. Concerning te velocity, te error is better by some factor for te Crouzeix Raviart element compared to te Taylor Hood element. Only for te smallest value of = 6 te error curves for te velocity are not yet saturated. However, te pressure beaves well also in tis case Instationary convection. In order to test bot elements wit respect to te dynamic beavior of te instationary Navier Stokes equations, we cose te example of an oscillating Bénard convection, compare also [3]. To tis end te Navier Stokes system (2.1) is augmented by a eat equation. Te examples as been solved in te unit square on very coarse grids in order to test te minimum level of grid refinement required to reproduce te oscillation. On a grid of refinement level 3 for te Taylor Hood element and of level 2 for te Crouzeix Raviart element, te expected oscillating pattern was visible, but eavily disturbed by oter interrupting flow patterns. On even coarser grids te solver did not converge at all. On finer grids te oscillation was qualitatively well reproduced. In a frequency analysis te results for te Crouzeix Raviart element on a given level of grid refinement turn out to be similar to tose from te Taylor Hood element on a grid tat is one level finer Comparison of u. As already mentioned, one virtue of te Crouzeix Raviart element is te local mass balance. Going a step furter, in tis section we

7 COMPARISON BETWEEN TAYLOR HOOD AND OUZEIX RAVIART ELEMENT 375 Quasi Stokes, = 6 Quasi Stokes, = 6 Quasi Stokes, = 6 p p u u EOC 3. u u H 1 (Ω) Quasi Stokes, = Quasi Stokes, = EOC 3. 1 Quasi Stokes, = p p u u u u H 1 (Ω) p p 5 Quasi Stokes, = u u 6 Quasi Stokes, = EOC 3. u u H 1 (Ω) 1 Quasi Stokes, = Figure 4.1. Quasi-Stokes problem wit µ = 1 and = 6 (top), = (center), and = (bottom) for a known exact solution (u, p) X Y. Discretization error p p L (left), u u L (center), and u u H 1 (Ω) (rigt) vs. number of degrees of freedom. compare te two elements concerning te pointwise solenoidal condition. To tis end, we first study an example wit a smoot solution, a stationary Bénard convection in te unit square. Te results are sown in Figure 4.2. Tere, te error in u is measured in te L 2 as well as in te L 8 -norm and plotted as a function of. Te reason for coosing te L 8 -norm is tat on one and computationally an L p -norm, 1 p <, is muc simpler to compute for iger order elements tan te L -norm and on te oter and for examples like tis one te L 8 -norm is rater close to te L -norm. As can be seen from te figures, te EOC takes on te expected value of 2 for bot norms. Somewat surprisingly, te error vs. ratio is quite close for te Crouzeix Raviart and te Taylor Hood element. Te next example is te backward facing step. Tis example admits a singular solution due to te reentrant step. Terefore one cannot expect te full order of convergence. As sown in Figure 4.3 te EOC is of order.5 for u in te L 2 - norm for bot elements. Moreover, also as expected, tere is divergence of u in te L 8 -norm, somewat stronger for te Taylor Hood element. In tis case of a

8 376 R. KRAHL, E. BÄNSCH stationary Bénard convection stationary Bénard convection u u 8 (Ω) Figure 4.2. Stationary Bénard convection. Beavior of u vs. number of degrees of freedom in te L 2 -norm (left) and in te L 8 -norm (rigt). backward facing step in 2D 1 backward facing step in 2D u EOC.5 u 8 (Ω) Figure 4.3. Backward facing step in 2D. Beavior of u vs. number of degrees of freedom in te L 2 -norm (left) and in te L 8 -norm (rigt). singular solution te performance of te Crouzeix Raviart element wit respect to te solenoidal condition is muc better tan te Taylor Hood element. 5. Comparison of preconditioners for te Quasi-Stokes problem. In order to test te performance of different preconditioners, we compare te number of iterations needed to solve te Quasi-Stokes problem. Te following preconditioners ave been tested: No precond.: Laplace precond.: see Section 2.2. S 1 p := p. S 1 p := µc 1 Lp p + p. Tis preconditioner was used for te Taylor Hood element only. Mass diagonal precond.: see Section 2.2. S 1 p := µ(b M 1 B T ) 1 p + p. Plain : Te matrix of te mass diagonal preconditioner was used witout adaptation to te parameters µ and of te problem. S 1 p := (B M 1 B T ) 1 p Tis preconditioner is only considered in order to demonstrate te difference of using (B M 1 B T ) 1 compared to te linear combination µ(b M 1 B T ) 1 + I. Oterwise tis preconditioner is of no practical use.

9 COMPARISON BETWEEN TAYLOR HOOD AND OUZEIX RAVIART ELEMENT , Level 4 Laplace precond , Level , Level 5 Laplace precond , Level , Level 6 Laplace precond , Level 5 5 Figure 5.1. Quasi-Stokes problem in 2D; µ = 1. Number of iterations vs. using different preconditioners for te Taylor Hood (left) and te conforming Crouzeix Raviart element (rigt). Level of refinement 4 (top) to 6 (bottom). Figure 5.1 sows te number of iterations needed to solve te Quasi-Stokes problem in te Scur complement formulation (2.5) wit a Conjugate Gradient metod for a given tolerance. Te parameter µ was fixed as µ = 1 and was varied in te range 6. Te Taylor Hood and te conforming Crouzeix Raviart element was tested wit different levels of refinement of te triangulation. Again we note tat te same beavior can be observed in 3D. Te corresponding figures are omitted for te sake of brevity. As expected, our tests confirm te teoretical result tat te Scur complement does not need preconditioning for large values of. For = 1 bot, te Laplace and te mass diagonal preconditioner, do not sow any visible effect compared to no preconditioning. On te oter and, te Laplace preconditioner is also robust in terms of number of iterations over te wole range of values of. It performs even better for smaller values of. Te mass diagonal preconditioner is able to keep te number of iterations witin an acceptable range in most situations, but it still got problems wit very small values of. It does not reac te performance of te Laplace

10 378 R. KRAHL, E. BÄNSCH preconditioner. 6. Conclusions. Te goal of tis paper was a comparison of te computational performance of te Taylor Hood and te conforming Crouzeix Raviart element in te finite element discretization of te Navier Stokes equations. As one sould expect from te additional number of degrees of freedom, te computational results for te Crouzeix Raviart element are better in all tests tan tose from te Taylor Hood element on te same grid. But also te costs in terms of computation time are iger. As a quite roug rule of tumb one can say, te results from te Crouzeix Raviart element are by one level of grid refinement better, but tey are also by one level of grid refinement more expensive tan te results from te Taylor Hood element. Tere is one major exception to tis rule: for nonsmoot solutions we observe tat te te local mass balance, measured in terms of te pointwise solenoidal function, is muc better fulfilled for te Crouzeix Raviart tan for te Taylor Hood element. Different preconditioners for te Quasi-Stokes problem ave been compared. Te Laplace preconditioner turns out to perform best in te wole parameter range. Te fact tat tis preconditioner is not (directly) available for te Crouzeix Raviart element is one important reason for te iger computational costs of tis element resulting from our computational approac. We note, owever, tat tis picture may cange, if one would use static condensation to reduce te degrees of freedom for te Crouzeix Raviart element or certain multigrid metods, wic may be computationally ceaper for te Crouzeix Raviart element tan for te Taylor Hood element. REFERENCES [1] T. Apel, S. Nicaise, and J. Scöberl, A non-conforming finite element metod wit anisotropic mes grading for te Stokes problem in domains wit edges, IMA J. Num. Anal., 21 (1), pp [2] F. H. Bertrand, M. R. Gadbois, and P. A. Tanguy, Tetraedral elements for fluid flow, Int. J. Numer. Metods Eng., 33 (1992), pp [3] E. Bänsc, Simulation of instationary, incompressible flows, Acta Mat. Univ. Comenianae, 67 (1998), pp [4] E. Bänsc and B. Hön, Numerical treatment of te Navier Stokes equations wit slip boudary condition, SIAM J. Sci. Comput., 21 (), pp [5] M. O. Bristeau, R. Glowinski, and J. Periaux, Numerical metods for te Navier Stokes equations. applications to te simulation of compressible and incompressible viscous flow, Comput. Pys. Reports, 6 (1987), pp [6] P. G. Ciarlet, Te Finite Element Metod for Elliptic Problems, Nort Holland, Amsterdam, [7] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element metods for solving te stationary Stokes equations, R.A.I.R.O. R3, 7 (1973), pp [8] C. Cuvelier, A. Segal, and A. A. van Steenoven, Finite Element Metods and Navier Stokes Equations, Reidel, Dordrect, [9] V. Girault and P.-A. Raviart, Finite Element Metods for Navier Stokes Equations, Springer, [] P. M. Greso and R. L. Sani, Isotermal Laminar Flow, volume 2 of Incompressible Flow and te Finite Element Metod, 3rd edition, Wiley, Cicester,. [11] P. Hood and C. Taylor, Navier Stokes equations using mixed interpolation, in Finite Element Metods in Flow Problems, J. T. Oden, R. H. Gallager, O. C. Zienkiewicz, and C. Taylor, eds., University of Alabama in Huntsville Press, 1974, pp [12] P. Knobloc and L. Tobiska, Te P1 mod element: a new nonconforming finite element for convection-diffusion problems, SIAM J. Numer. Anal., 41 (3), pp [13] R. Kral, Das Crouzeix Raviart Element bei der numeriscen Lösung der Navier Stokes Gleicung mit FEM, Diplomarbeit, Universität Bremen, Studiengang Matematik, 2.

11 COMPARISON BETWEEN TAYLOR HOOD AND OUZEIX RAVIART ELEMENT 379 [14] M. Ol sanskii, On te Stokes problem wit model boundary conditions, Sbornik: Matematics, 188 (1997), pp [15] S. Turek, On discrete projection metods for te incompressible Navier Stokes equations: an algoritmal approac, Comp. Metods Appl. Mec. Engrg., 143 (1997), pp