37-41 Mortimer Street, London W1T 3JH, UK
Numerical Functional Analysis and Optimization
Publication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lnfa20
A Posteriori Least-Squares Finite Element Error
Analysis for the Navier–Stokes Equations
Jang Jou a & Jinn-Liang Liu b a
Department of Applied Statistics and Information Science , Ming Chuan University , Kwei Shan, Taoyuan, Taiwan
b
Department of Applied Mathematics , National Chiao Tung University , Hsinchu, Taiwan Published online: 31 Aug 2006.
To cite this article: Jang Jou & Jinn-Liang Liu (2003) A Posteriori Least-Squares Finite Element Error Analysis for the Navier–Stokes Equations, Numerical Functional Analysis and Optimization, 24:1-2, 67-74, DOI: 10.1081/NFA-120020245
To link to this article: http://dx.doi.org/10.1081/NFA-120020245
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no
representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.
This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any
form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions
Vol. 24, Nos. 1 & 2, pp. 67–74, 2003
A Posteriori Least-Squares Finite Element Error
Analysis for the Navier–Stokes Equations
Jang Jou1 and Jinn-Liang Liu2,*
1
Department of Applied Statistics and Information Science, Ming Chuan University, Kwei Shan, Taoyuan, Taiwan
2
Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan
ABSTRACT
A residual type a posteriori error estimator is presented for the least-squares finite element solution of stationary incompressible Navier–Stokes equations based on the velocity–vorticity–pressure formulation with nonstandard and standard boundary conditions. Using the coerciveness of the corresponding Stokes opera-tor and the special feature of the nonlineariry of the formulation, it is shown that the error estimator is exact for the Stokes problem and is asymptotically exact for the Navier–Stokes problem in an energy-like norm. The resulting adaptive method is highly parallel because it does not require to assemble the global matrix and the error estimation can be completely localized without using any information from neighboring elements.
1. INTRODUCTION
A posteriori error estimation is now a standard component in adaptive methods (Oden et al., 1989; Verfu¨rth, 1996; Zienkiewicz, 1992). The least-squares finite
*Correspondence: Jinn-Liang Liu, Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan; E-mail: jinnliu@math.nctu.edu.tw.
67
DOI: 10.1081/NFA-120020245 0163-0563 (Print); 1532-2467 (Online)
Copyright & 2003 by Marcel Dekker, Inc. www.dekker.com
element method (LSFEM) has been recognized as an attractive method in many applications (Bochev, 1997; Cai et al., 1994, 1997; Jiang, 1998; Jiang and Chang, 1990). It is shown here that, for the Stokes problem in first-order least-squares formulation, the residual type error estimator is locally as well as globally equal to the exact error in the norm induced by the least-squares functional. For Navier–Stokes equations, the error estimator is proved to be asymptotically exact. In other words, the error estimator is perfectly reliable for the LSFE approximation of the Stokes problem and very reliable for the Navier–Stokes problem. Moreover, the computation of the estimator is completely localized without any restriction on the approximation order and without requiring any information from the neighboring elements and therefore very efficient for parallel computations. These advantageous properties can be regarded as an additional appealing feature of LSFEM.
We consider the steady, incompressible Navier–Stokes equations
1
Reu þ ðu r Þ u þ rp ¼ f in , ð1Þ
5 u ¼ 0in , ð2Þ
B ðu, pÞ ¼ 0 on @: ð3Þ
where the symbols , r, and r stand for the Laplacian, gradient, and divergence operators, respectively, is an open bounded connected domain in R2 with the boundary @, Re > 0the Reynolds number, u ¼ ðu1, u2ÞT2 ½H1ðÞ 2 the velocity
field, and f ¼ ð f1, f2ÞT 2 ½L2ðÞ 2 the body force. The pressure p 2 L2ðÞ if the
admissible homogeneous boundary operator B describes the pressure on @, other-wise p 2 L20ðÞ. Here HsðÞ, s 2 R, denotes a usual Sobolev space equipped with the
norm kks and L20ðÞ ¼ fq 2 L2ðÞ j ðq, 1Þ0¼0g, where ðu, vÞ0:¼
R
uv d. We
denote eHHsðÞ ¼ HsðÞ \ L20ðÞ.
For least-squares formulation, one usually reduces the second-order PDE to a first-order system by introducing some suitable new variables. The standard velocity– vorticity–pressure formulation is given in Sec. 2. We are interested in the coercivity of the linear operator obtained from this particular formulation. A priori error analysis of the LSFE approximation based on this formulation has been thoroughly studied by Bochev, Gunzburger, and Jiang, see e.g., Bochev, 1997; Bochev and Gunzburger, 1998; Jiang, 1998. We are concerned here with the a posteriori error analysis which is given in Sec. 3. The analysis is mainly based on the coerciveness of the first-order Stokes operator and the special feature of the nonlinear term in the formulation.
2. VELOCITY–VORTICITY–PRESSURE FORMULATION
In two space dimensions, with the vorticity ! ¼ r u ¼ @u2=@x @u1=@y
and the Bernoulli pressure or the total pressure r ¼ p þ ð1=2Þjuj2, the Navier–Stokes Eqs. (1)–(3) can be reduced to the first-order system (Bochev, 1997; Jiang and Chang, 1990)
NU ¼ F in , ð4Þ
68 Jou and Liu
with the boundary condition RU ¼ 0 on @, ð5Þ where NU ¼ u ! þ 1 Re 5 ! þ r r ! 5 u 5 u 8 > > < > > : ð6Þ
U ¼ ðu, !, rÞT and F ¼ ðf, 0 , 0 ÞT: Note that the cross product u ! is defined by embedding u and ! into three-dimensional vectors ðu1, u2, 0ÞT and ð0 , 0 , !ÞT, i.e.,
u ! ¼ ðu2!, u1!ÞT:The corresponding first-order Stokes operator is
LV ¼ 1 Re 5 þ r 5 v 5 v: 8 > < > : ð7Þ Let V ¼ fV ¼ ðv, , Þ 2 ½H1ðÞ 2H1ðÞ H1ðÞ; RV ¼ 0 on @g; ð8Þ
where H1ðÞ denotes the space H1ðÞ whenever the boundary operator R prescribes the pressure r on @, and eHH1ðÞ otherwise. We shall consider five different types of the boundary condition RU described as in Lemma 1.
The nonlinear term u ! in Eq. (6) is of zero order and thus is not related to any derivatives while the rest of the terms constitute the linear Stokes operator. Therefore, the nonlinear term has no effect on the classification of the Navier– Stokes equations and the boundary conditions for the Stokes equations are valid for the Navier–Stokes equations. And it does not matter how large the Reynolds number is, the whole system is elliptic. For this reason, the permissible boundary conditions for Navier–Stokes equations are those for Stokes equations.
The coercivity of the Stokes operator L on function space V for a large number of boundary operators R was studied by Bochev (1997) and Jiang (1998). In Bochev (1997), according to the elliptic regularity theory of Agmon et al. (1964), Bochev examined the complementing condition of Agmon (1964), which is both necessary and sufficient for such coercivity to hold for the operators L and R: Jiang proved the same coercivity based on the bounded inverse theorem and the Friedrichs inequalities related to grad, div, and curl operators. For the sake of simplicity, we consider only homogeneous boundary conditions. These results can be extended to mixed and inhomogeneous boundary conditions without difficulty. Following Jiang (1998), we summarize these results as follows.
Lemma 1. For the first-order Stokes operator L of Eq. (7), let the boundary operator R be of the following five types:
The Navier–Stokes Equations 69
RV ¼ v n RV ¼ v n RV ¼ v t RV ¼ v t ð9Þ RV ¼ v n v t
where n and t are the outward normal and tangential unit vectors to @ , respectively. Then there exists a positive constant C depending on the Reynolds number such that, for types ( i )–(iv),
LV k k20C k kv 12þk k 21þk k 2 1 8V 2 V ð10Þ
and that, for type (v),
LV
k k20C k kv 21þk k 20þk k 20 8V 2 V: ð11Þ For V 2 V, define the functional:
JðVÞ ¼1 2 v þ 1 Re 5 þ r f 2 0 þk 5 vk20þ 5 k vk20 ! :
A necessary condition that the solution Un2 V of Eq. (4) be a minimizer of the functional J is lim t!0 d dtJðUnþtVÞ ¼ 0 8V 2 V, which is equivalent to BnðUn, VÞ ¼0 8V 2 V, ð12Þ where
BnðUn, VÞ ¼ BsðUn, VÞ FðVÞ þ NðUn, VÞ ð13Þ
BsðUn, VÞ ¼ ðLUn, LVÞ0 ð14Þ F ðVÞ ¼ ðF, LVÞ0 ¼ ðf, 1 Re 5 þ r Þ0 ð15Þ NðUn, VÞ ¼ ðu ! þ 1 Re 5 ! þ r r f, v ! u Þ0 þ ðu !, 1 Re 5 þ rÞ0: ð16Þ Similarly, corresponding to the Stokes problem, we have the variational formulation
BsðUs, VÞ ¼F ðVÞ, ð17Þ
where Us is the solution of Eq. (4) in which the operator N is replaced by L.
70 Jou and Liu
With respect to Eqs. (17) and (12), the finite element problems are to seek the solutions Us, h2Sh and Un, h2Sh such that
BsðUs, h, VhÞ ¼F ðVhÞ 8Vh2Sh ð18Þ BnðUn, h, VhÞ ¼0 8Vh2Sh ð19Þ
where Sh is a finite element subspace of V parametrized by the mesh size h of some
triangulation (denoted by Th) on the domain . The abstract approximation theory
for branches of nonsingular solutions developed by Brezzi et al. (1980) allows us to address existence, uniqueness, and a priori error estimates for LSFEM solutions of the Navier–Stokes equations by using the results established in the context of the linear Stokes equations (Bochev, 1997). The subspace Sh can be constructed by the standard finite elements for all variables in the vector-valued function U. For example, the velocity components u1, u2, vorticity !, and total pressure r can all
be approximated by the same piecewise linear polynomials. Newton’s iteration on Eq. (19) always results in symmetric positive definite systems of linear algebraic equations independent of the Reynolds number provided that the initial guess of the iteration is sufficiently close to the solution.
3. ERROR ESTIMATION
Once an approximate solution Us, h or Un, h is available, one of the major concerns in practice is to assess the reliability of the approximation, i.e., to estimate the exact error Es¼UsUs, h or En¼UnUn, h in some suitable norm
for which, following the a priori estimates Eqs. (10) and (11), we choose the norm LV
k k0, 8V 2 ½H1ðÞ 4. For LSFE approximation, residual type of error estimation
is a natural choice. Define the local residual norms Es, i ¼ kF LUs, hk0, ti, En, i¼ F N Un, h
0, ti ð20Þ
on each element ti2Th and the estimators
Es¼ X ti2Th E2s, i !1=2 , En¼ X ti2Th E2n, i !1=2 , ð21Þ
where the norm k k0, ti is the L2 norm restricted to the element ti:
The error indicators Es, i and En, i are readily computable without any jump
conditions across inter-element boundaries and hence the resulting computations are highly efficient and very suitable for parallel implementation. Together with the symmetric property of the algebraic system, the resulting adaptive procedure of LSFE computations can be completely parallel if a conjugate gradient solver is used because there is no need for a global assembly of the system and the iterative process can be done locally (Jiang and Carey, 1987). Moreover, for the Stokes problem, the error estimator and error indicators are perfectly reliable and effective.
Theorem 1. Let Es¼UsUs, hwhere Us and Us, hare the solutions of problems (17)
and(18), respectively. Then
The Navier–Stokes Equations 71
Es, i¼ LE s 0, t i, 8ti2Th, ð22Þ Es¼ LE s 0: ð23Þ Proof. E2s, i¼ F LUs, h 20, t i ¼ LU s LUs, h 2 0, ti:
Hence, we have Eqs. (22) and (23).
We now study the estimator for the Navier–Stokes equations. For this we make the following assumption
ku uhk0 þ k! !hk0 Ch ku uhk1, >0, ð24Þ
where C is a generic positive constant independent of h. The assumption essentially states that the convergence rate of the approximate velocity uh and vorticity !h in L2 norm is of order > 0which is higher than that in H1 norm. This kind of convergence is commonly observed in numerical experiments on finite element com-putations of second-order partial differential equations (see Bochev (1997), Bochev and Gunzburger (1998), Jiang (1998) for Navier–Stokes equations).
Theorem 2. Let En¼UnUn, hwhere Unand Un, hare the solutions of Problems(12)
and (19), respectively. If assumption (24) holds, then the error estimator En is
asymp-totically exact, i.e.,
ð1 Oðh ÞÞ LE n 0 En ð1 þ Oðh ÞÞ LE n 0: ð25Þ Proof. Let KUn¼ NUn LUn 8Un2V: Then, KUn KUn, h 2 0¼ u ! 0 0 0 B @ 1 C A uh!h 0 0 0 B @ 1 C A 2 0 ¼ u2! u1! 0 0 0 B B B @ 1 C C C A u2, h!hÞ u1, h!h 0 0 0 B B B @ 1 C C C A 2 0 ¼ u2! u2, h!h 2 0þ u1! u1, h!h 2 0 ¼ u2ð! !hÞ þ!hðu2u2, hÞ 2 0þ u1ð! !hÞ þ!hðu1u1, hÞ 2 0 4 k ku 20 ! !h 2 0þ !h 2 0 u uh 2 0
The convergence assumption implies that k!hk0 is bounded independently of h
and hence
72 Jou and Liu
u k k20 ! !h 2 0þ !h 2 0 u uh 2 0C u uh 2 0þ ! !h 2 0
It follows Eq. (24) that
KUn KUn, h 0Cð u u h 0þ ! !h 0Þ Ch u uh 1 Ch LðUnUn, hÞ 0: Since En¼ kF N Un, hk0 ¼ N Un NUn, h 0 ¼ LUn LUn, hþ KUn KUn, h 0 ¼ LE nþ ðKUn KUn, hÞ 0 we have LEn 0 KU n KUn, h 0 En LE n 0þ KU n KUn, h 0
and therefore Eq. (25).
4. CONCLUSION
It is well known that the LSFEM provides very attractive properties in applica-tions. For example, a single piecewise polynomial finite element space may be used for all test and trial functions, it always leads to symmetric positive definite systems, and it does not require the inf–sup condition when compared with the mixed finite element method. However, it usually results in more degrees of freedom in the systems due to extra state variables. Adaptive methods with effective mesh refine-ment can dramatically reduce DOFs especially for the singular problems.
The a posteriori error analysis presented in this article shows that the error indicator is perfectly reliable for the guidance of mesh refinement at least for the Stokes problems and is very effective for the Navier–Stokes problem. And the error estimator is also highly reliable for feedback error control in self-adaptive automatic computations. The implementation of the residual estimator is very simple. The error indicators can be computed strictly within each element without using any information from neighboring elements because they do not involve jump conditions across element boundaries and local boundary conditions. Therefore, together with the symmetric property of the algebraic system in a neighborhood of a solution (Bochev and Gunzburger, 1993), the adaptive procedure of least squares computations for the Navier–Stokes equations can be completely parallel on an element-by-element basis if a conjugate gradient solver is used (Jiang and Carey, 1987). For more numerical results of adaptive LSFE computations, we also refer to Hsieh et al. (1999).
The Navier–Stokes Equations 73
ACKNOWLEDGMENTS
This work was supported by NSC under grant 89-2115-M-130-001, Taiwan.
REFERENCES
Agmon, S., Douglis, A., Nirenberg, L. (1964). Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math. 17:35–92.
Bochev, P. B. (1997). Analysis of least-squares finite element methods for the Navier–Stokes equations. SIAM J. Numer. Anal. 34:1817–1844.
Bochev, P. B., Gunzburger, M. D. (1993). Accuracy of least-squares methods for the Navier–Stokes equations. Compu. & Fluids 22:549–563.
Bochev, P. B., Gunzburger, M. D. (1994). Analysis of least-squares finite element methods for the Stokes equations. Math. Comp. 63:479–506.
Bochev, P. B., Gunzburger, M. D. (1998). Finite element methods of least-squares type. SIAM Rev. 40:789–837.
Brezzi, F., Rappaz, J., Raviart, P. A. (1980). Finite-element approximation of nonlinear problems, Part I: branches of nonsingular solutions. Numer. Math. 36:1–25.
Cai, Z., Lazarov, R, Manteuffel, T. A., McCormick, S. F. (1994). First-order system least squares for second-order partial differential equations: Part I. SIAM J. Numer. Anal.31:1785–1799.
Cai, Z., Manteuffel, T. A., McCormick, S. F. (1997). First-order system least squares for second-order partial differential equations: Part II. SIAM J. Numer. Anal. 34:425–454.
Hsieh, M.-C., Lin, I.-J., Liu, J.-L. (1999). An adaptive least squares finite element method for Navier–Stokes equations. In: Lin, C.A., et al., ed. Parallel compu-tational fluid dynamics: development and applications of parallel technology. Elsevier, pp. 443–450.
Jiang, B.-N. (1998). The Least-Squares Finite Element Method. Springer.
Jiang, B.-N., Carey, G. F. (1987). Adaptive refinement for least-squares finite elements with element-by-element conjugate gradient solution. Int. J. Numer. Methods Eng.24:569–580.
Jiang, B.-N., Chang, C. L. (1990). Least-squares finite elements for the Stokes problem. Comput. Meth. Appl. Mech. Engng. 78:297–311.
Oden, J. T., Demkowicz, L., Rachowicz, W., Westermann, T. A. (1989). Toward a universal h–p adaptive finite element strategy, Part 2. A posteriori error estima-tion. Compt. Meth. Appl. Mech. Engng. 77:113–180.
Verfu¨rth, V. (1996). A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Stuttgart: Teubner-Wiley.
Zienkiewicz, O. C. (1992). Computational mechanics today. Int. J. Numer. Meth. Engrg.34:9–33.
74 Jou and Liu
