A posteriori error estimates applied to flow in a channel with corners

Mathematics and Computers in Simulation 61 (2003) 375 383 A posteriori error estimates applied to flow in a channel with corners Pavel Burda a,, Jaroslav Novotný b, Bedřich Sousedík a a Department of Mathematics, Faculty of Mechanical Engineering, Czech University of Technology, Karlovo nám. 13, CZ-121 35 Praha 2, Czech Republic b Institute of Thermomechanics AS CR, Dolejškova 5, CZ-18200 Prague 8, Czech Republic Abstract The paper consists of three parts. In the first part, we investigate a posteriori error estimates for the Stokes and Navier Stokes equations on two-dimensional polygonal domains. Special attention is paid to the sources of the constants in the estimates, as these play a crucial role in practical applications to adaptive refinements, as we also show. In the second part, we deal with the problem of determining accurately the constants that appear in the estimates. We present a technique for calculating the constant with high accuracy. In the third part, we apply the a posteriori error estimates with the constants found numerically to the technique of adaptive mesh refinement we solve an incompressible flow problem in a domain with corners that cause singularities in the solution. 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. MSC: 65M50; 76D05 Keywords: A posteriori error estimate; Stokes problem; Taylor Hood element; Singularities; Adaptive mesh refinement 1. Introduction At present various a posteriori error estimates for the Stokes problem are available. Let us mention, e.g. Ainsworth and Oden [1], Carstensen and Verfürth [7], Johnson et al. [10], Verfürth [13], and other references in [5]. The aim of the first part of the paper is to focus on the aspect of the constant that appears in the estimate because it plays a significant role in the adaptive mesh refinement, cf. also [12]. That is why we derive our own a posteriori estimate and trace carefully the role of different constants and their sources. In [5], we derived an a posteriori error estimate for the Stokes problem in a two-dimensional polygonal domain. In [6], we used similar technique to derive a posteriori error estimates also for three-dimensional domains. Corresponding author. E-mail addresses: burda@fsik.cvut.cz (P. Burda), novotny@bivoj.it.cas.cz (J. Novotný), sousedik@student.fsid.cvut.cz (B. Sousedík). 0378-4754/02/$ see front matter 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S0378-4754(02)00090-3

376 P. Burda et al. / Mathematics and Computers in Simulation 61 (2003) 375 383 The outline of the paper is as follows. In Section 2, we recall the Stokes problem and its finite element approximation with Taylor Hood elements. The theorem on a posteriori error estimate for the Stokes problem is in Section 3. In Sections 4 6, we describe implementation and testing of a posteriori error estimates of the discretization error for Navier Stokes equations. These a posteriori estimates are computed using approximate numerical solution on an initial finite element mesh. On each element they give us information about the discretization error reached on it. This enables us to consider the quality of the mesh, and also to refine the elements of the mesh, where the discretization error is too high and then compute new solution on that new mesh. This way we can continue, until the prescribed accuracy is reached. Numerical results are demonstrated in Section 7 on a computation of fluid flow in domain with corner singularity. 2. The Stokes problem and its finite element approximation Let us consider the Stokes problem on a bounded Lipschitzian domain Ω R d, with either d = 2or d = 3: given f L 2 (Ω), find {v,p} H 1 (Ω) d L 2 0 (Ω) such that, in the weak sense, ν v + p = f in Ω, div v = 0 in Ω, v = 0 on Ω, (1) where v is the velocity vector, p the pressure, and ν>0 is the viscosity. L 2 0 (Ω) is the space of L2 functions having mean value zero. Let us denote (, ) 0 the scalar product in L 2, and let V = H 1 0 (Ω)d L 2 0 (Ω). Problem (1) is known to have a unique solution, cf. [9], and consists in: find {v,p} V such that ν( v, v ) 0 (p, div v ) 0 + (p, div v) 0 = (f, v ) 0, {v,p } V. (2) For the finite element approximation, we suppose Ω to be a polygon in R 2 or a polyhedron in R 3, for simplicity. Let {T h } h 0 be a regular (cf. [9]) family of triangulations of Ω. InR 3 we decompose Ω into tetrahedra. Let P m be polynomials of degrees m>0. We let X h,m h be the finite element spaces of Taylor Hood elements (cf. [3]), i.e. X h ={v H0 1(Ω)d C(Ω),v/K P 2 (K) d,k T h }, M h ={p L 2 0 (Ω), p/k P 1 (K), K T h }, (3) where d = 2ord = 3 as in (1), cf. Fig. 1 for the case d = 2. Velocity values are given in corner nodes and midside nodes of the quadrilateral or the triangle, and pressure values only in corner nodes, in order to satisfy Babuška Brezzi condition [3]. Then velocities and pressure are approximated as continuous functions of spatial variables. The finite element approximation of the Stokes problem consists in finding {v h,p h } X h M h such that, {v h,ph } Xh M h, ν( v h, v h ) 0 (p h, div v h ) 0 + (p h, div vh ) 0 = (f, v h ) 0. (4)

P. Burda et al. / Mathematics and Computers in Simulation 61 (2003) 375 383 377 Fig. 1. Referential elements. 3. A posteriori error estimate for the Stokes problem Applying the ideas of Eriksson et al. [8], and those of Babuška and Rheinboldt [2] we proved in [5] the estimate for the Taylor Hood elements on the Stokes problem. We define the residual components on the elements K T h, by the relations R 1 (v h,p h ) = f + ν v h p h, R 2 (v h,p h ) = div v h. (5) The error components are defined on Ω by e v = v v h, e p = p p h, where {v,p} is the exact solution defined in Eq. (2), {v h,p h } is the approximate solution, by Eq. (4). The V norm of {e v,e p } is {e v,e p } 2 V = (e v, e v ) 1 + (e p,e p ) 0. Using the Poincaré Friedrichs inequality, Galerkin orthogonality, the Schwarz inequality, the interpolation properties of X h, M h, and the estimate of the solution of the dual problem, we get the following theorem (proof in [5,6]). Theorem 1. Let Ω be a polygon in R 2 or a polyhedron in R 3, with Lipschitz continuous boundary. Let T h be a regular family of triangulations of Ω. Let {v h,p h } be the Hood Taylor approximation of the solution {v,p} of the Stokes problem. Then the error {e v,e p } satisfies the following a posteriori estimate: e v 1 + e p 0 ( 2C P C I C R h K R 1 (v h,p h ) 0,K + R 2 (v h,p h ) 0,K + h 1/2 1 K 2 K T h where C P,C I,C R are positive constants. l K [ν vh n ] l 0,l ), (6)

378 P. Burda et al. / Mathematics and Computers in Simulation 61 (2003) 375 383 3.1. Remarks The constants C P,C I, and C R in Theorem 1 come, in turn, from the Poincaré inequality, the interpolation properties of X h,m h, and the regularity of the dual problem, respectively. Our result in Theorem 1 is in agreement with that of Verfürth [13], though the technique of the proof is different, and we do not require any regularity. Let us note that an algorithm that generates local refinements of tetrahedral meshes has been proposed by Křížek and Strouboulis [11]. 4. Steady Navier Stokes equations in 2D Let Ω R 2 be a bounded domain with boundary Ω, f = (f 1,f 2 ) the volumetric loads, and ν the kinematic viscosity coefficient. The steady Navier Stokes problem for the incompressible fluid consists in finding the velocity v = (v 1,v 2 ), and pressure p defined in Ω and satisfying (v )v ν v + p = f, div v = 0 (7) in Ω together with boundary conditions on disjoint parts of the boundary Γ in, Γ wall and Γ out (meaning, in turn, the inlet, the wall, and the outlet parts), v = g on Γ in Γ wall, ν v n pn = 0 on Γ out ( do nothing boundary condition). (8) For the finite element discretization we use again Taylor Hood elements P 2/P 1. 5. A posteriori estimates for the Navier Stokes equations in 2D Suppose, that exact solution of the problem is denoted by (v 1,v 2,p) (v x,v y,p)and the approximate finite element solution by (v h 1,vh 2,p h). The exact solution differs from the approximate solution in the error (e v1,e v2,e p ) = (v 1 v h 2,v 2 v h 2,p p h). For the solution (v 1,v 2,p), we denote (v 1,v 2,p) 2 V = (v 1,v 2 ) 2 1,Ω + p 2 0,Ω ( ( ) 2 = v1 2 + v2 2 + v1 + x Ω ( ) 2 v1 + y ( ) 2 v2 + x The estimate in Theorem 1 can be generalized to the Navier Stokes equations: ( ) ) 2 v2 dω + p 2 dω. y Ω (e v1,e v2 ) 2 1,Ω + e p 2 0,Ω E2 (v h 1,vh 2,ph ), (9) where (cf. [13]) [ E 2 (v1 h,vh 2,ph,Ω)= C h 2 (r1 2 + r2 2 ) dω + ] r3 2 dω, (10) l K Ω l l K Ω l

P. Burda et al. / Mathematics and Computers in Simulation 61 (2003) 375 383 379 where h denotes the diameter of the finite element and r i stand for the residuals ( r 1 = f x v1 h v1 h x + v v h ) ( 1 2 v1 h 2 + ν y x + 2 v h ) 1 ph 2 y 2 x, ( r 2 = f y v1 h v2 h x + v h ) ( vh 2 2 v2 h 2 + ν y x + 2 v h ) 2 ph 2 y 2 y, r 3 = vh 1 x + vh 2 y. Let us note that according to our practical experience we use only the element residuals in (10). In this respect cf. also [7]. Qualitatively, the value of the constant C is not simple to determine, the sources are seen in Theorem 1.It is important, that this constant does not depend on the mesh size and so can be determined experimentally for general situation, cf. [12]. By computing the estimates (9), we obtain absolute numbers that will depend on given quantities in different problems. We are mainly interested in the error related to the computed solution, i.e. relative error. This is given by the ratio of absolute norm of the solution error, related to unit area of the element Ω l, (1/ Ω l )E 2 (v1 h,vh 2,ph,Ω l ), and the solution norm on the whole domain Ω, related to unit area (1/ Ω l ) (v1 h,vh 2,ph ) 2 V,Ω, i.e. R 2 (v1 h,vh 2,ph,Ω l ) = Ω E2 (v1 h,vh 2,ph,Ω l ) Ω l (v1 h,vh 2,ph ) 2. (11) V,Ω 6. Example and determination of the constant C in the estimates 6.1. Example Steady Stokes equations on one element, Ω = [0; 2] [0; 2]. Let the exact solution and corresponding boundary conditions be given as follows: v 1 = 1 + x 3, v 2 = 3x 2 y, p = 3νx 2 3νy 2, x [0; 2],y [0; 2], Fig. 2. Geometry of the channel.

380 P. Burda et al. / Mathematics and Computers in Simulation 61 (2003) 375 383 and the approximate solution as v h 1 = 3x2 2x + 1, v h 2 = 3x2 y, p h = 0.04xy + bx + cy, where b = 0.1133333,c = 0.0333333. Fig. 3. Initial finite element mesh. Fig. 4. Relative errors on elements of initial mesh. Fig. 5. (a) Finite element mesh after the first refinement; (b) finite element mesh after second refinement; (c) finite element mesh after third refinement.

P. Burda et al. / Mathematics and Computers in Simulation 61 (2003) 375 383 381 For determining the constant C in estimates (9), (10) we compute error estimate of problem, where exact solution is known and also the norm of the error between exact and approximate solution. The required constant is then given by ratio of these numbers. Then, we have: } E 2 (v1 h,vh 2,ph ) = C {h 2 [(ay+b 6ν) 2 + (ax + c + 6νy) 2 ]dx dy + (3x 2 6x + 2) 2 dx dy Ω Ω = C 3.2838824. Fig. 6. (a) Streamlines near the corners after third refinement; (b) velocity v x after third refinement; (c) velocity v y after third refinement; (d) pressure p after third refinement.

382 P. Burda et al. / Mathematics and Computers in Simulation 61 (2003) 375 383 Our computer code gives using three-point Gaussian integration E 2 computed (vh 1,vh 2,ph ) = C 3.2839114. For determining the constant C we now have to compute the exact error norm between approximate and exact solution, which is available in this special case. For the error between approximate and exact solution we can write: v 1 v1 h = x3 3x 2 + 2x, v 2 v2 h = 0, p p h = 0.03x 2 0.03y 2 axy bx cy. Substituting, we get (v 1 v h 1,v 2 v h 2,p ph ) analytical = 3.5129774. And so we get C = 1.069754. 7. Numerical results and application of estimates to the construction of adaptive meshes Consider two-dimensional flow of viscous, incompressible fluid described by Navier Stokes equations in domain with corner singularity, cf. Fig. 2. Due to symmetry, we solve the problem only on half of the channel, cf. Fig. 3. On the inflow we consider parabolic velocity profile, at the outflow do nothing boundary condition. On the upper wall, no-slip condition and on the lower wall, condition of symmetry (i.e. only y component of velocity equals zero). We consider the following parameters: ν = 0.0001 m 2 /s, v in = 1 m/s. The initial mesh is in Figs. 3 and 4. Elements, where the relative error exceeds 3% are refined, and new solution together with new error estimates is computed. In Fig. 5a c, we show three steps of adaptive refinement of the mesh. In Fig. 6a d, we give the results reached on the third level of the mesh refinement. The singularity in v y component of velocity and in pressure is well seen in Fig. 6c and d, respectively. 8. Conclusions In the estimate (6) only the constant C P can be evaluated directly. The others are not known. But in the application to the adaptive mesh refinement we need the value of the constant. We developed a technique for calculating the constant with high accuracy. Of course, this approach needs some improvement, and it will be a subject of future research. Let us note that another way of local mesh refinement near the singularity has been suggested in [4]. This is based on the asymptotic behavior of the solution near the singularity, not on a posteriori error estimates.

Acknowledgements P. Burda et al. / Mathematics and Computers in Simulation 61 (2003) 375 383 383 This research has been supported partly by the State Research Project No. J04/98/210000003 and partly by the State Research Project No. J04/98/210000010. References [1] M. Ainsworth, J.T. Oden, A posteriori error estimators for the Stokes and Oseen problems, SIAM J. Numer. Anal. 34 (1997) 228 245. [2] I. Babuška, W.C. Rheinboldt, A posteriori error estimates for the finite element method, Int. J. Numer. Meth. Eng. 12 (1978) 1597 1615. [3] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, Berlin, 1991. [4] P. Burda, On the F.E.M. for the Navier Stokes equations in domains with corner singularities, in: M. Křížek, P. Neittaanmäki, R. Stenberg (Eds.), Finite Element Methods, Supeconvergence, Post-Processing and A Posteriori Estimates, Marcel Dekker, New York, 1998, pp. 41 52. [5] P. Burda, An a posteriori error estimate for the Stokes problem in a polygonal domain using Hood Taylor elements, in: P. Neittaanmäki, T. Tiihonen, P. Tarvainen (Eds.), Proceedings of the 3rd European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 99, Jyväskylä, Finland, 26 30 July 1999, World Scientific, Singapore, 2000, pp. 448 455. [6] P. Burda, A posteriori error estimates for the Stokes flow in 2D and 3D domains, in: P. Neittaanmäki, M. Křížek (Eds.), Finite Element Methods, 3D Problems, GAKUTO Int. Ser., Math. Sci. Appl. 15 (2001) 34 44. [7] C. Carstensen, R. Verfürth, Edge residuals dominate a posteriori error estimates for low order finite element methods, SIAM J. Numer. Anal. 36 (5) (1999) 1571 1587. [8] K. Eriksson, D. Estep, P. Hansbo, C. Johnson, Introduction to adaptive methods for differential equations, Acta Numerica, CUP, 1995, pp. 105 158. [9] V. Girault, P.G. Raviart, Finite Element Method for Navier Stokes Equations, Springer, Berlin, 1986. [10] C. Johnson, R. Rannacher, M. Boman, Numerics and hydrodynamic stability: towards error control in computational fluid dynamics, SIAM J. Numer. Anal. 32 (1995) 1058 1079. [11] M. Křížek, T. Stroboulis, How to generate local refinements of unstructured tetrahedral meshes satisfying a regularity ball condition, Numer. Meth. Partial Differential Equations 13 (1997) 201 214. [12] B. Sousedík, P. Burda, A. Szölös, J. Novotný, Application of a posteriori error estimates to fluid flow in a domain with singularity, in: K. Kozel, J. Příhoda, M. Feistauer (Eds.), Proceedings of the 4th Seminar on Euler and Navier Stokes Equations, IT AS CR Praha, 2001, pp. 125 128. [13] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley and Teubner, Chichester, 1996.