A Mixed Formulation of the Bingham Fluid Flow Problem

Comput. Methods Appl. Mech. Engrg. 200 (2011) 2434–2446

Contents lists available at ScienceDirect

Comput. Methods Appl. Mech. Engrg.

journal homepage: www.elsevier.com/locate/cma

A mixed formulation of the Bingham ﬂuid ﬂow problem: Analysis and numerical solution ⇑ Alexis Aposporidis a, , Eldad Haber b, Maxim A. Olshanskii c,1, Alessandro Veneziani a a Department of Mathematics and Computer Science, Emory University, United States b Department of Mathematics, University of British Columbia, Canada c Department of Mechanics and Mathematics, Moscow M.V. Lomonosov State University, Russia article info abstract

Article history: In this paper we introduce a mixed formulation of the Bingham fluid flow problem. We consider both the Received 26 October 2010 original and a regularized version of the problem, where a parameter e is introduced, forcing the entire Received in revised form 28 February 2011 domain to be formally a fluid region. In general, common solvers for the regularized problem experience a Accepted 5 April 2011 performance degradation when the parameter e gets smaller. The method studied here introduces an Available online 12 April 2011 auxiliary tensor variable and shows enhanced numerical properties for small values of e. A good performance is also observed for the non-regularized case. The well posedness for the regularized problem and Keywords: the equivalence – at the continuous level – between the original (primitive variables) and the mixed for- Non-Newtonian flow mulation are demonstrated. We analyze properties of linearized problems that are relevant for the con- Bingham fluid Mixed formulation vergence of numerical solvers. A finite element method for the mixed formulation is discussed. Numerical Augmented formulation results confirm the predicted better performances of the mixed formulation when compared to the Regularization primitive variables formulation. A comparison to a non-regularized solver based on the augmented Duvaut–Lions–Glowinski formulation of the problem is carried out as well. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction where div denotes the divergence operator for tensors, u, p are the unknown velocity and pressure. For Bingham ﬂuids the domain X is

The Bingham plastic is a material that behaves like a rigid split into two subdomains, the fluid region Xf and the rigid (or plug) medium for stresses s not exceeding a certain critical value ss region Xr. The constitutive relation for the stress deviator tensor s (called the yield stress) and behaves like an incompressible fluid and the strain rate tensor reads if the stresses are equal to or exceed s . The viscosity of the fluid 8 s < 0 for jsj 6 s ðrigid regionÞ; depends on the shear rate, thus the Bingham flow represents an s Du ¼ : ss s ð1:2Þ example of a non-Newtonian fluid. Bingham fluids occur in many 1 for jsj > ss ðfluid regionÞ: situations of geophysical as well as industrial interest, see [6] for jsj 2l a comprehensive review, and more recently [37] for the applica- where the plastic viscosity l > 0 and the yield stress ss P 0 are tions in hemodynamics. given constants. These equations can be observed as a generaliza- Let Du ¼ 1 ðru þ ruT Þ denote the strain rate tensor and let pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 tion of the classical Stokes equation having in Xf a shear dependent Du Du : Du be the Frobenius norm of Du. The conservation ss j j¼ viscosity l^ ¼ 2l þ that reduces to the Stokes equations with of momentum in the steady case for an incompressible fluid reads jDuj constant viscosity for ss = 0. One of the difficult features of the prob- divs þ rp ¼ f in X; ð1:1Þ lem is that the two regions are unknown a priori and finding them is r u ¼ 0 a part of the problem; also l^ becomes singular in the plug zone. A common way to avoid this difficulty is to regularize l^. This can be done in different ways, see e.g., [4,32,18]. Here we consider the ^ Bercovier–Engelman regularizationp[4]ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: in the definition of l the ⇑ 2 Corresponding author. norm jDuj is replaced with jDuje ¼ Du : Du þ e . Extension of the E-mail addresses: [email protected], [email protected] (A. Aposporidis), approach presented hereafter to other forms of regularization can [email protected] (E. Haber), [email protected] (M.A. Olshanskii), be considered as well. The regularization ensures l^ to be nonsingu- [email protected] (A. Veneziani). lar even in presence of plug regions and the fluid equations can be 1 The work of this author was supported in part by the Russian Foundation for Basic Research grants 09-01-00115, 11-01-00971 and by a Visiting Fellowship granted by posed in the entire domain: the Emerson Center for Scientific Computation at Emory University.

0045-7825/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2011.04.004 A. Aposporidis et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2434–2446 2435 8 < ss The mixed formulation (1.4) and (1.5) is closely related with the div 2l þ Du þ rp ¼ f : jDuje in X: ð1:3Þ approach of Cea and Glowinski [7] (see also Sections 5–7 in [13]). r u ¼ 0 In that approach a symmetric tensor W satisfying W : Du ¼j Du j was introduced in the numerical formulation through the relation The regularized model can be treated as the model of a quasi- W ¼ PðW þ rDuÞ 8r P 0; ð1:6Þ Newtonian fluid flow and its numerical implementation becomes relatively simple within an existing CFD code. A variety of well- with the projector P on the convex set of tensor functions established computational techniques, including parameter free Z 2 (L2(X))dd satisfying jZj < 1. The projector is defined by iterative algorithms as Newton method and Krylov subspace meth- PðZÞðxÞ :¼ ZðxÞ½maxf1; j ZðxÞ jg1. The equations (1.5) and (1.6) ods, can be used to treat the regularized equations numerically. were solved numerically with the Uzawa type method with W However, the regularization prevents finding the ‘exact’ visco-plas- serving for the primal iterated variable and r as a relaxation param- tic solution. In particular, finding arrested states and defining plug eter. Further, a special regularization was introduced in [13] to facil- regions with e > 0 become non-trivial tasks, see [35] and also itate the application of a variant of the Newton method. While [38,39] (the latter papers deal with compressible fluids). Therefore formulation (1.4) and (1.5) is formally equivalent to (1.5) and (1.6) accurate and predictive computations demand using small values for e = 0, it leads to a different variational formulation and finite of the regularization parameter e (see e.g., [18,31,12]). Using small element solutions, coupled iterative algorithms of Picard and (for values of e in (1.3) gives rise to several computational issues. For e > 0) Newton may be directly applied. Moreover (1.4) and (1.5) is example, the Newton method applied to the regularized problem amenable to common regularizations like the one used in this paper. (1.3) is not robust with respect to e (see [12] and numerical results The remainder of the paper is organized as follows. Necessary in [25,26]). The domain of convergence for the Newton method notations and preliminaries are given in Section 1.1. In Section 2, shrinks as e ? 0. Indeed, the norm of the matrix of the second deriv- we consider the weak formulations of (1.3) and (1.4)–(1.5) and atives grows like O(e1) [12] implying that to ensure convergence prove some well-posedness results. Some linearized problems are the initial guess for the Newton method should belong to an O(e)- studied here as well. We prove that the weak formulations of neighborhood of the (unknown) solution. One possibility to over- (1.3) and (1.4)–(1.5) are equivalent in the sense that they share come the issue is to apply a continuation method in e. This means the unique solution. At the same time, equivalence does not neces- that e is selected dynamically and it gets smaller along the itera- sarily hold for the corresponding numerical discretizations. In Sec- tions. Another option is to perform a number of more robust Picard tion 3 we introduce non-linear iterative methods of Picard and iterations and switch to the Newton method when a sufficiently Newton for solving (1.3) and (1.4)–(1.5). Several convergence esti- good approximation to the solution is found. In the latter case, how- mates for the case e > 0 are proven which suggest the superior ever, the required number of Picard iterations still grows as e ? 0 properties of (1.4) and (1.5) in building efficient solvers. A finite [25]. element discretization method is considered in Section 4, including Both mathematical and numerical difficulties forced several the discussion of algebraic properties of resulting discrete systems. authors to consider different formulations of the problem (1.1) In Section 5 we briefly recall another method for numerical treat- and (1.2). One approach is based on the variational inequality for- ment of the Bingham problem (1.1) based on variational inequali- mulation of Duvaut and Lions [15] and has been proposed by Glo- ties and augmentation. Several numerical results are presented in winski and coauthors (see Section 8 of the review paper [13] and Section 6. These results show that the mixed formulation (1.4) refereneces therein). The formulation has been studied mathemat- and (1.5) leads to much better convergence rates than the primi- ically and used to solve the problem numerically with Uzawa-like tive variables formulation (1.3). Moreover, it appears that the Pi- iterative schemes. The iterations are proven to be convergent upon card method for solving (1.4) and (1.5) is applicable with e =0 the introduction of a relaxation parameter (see [12]), however may and demonstrates fast convergence even in this limit case. Thus exhibit a slow convergence rate. Nevertheless, the approach is for e = 0, we include few results of comparison with the Uzawa attractive for solving practical problems when it is necessary to method for the augmented saddle-point formulation of Glowinski compute the ‘true’ visco-plastic solution and find the plug region et al. In this section, we also consider a continuation Newton meth- (see e.g., [35,34]). We briefly review this approach in Section 5. od based on the mixed formulation. Section 7 contains some clos- In this paper, we consider a different formulation intended to ing remarks. enhance the numerical properties of the regularized formulation (1.3). We introduce an auxiliary symmetric tensor W such that 1.1. Notations and preliminaries jDujeW Du ¼ 0: ð1:4Þ In what follows, we use the standard notation for the functional spaces we need: for 1 6 p 6 1 and k >0, Lp(X), Hk(X) Equations (1.3) in X with the auxiliary variable read 2 are standard Lebesgue and Sobolev spaces. Also L0ðXÞ denotes 2 the subspace of L (X) of functions with zero mean over X, 1 1 divð2lDu þ ssWÞþrp ¼ f ; H ðXÞ is the space of functions in H (X) with vanishing trace ð1:5Þ 0 r u ¼ 0: on @X. The corresponding spaces for (2D or 3D) vectors are de- 2 k 1 1 noted in bold, e.g. L (X), H (X)orH0ðXÞ. The subspace of H0ðXÞ System (1.4) and (1.5) represents the mixed formulation we inves- of divergence free vector-functions is denoted by V. We use the tigate in this paper. We will show that this formulation is efficient notation HkðXÞ for tensors whose components are Hk(X) func- for solving the regularized problem. For a given e the number of tions. For symmetric tensors, this particularizes to p k nonlinear iterations required for convergence is significantly re- Ls ðXÞ; Hs ðXÞ. When there is no possibility of confusion, we duced compared to solving the original problem (1.3) in the primi- omit the indication of the domain X. The norm in Hk is denoted 2 tive variables. While most analysis of (1.4) and (1.5) is carried out in by kkk, the scalar product and the norm in L is denoted by (,) this paper for e > 0, numerical results show that the method re- and kk, respectively, the same norm and product notation is mains efficient even for the case e = 0. In this case, the approach used for the vector and tensor counterparts of Hk and L2. and the resulting iterative method compares favorably with the From the vector identities 2divD = D + rr and rr = D + Uzawa type algorithm for the augmented Lagrangian saddle-point r r with the help of integration by parts one immediately gets formulation of Glowinski et al. the following Korn type inequalities 2436 A. Aposporidis et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2434–2446 pffiffiffi 6 6 1 aeðu; Þ¼ðf ; Þ 8 2 V: ð2:6Þ kDuk kruk 2kDuk 8u 2 H0; ð1:7Þ v v v 2 1 stating the equivalence between the L norms of the gradient and its Thanks to (2.1)–(2.3) and V H0 one applies the Browder–Minty symmetric part. We shall also refer to the Friedrichs inequality method of strictly monotone operators (see e.g., [16], Section 9.1) to prove the existence and uniqueness of u 2 V solving (2.6). The 6 1 kuk CF kruk 8u 2 H0: ð1:8Þ equivalence of (2.6) and (2.4) together with the existence and uniqueness of the pressure as a Lagrange multiplier corresponding 2. Some well-posedness results for the regularized problem to the div-free constraint can be shown by a standard argument, see [23]. To prove the estimate (2.5) for velocity one sets in (2.4) We assume that the domain is polygonal or @X 2 C1;1 and that v = u and q = p and applies the (f,v) 6 kfk krvk inequality to esti- f 2 L2. Moreover, for the sake of analysis, we assume homogeneous 1 mate the right-hand side. The bound for the pressure follows thanks Dirichlet boundary conditions to hold (i.e. u = 0 on oX). The gener- to the Necˇas inequality alization to mixed Dirichlet/Neumann boundary problems is straightforward. ðr v; pÞ kpk 6 c sup : ð2:7Þ Our primary goal is to develop and study a formulation of the v2H1 krvk regularized Bingham problem with enhanced robustness proper- 0 ties with respect to a small regularization parameter. Thus, if not Indeed, setting in (2.4) q = 0, dividing the equality by krvk and 1 explicitly stated otherwise, we consider further only the regular- exploiting kDuje Duj < 1 and the Korn and Cauchy inequalities ized version of the problem, i.e. e > 0. However, we shall also one obtains address some properties of the problem for e ? 0. 1 ðr v; pÞ 2lðDu; DvÞþssðjDuj Du; DvÞðf ; vÞ Let us introduce the following bilinear forms, ¼ e Z krvk krvk 1 1 2lkrukkr kþs ð1; jD jÞ þ kf k kr k aðu; vÞ¼ 2lDu : Dv on H0 H0; 6 v s v 1 v XZ krvk 2 1 bðp; Þ¼ pr on L H : 1 v v 0 0 6 2 : X 2lkrukþssjXj þkfk1 For the form Passing to the upper limit with respect to v and using (2.7) yields Z 1 ss 2 1 1 kpk 6 cð2lkrukþssjXj þkfk Þ: aeðu; vÞ :¼ aðu; vÞþ Du : Dv on H0 H0; 1 X jDuje The estimate one readily checks (thanks to the Korn and the Friedrichs inequali- s ties (1.7) and (1.8) the coercivity kpk 6 c 2l þ s krukþkfk : e 1 2 1 aeðu; uÞ P ckuk1 8u 2 H0 ð2:1Þ is proven by the same arguments through the use of (2.2). Combin- h and continuity ing both estimates leads to the pressure estimate in (2.5). Notice that for the limit case e = 0 we do not have the implica- 6 ss 1 aeðu; vÞ 2l þ kuk1kvk1 8u; v 2 H0: ð2:2Þ tion f ? 0 ) p ? 0 because in the model the kinematic pressure is e under-determined in the rigid zone. Moreover, one can show the strict monotonicity: To handle the mixed form (1.4) and (1.5) we define the follow-

2 1 ing bilinear and non-linear forms aeðu; u vÞaeðv; u vÞ P cku vk1 8u; v 2 H0: ð2:3Þ Z Z 1 2 Indeed, it holds cðu; ZÞ¼ ssDu : Z on H0 Ls ; gðjDuje; W; ZÞ¼ ssjDujW Z X X 2 DuDv : Z on H1 L2 L1: aeðu;uvÞaeðv;uvÞ¼ 2ljDuDvj þss 0 s s jDuj X Z e Note that g(jDuj ,W,Z) is well defined on H1 2 1 and it 1 1 2 e 0 Ls Ls þ Dv : ðDuDvÞ¼ 2ljDuDvj holds jDuje jDvje X 6 ss 2 jDuj jDvj jgðjDuj ; W; ZÞj kuk1kWkkZkL1 : þ jDuDvj e e Dv : ðDuDvÞ e jDuj jD j Z e v e The weak formulation of (1.4) and (1.5) reads as follows: Find 1 2 1 2 2 ss 2 jDuDvj u 2 H ; p 2 L and W 2 2 such that for any 2 H ; q 2 L and P 2ljDuDvj þ jDuDvj Dv : ðDuDvÞ : 0 0 Ls v 0 0 jDuj jD j 1 X e v e Z 2 Ls Monotonicity (2.3) follows from (1.7) applied to the first term in the aðu; vÞbðp; vÞþcðv; WÞþbðq; uÞþcðu; ZÞ last inequality and noting that since kD j1D j 6 1, the second term v e v gðjDuj ; W; ZÞ is non-negative. e The weak formulation of the regularized problem (1.3) reads: ¼ðf; vÞ: ð2:8Þ 1 2 1 2 find u 2 H0; p 2 L0 such that for any v 2 H0; q 2 L0 Now we are in position to prove the following well-posedness result: aeðu; vÞbðp; vÞþbðq; uÞ¼ðf; vÞ: ð2:4Þ Theorem 1. The problem (2.8) has a unique solution {u,W,p} from 1 2 2 Proposition 1. The problem (2.4) has a unique solution fu; pg2 H0 Ls L0 such that H1 L2 satisfying the estimate 0 0 2 2 6 kuk1 þ esskWk kf k1; kpk 6 1 6 1 kruk l kfk1; kpk cðkfk1 þ ss minf1; e kfk1gÞ: ð2:5Þ 6 1 cðkfk1 þ ss minf1; e kfk1gÞ: ð2:9Þ 1 Moreover W 2 Ls and Proof. First, consider (2.4) restricted to the divergence-free sub- kWk 1 6 1: ð2:10Þ space V: find u 2 V such that L A. Aposporidis et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2434–2446 2437

Proof. The proof of the well-posedness is based on showing the 3. Iterative methods equivalence of (2.8) and (2.4) and applying Proposition 1. Indeed, 1 2 1 assume u, p from H0 L0 solves (2.4) and define W :¼jDuje Du. 3.1. Two auxiliary linear problems 2 Since Du 2 Ls we conclude that the components of W are measur- 1 able functions (as products of such functions). Moreover, since for Let b 2 H0 be given, we look for u, p such that 1 6 1 8 any x 2 X; kDuðxÞje DuðxÞj 1, we have W 2 Ls and both equal- 1 1 2 < ss ities WjDuje ¼ Du and W ¼jDuje Du hold in Ls . Therefore, in div 2l þ Du þ rp ¼ f 1 jDbje in X ð3:1Þ view of (2.4) and noting that for W ¼jDuje Du it holds : r u ¼ 0 aðu; vÞþcðv; WÞ¼aeðu; vÞ; with u = 0 on @X for simplicity. Let us introduce the bilinear form we conclude that the triple {u,W,p} satisfies (2.8) for any {v,Z,q} on V V, Z from H1 L1 L2. Thus the existence of a solution to (2.8) follows 0 s 0 ss abðu; vÞ :¼ 2l þ Du : Dv: from Proposition 1. Now assume that some {u,W,p} solves (2.8). jDbj 1 X e Setting v =0, q = 0ÀÁ and varying Z 2 Ls we conclude that 1 0 1 The weak formulation of (3.1) reads: find u; p H1 L2 such that jDuje W=Du holds in Ls Ls . Hence for any solution of (2.8) ð Þ2 0 0 it holds 1 2 abðu; vÞþbðp; vÞbðq; uÞ¼ðf; vÞ 8ðv; qÞ2H0 L0: ð3:2Þ 1 W ¼jDuje Du a:e: ð2:11Þ The following proposition can be promptly proven: Using this in the third term of (2.8) and setting Z = 0 we deduce that u, p satisfy (2.4). The uniqueness of the solution to (2.8) follows Proposition 2. Given f 2 L2 and b 2 H1, there exists a unique solution from Propositions 1 and 2.11. Finally, the first two estimates in 0 {u,p} to (3.2). (2.9) follow by the same arguments as in (2.5), i.e. by taking v = u, 1 h q = p, Z = W, and W 2 Ls with (2.10) follows from (2.11). Although the formulations (2.8) and (2.4) are equivalent, their Proof. Thanks to the Korn and Friedrichs inequality, the bilinear (finite element) discretizations are not necessarily equivalent and form ab(,) is coercive, feature different numerical properties. a ðu; uÞ P lkruk2 P ckuk2: We are not aware of possible extensions of the results of Theo- b 1 rem 1 or Proposition 1 to the limit case of e = 0. Some well-posede- It is also straightforward to check that ab(,) is continuousn on V V. 1 ness results for the (non-regularized) Bingham problem are based Since V can be equivalently defined as V :¼ v 2 H0 : 2 on the reformulation of the problem as a variational inequality, see bðq; vÞ¼0 8q 2 L0g, the result of the Proposition follows from [15].In[2] a non-regularized nonhomogeneous problem is analyzed Corollary 5.1 from [23]. h as a limit case of a regularized problem with an existence result for The mixed formulation of (3.1) reads 2D periodic problems. Somewhat related results can be found in 8 > divð2lDu þ WÞþ p ¼ f [5], where the Navier–Stokes unsteady equations are considered < r r u ¼ 0 in X: ð3:3Þ for fluids featuring a stress tensor of the form :> s2 jDbje W Du ¼ 0 s ¼ lðd þjDujÞ Du; s 2ð1; 1Þ: R Denote g ðW; ZÞ s jDbj W : Z. The weak form of (3.3) reads: The extension to the case s = 1 and to the Bingham fluid problem is e;b X s e find fu; p; Wg2H1 L2 L2 such that still an open problem. 0 0 s

aðu; vÞþbðp; vÞþcðv; WÞbðq; uÞcðu; ZÞþge;bðW; ZÞ¼ðf; vÞ Remark 1. Once we proved that the solution W of (2.8) is actually ð3:4Þ from L1 one can extend the space of admissible test functions Z in s 1 2 1 1 2 for any ðv; q; ZÞ2H0 L0 Ls . (2.8) from Ls to Ls . The mixed formulation and convex programming. The velocity Proposition 3. Given f L2 and b V, there exists a unique solution solution to the Bingham problem (1.1) satisfies 2 2 {u,p,W} to (3.4). u ¼ arg min JðvÞ; v2V R R 2 Proof. The proof is carried out by showing the equivalence of the where the functional JðÞ reads JðvÞ¼lkDvk þ ss X jDvj X f v. Notice that primitive variables and mixed weak formulations (3.2) and (3.4) Z Z and then applying Proposition 2. The arguments are largely the 1 jDvj¼ sup Dv : W for v 2 H0: same as in the nonlinear case, see the proof of Theorem 1. The solu- X jWj61 X 1 2 tion of (3.2) {u,p} together with W :¼jDbje Du 2 Ls solves (3.4). By the duality theory for convex programming, the mixed formula- Reverse implication: if {u,p,W} solves (3.2), then setting v =0,q =0 1 1 tion (1.4) and (1.5) can be observed therefore as the Euler–Lagrange andÀÁ varying Z 2 Ls one finds that jDbje W ¼ u holds in 1 0 1 1 system corresponding to the saddle point problem Ls Ls . Thus W ¼jDbje Du holds almost everywhere in X. Z Z Inserting this in the third term of (3.4) and letting Z=0 we find that 2 min sup flkDvk þ ss Dv : W f vg: u, p solves (3.2). h v2V 6 jWj 1 X X On the numerical level, problem (3.2) or (3.3) has to be solved at This introduces W from (1.4) as a dual variable. The argument has each iteration of the Picard algorithm for (1.3) or (1.4) and (1.5), been advocated in [11] for an image restoration problem and gave respectively. additional stimulus to the present work. In that paper, the scalar variable u is the grey-levelR of an image and the minimization in- 3.2. Picard iteration volves the total variation X jruj. The analogies between image restoration problems and visco-plastic fluids have been already Given the well-posedness results for the auxiliary linear prob- exploited in [19,20]. lems (3.1) and (3.3) we now consider iterative methods of Picard 2438 A. Aposporidis et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2434–2446 type to iterate for the solutions of nonlinear problems (1.3)–(1.5). The previous theorem quantifies the impact of small values of ðkÞ 1 For the original problem (1.3) one iteration reads: given u 2 H0, the regularization parameter e on the Picard iteration in the prim- ðkþ1Þ 1 ðkþ1Þ 2 find u 2 H0; p 2 L0 such that itive variables formulation. In general, small values of e slow down 8 the convergence, as was mentioned in the introduction and <> ss pointed out in [25,26]. div 2l þ Duðkþ1Þ þ rpðkþ1Þ ¼ f ; DuðkÞ Let us consider now a similar analysis for the mixed > j je ð3:5Þ : formulation. r uðkþ1Þ ¼ 0: Proposition 5. The error of the iteration (3.6) satisfies Likewise, for the mixed problem (1.4) and (1.5) one iteration reads: ðkÞ 1 ðkþ1Þ 1 ðkþ1Þ 2 ðkþ1Þ 2 ðkþ1Þ 1 ðkþ1Þ 1 ðkÞ given u 2 H , find u 2 H ; p 2 L ; W 2 such that 2 6 2 0 0 0 L ke k1 þ e kE k Ce ke k1: ð3:12Þ 8 <> divð2lDuðkþ1Þ þ s Wðkþ1ÞÞþrpðkþ1Þ ¼ f ; s Proof. After a memberwise subtraction of (3.6) and (4.1) and ðkþ1Þ r u ¼ 0; ð3:6Þ (k+1) (k+1) (k+1) :> standard manipulations with v = e , q = e and Z = E (test DuðkÞ Wðkþ1Þ Duðkþ1Þ 0: 2 j je ¼ function can be taken from Ls , cf. Remark 1), we get ÀÁ ðkþ1Þ ðkþ1Þ ðkÞ ðkþ1Þ These problem are of the form (3.1) and (3.3) with b = u(k) and a e ; e þ ge jDu je jDuje; W; E u = u(k+1) and should be understood in the weak sense of (3.2) and þ g jDuðkÞj ; Eðkþ1Þ; Eðkþ1Þ ¼ 0: ð3:13Þ (3.4). Thus, due to Propositions 2 and 3 a single Picard iteration is e e well-defined. 1 2 For the mapping f(v) :¼jDvje from H to L we find the Frechet derivative operator 3.2.1. Analysis of the Picard iterations 1 6 1 Denote by u, p, W the solution to (2.8) (u, p also solve (2.4) and dðf Þja ¼jDaj Da : D )kdðf Þjak 1 2 1 8a 2 H : ð3:14Þ e H0 !L 0 e(k) u(k) u, E(k) W(k) W and e(k) p(k) p. Consider first the iterations (3.5) for the original formulation of the problem (2.4). Therefore it holds Eqs. (2.4) and (3.5) yield the following error equation ðkÞ 6 ðkÞ Z kjDu je jDujek kDe k: ð3:15Þ ðkþ1Þ ðkþ1Þ Du Du ðkþ1Þ ðkþ1Þ (k) (k+1) aðe ;vÞþ ss : Dv þbðv;e Þbðe ;qÞ¼0; Recalling kWk 1 6 1 (cf. Theorem 1) and g (jDu j ,E , jDuðkÞj jDuj L e e X e e E(k+1)) P ekE(k+1)k2, we obtain from (3.13) and (3.15) the inequality ð3:7Þ ðkþ1Þ 2 ðkþ1Þ 2 6 ðkÞ ðkþ1Þ 2 lkre k1 þ ekE k sske k1kE k for all v 2 V and q 2 L0. s2 e 6 s keðkÞk2 þ kEðkþ1Þk2: ð3:16Þ Proposition 4. The velocity error of the iteration (3.5) satisfies 2e 1 2

Thus we get k 1 1 k k 2 ð þ Þ 6 ð Þ ð Þ 2 ðkþ1Þ 2 2 ke k1 Ce ke k1 þ O ke k1 : ð3:8Þ ðkþ1Þ 6 2 1 ðkÞ ke k1 þ ekE k Css e ke k1:

Proof. We rewrite the second term in (3.7) as Remark 2. Notice that (3.12) enjoys a milder dependence on e Z than (3.8); also the higher order terms disappear. At the same time, Duðkþ1Þ Du Du Du the velocity (and pressure) iterations from (3.5) and (3.6) are the : D ss ðkÞ ðkÞ þ ðkÞ v X jDu je jDu je jDu je jDuje same (thanks to the equivalence of the auxiliary linear systems dis- Z Z cussed in Section 3.1). Thus the velocity error e(k) from (3.5) should Deðkþ1Þ : Dv 1 1 ¼ ss þ ss Du : Dv: also satisfy the improved bound (3.12). However, the argument to jDuðkÞj jDuðkÞj jDuj X e X e e show this improved bound is indirect and resorts to the mixed for- Upon Frechét linearization, which is always possible for e >0,we mulation. Moreover, such an indirect argument may not be valid in have the discrete case, since the equivalence does not necessarily hold any longer, apart from some special choice of the discretization 1 1 Du : DeðkÞ ¼ þ h:o:t: ð3:9Þ scheme (see Section 6). Testing both formulations numerically ðkÞ Du 3 jDu je j je jDuje shows that iterative methods for the mixed one are less sensitive Let us now select v = e(k+1) and q = e(k+1) in the error Eq. (3.7), so that to small values of e. we have A relaxed formulation. The method of the previous section can be Z w ðkþ1Þ ðkþ1Þ generalized by introducing a relaxation parameter # 2 [0,1]. If u ðkþ1Þ ðkþ1Þ De : De aðe ; e Þþ ss denotes the solution to (3.5) or (3.6), we set jDuðkÞj X !e Z ðkþ1Þ H ðkÞ ðkÞ u ¼ #u þð1 #Þu : Du : De ðkþ1Þ ¼ ss 3 þ h:o:t: Du : De : ð3:10Þ X jDuje Relaxation can be either static or dynamic, i.e. with # depending on k. Exploiting the coercivity of the bilinear form a(,), inequality (1.7), 1 Given b 2 H0 define the following norms: the bound (2.9) on u (independent of e) and noting that pffiffiffiffiffiffiffiffiffiffiffiffiffix2 6 ðx2þe2 Þ3 1 2ffiffiffiffi 1 1 2 p e for any x 2 R, we obtain the inequality 2 2 2 1 27 kukb :¼kruk þ sskjDbje Duk on H0; ð3:17Þ ðkþ1Þ 2 C ðkÞ ðkÞ 2 ðkþ1Þ 1 6 1 2 lkre k ke k1 þ O ke k1 ke k1 ð3:11Þ 2 2 2 1 1 e ku; Wkb :¼kruk þ sskjDbjeWk on H0 Ls : where C here and after is a constant independent of e. The latter inequality yields (3.8). h We prove the following theorem. A. Aposporidis et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2434–2446 2439

Theorem 2. The relaxed versions of the Picard iterations (3.5) and 3.3. The Newton method (3.6) admit the following error reduction relations:

3 For the primitive variables formulation (1.3) one step of the ðkþ1Þ 2 ðkÞ ðkÞ 2 2 : ðkÞ 6 ðkÞ ; # ; (k) (k+1) for ð3 5Þke ku ð1 ce Þke ku þ Oðke kuðkÞ Þ with ¼ ce Newton method can be written as follows: given u find u , ðkþ1Þ ðkþ1Þ ðkÞ ðkÞ p(k+1) solving for ð3:6Þke ;E k ðkÞ 6 ð1 ceÞke ;E k ðkÞ ; with # ¼ ce; u u "#! ðkÞ ðkÞ ðkÞ ðkÞ 2 ð3:18Þ ss Du Du : ðkþ1Þ ðkþ1Þ Du jDu j div 2lþ 1 Du þrp ¼ f ssdiv ; jDuðkÞj jDuðkÞj2 jDuðkÞj3 and sufficiently small positive constant c independent of e. e e e ruðkþ1Þ ¼ 0: ð3:21Þ Proof. The proof of the first estimate in (3.18) is a simple variation of the arguments used in proving Proposition 4. We show the proof For the mixed formulation one step of the Newton method reads: (k) (k) (k+1) (k+1) (k+1) below. The error equation reads (compare to (3.7)) given u , W find u , p , W solving Z ÀÁH ðkþ1Þ ðkþ1Þ ðkþ1Þ ðkþ1Þ ðkÞ Du Du div 2lDu þ ssW þ rp ¼ f ; a e ð1 #Þe ; v þ ss# : Dv jDuðkÞj jDuj X e e r uðkþ1Þ ¼ 0; ðkþ1Þ ðkþ1Þ þ bð ; e Þbðe ; qÞ¼0; ð3:19Þ 2 v DuðkÞ : Duðkþ1Þ jDuðkÞj Duðkþ1Þ DuðkÞ W ðkþ1Þ WðkÞ WðkÞ: j je ðkÞ ¼ ðkÞ for all v 2 V and q 2 Q. To produce the last term on the left-hand jDu je jDu je w side of (3.19) we use that u(k), u 2 V yields u(k+1) 2 V. We rewrite ð3:22Þ the second term in (3.19) as As stated in the Introduction, the degradation of performance of the Z H Du Du Du Du Newton method when e gets smaller can be remedied in different #ss þ : Dv jDuðkÞj jDuðkÞj jDuðkÞj jDuj ways. One possibility is to use a mix of Picard and Newton methods: X Z e e e e Dðeðkþ1Þ ð1 #ÞeðkÞÞ : Dv One performs a few iterations of more robust Picard method and ¼ ss ðkÞ starts Newton when a good initial guess becomes available. Another X jDu j Z e possibility is a continuation strategy that corresponds to a non-sta- 1 1 # Du : D : tionary selection of e, such that e = e(k) and limk?1e(k) = 0. Numer- þ ss ðkÞ v X jDu je jDuje ical results with these strategies will be presented in Section 6. Set v = e(k+1) and q = e(k+1) in Eq. (3.19) and use (3.9) to obtain Z Remark 3. While the Picard iterations formally produce the same Dðeðkþ1Þ ð1 #ÞeðkÞÞ : Deðkþ1Þ approximations for velocity and pressure for both formulations, cf. aðeðkþ1Þ ð1 #ÞeðkÞ; eðkþ1ÞÞþ s s ðkÞ Remark 2, the Newton method approximations are in general X !jDu je Z different for primitive variables and mixed formulations both on Du : DeðkÞ ðkþ1Þ continuous and discrete levels. This can be seen by eliminating ¼# ss 3 þ h:o:t: Du : De X jDuje W(k+1) from (3.22) with the help of the third relation. Doing this Z ! ðkÞ one finds that (3.21) and (3.22) would be equivalent only if Du : De ðkþ1Þ ðkÞ ¼# s þ h:o:t: Du : ðDe ð1 #ÞDe Þ ðkÞ ðkÞ 1 ðkÞ s 3 W ¼jDu je Du . The latter is not expected to be true for a X jDuje ðkÞ 1 ÀÁ general k, since setting W ¼jDuðkÞj DuðkÞ in (3.22) does not Z 2 e Du : DeðkÞ imply Wðkþ1Þ ¼jDuðkþ1Þj1Duðkþ1Þ. Numerical experience shows #ð1 #Þ s þ O keðkÞk3 : ð3:20Þ e s 3 1 that for small values of e (3.22) is considerably less sensitive to X jDuje the choice of the initial guess than (3.21). Now we note that the second integral term on the right-hand side of (3.20) is negative, exploit the simple identity 2(a b,a)= 4. Finite element approximation 2 2 2 2 kak kbk + ka bk as well as pffiffiffiffiffiffiffiffiffiffiffiffiffix 6 p2ffiffiffiffi e1 and the Cauchy ðx2 þe2 Þ3 27 inequality to obtain the inequality There are different ways to discretize (1.1), examples are the MAC discretization on staggered grids and collocated finite differ- ðkþ1Þ 2 ðkþ1Þ ðkÞ 2 2 ðkÞ 2 ke k ðkÞ þke ð1 #Þe k ðkÞ 6 ð1 #Þ ke k ðkÞ ence methods [27,30], finite volume [36] or LBB-stable finite ele- u u u C ments [13]. In this paper we consider Galerkin finite element þ #kDeðkÞkkDðeðkþ1Þ ð1 #ÞeðkÞÞk þ O keðkÞk3 e 1 discretization methods, although the approach is essentially independent of a specific discretization method. with C independent of e. Now we apply the Young inequality to the Denote by H H1; Q L2 and W L2 the finite dimen- handle the second term on the right-hand side and get after a h 0 h 0 h s sional subspaces for the velocity, pressure and W, respectively. cancellation We assume that the pair of spaces Hh, Qh is LBB stable [23]. The 2 C finite element method for (2.8) reads: find uh 2 Hh; ðkþ1Þ 2 6 2 ðkÞ 2 2 ðkÞ 2 ðkÞ 3 ke kuðkÞ ð1 #Þ ke kuðkÞ þ # kDe k þ O ke k1 : 4e2 ph 2 Q h; Wh 2 Wh such that 2 The latter inequality proves the first estimate in (3.18) for # = ce , aðuh; vhÞþbðph; vhÞþcðvh; W hÞbðqh; uhÞcðuh; ZhÞ with a sufficiently small positive constant c independent of e. Sim- þ geðuh; Wh; ZhÞ¼ðf; vhÞð4:1Þ ilar changes should be applied to the arguments of Proposition 5 to h show the second estimate in (3.18). for any vh 2 Hh, qh 2 Qh and Zh 2 Wh. We note that the error norms in (3.18) depend in general on the Now we address the well-posedness and stability of (4.1). Un- iteration number k. Thus we consider (3.18) as an error reduction like the continuous case (see Theorem 1) the discrete problem property rather than a convergence result. However, comparing (4.1) is not in general equivalent to the finite element counterpart the reduction factors for both formulations we note that the mixed of the original problem (2.4). Thus the well-posedness for (4.1) formulation still leads to a milder dependence on e than the origi- would not follow directly from the theory of monotone operators nal one. applied to the Galerkin formulation of (2.4). The proof of the 2440 A. Aposporidis et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2434–2446 well-posedness for (4.1) relies on the Schaefer’s extension of the 4.1. Algebraic properties Brouwer theorem (a.k.a. the Leray-Schauder theorem, see e.g.,

[16], Sections 8.1.4 and 9.2.2). Denote by Nvel, Np and Nw the number of degrees of the freedom of each velocity component in Hh, the pressure in Qh and each com-

Theorem 3. The problem (4.1) has a solution {uh,Wh,ph} from ponent of the symmetric tensors in Wh, respectively. Let {ui} for Hh Wh Qh such that i =1,...,Nvel be the basis functions for each velocity component, {q } for i =1,...,N the basis functions of Q , and {z } for 2 2 6 6 1 i p h i kuhk1 þ esskWhk0 kf k1; kphk cð1 þ sse Þkfk1: ð4:2Þ i =1,...,Nw the basis functions for each entry of the symmetric 2 If f is sufficiently small or l,e are sufficiently large then the solution is tensors. We introduce to the following notation for 2D problems : unique. Matrix A is 2Nvel 2 Nvel s.t. 8 R > 11 @uj @u @uj @u "#> A ¼ l i þ 1 i ; > ij X @x1 @x1 2 @x2 @x2 Proof. Define the discrete divergence-free space 11 12 < R A A 12 21 @u @u A ; j i 4:6 V :¼f 2 H : bðq ; Þ¼0 8q 2 Q g. For arbitrary k 2 [0,1] ¼ 21 22 Aij ¼ Aji ¼ X l @x @x ; ð Þ h vh h h vh h h A A > 1 2 k k > R consider the problem: find u 2 V , W 2 W such that : 22 @uj @u @uj @u h h h h A ¼ l i þ 1 i : ij X @x2 @x2 2 @x1 @x1 k k aðuh; vhÞþkcðvh; WhÞ¼kðf; vhÞ; ð4:3Þ Matrix B is 2Nvel Np s.t. k k k ÂÃ R R geðuh; Wh; ZhÞkcðuh; ZhÞ¼0; @u @u B ¼ B1 B2 ; B1 ¼ q j ; B2 ¼ q j : ð4:7Þ ij X i @x1 ij X i @x2 for any vh 2 Vh and Zh 2 Wh. For k = 1 the problem (4.3) is equiva- We assume that the finite elements for velocity and pressure are lent to (4.1). To apply the Leray–Schauder fixed point theorem it T 3 is sufficient to show: (i) the set of solutions to (4.3) is bounded uni- inf-sup compatible, so that B is a full-rank matrix. old old Matrix C is 2Nvel 3Nw s.t. formly with respect to k and (ii) the mapping fuh ; Wh g! "# new unew; W defined by 1;1 2;1 12;1 f h h g C C C C ; ÀÁ ¼ 1;2 2;2 12;2 new old C C C a uh ; vh ¼ðf; vhÞc vh; Wh 8vh 2 Vh; 8 R R ÀÁÀÁ ð4:4Þ < 1;1 @ui 2;1 12;1 @ui new C ¼ sszj ; C ¼ 0; C ¼ sszj ; old old ij X @x1 ij X @x2 ge uh ; Wh ; Zh ¼ c uh ; Zh 8Zh 2 Wh R R ð4:8Þ : 1;2 2;2 @ui 12;2 @ui C ¼ 0; Cij ¼ X sszj @x ; Cij ¼ X sszj @x : is continuous and bounded (all spaces are of finite dimension, so the 2 1 boundedness implies compactness). Finally, we define matrix N 2 R3Nw3Nw s.t. k k k k R To find a bound for u , W we set in (4.3) vh ¼ u , Zh ¼ W . h h h h N ¼ blockdiagðN1; N2; N12Þ; Nl ¼ jDuðkÞj z z ; l ¼ 1; 2; 3: Summing up the equalities gives ij X ss h e j i ð4:9Þ k 2 k 2 minð2l; sseÞkruhk þkWhk Notice that N is a weighted type mass matrix, so in computations it 6 k k k k k k aðuh; uhÞþgeðuh; Wh; WhÞ¼kðf; uhÞ can often be replaced by a lumped (diagonal) matrix. 1 Each iteration of the Picard method for the mixed formulation 6 minð2l; s eÞkruk k2 þ maxðl1; s1e1Þkfk2 : 2 s h s 1 requires solving a system of the form 2 3 2 3 2 3 Thus U b ACBT 6 7 6 7 6 7 ÀÁ A4 W 5 ¼ 4 0 5; with A ¼ 4 CT N 0 5: ð4:10Þ kruk k2 þkWk k2 6 max l2; s2e2 kf k2 8k 2½0; 1: h h s 1 P 0 B 00 Now we check that the mapping defined by (4.4) is bounded and T T and U ¼½U1; U2 , W =[W1,W2,W12] and P are the vectors of the new continuous. To see the boundedness we set in (4.4) vh ¼ uh , nodal values of the unknowns. Let us denote for convenience new Zh ¼ Wh and get through the Cauchy and Friedrichs inequalities: "# AC DBT new 6 old new 6 old D ¼ T ; B ¼ ½B 0 ; then A ¼ : kruh k cðsskWh kþkf k1Þ and ekWh k kruh k: C N B 0

The continuity follows from the observation that forms in (4.4) are Proposition 6. For e > 0, system (4.10) is non-singular for any choice continuous with respect to every argument. Hence the Leray- of the ﬁnite element subspace W . Schauder theorem provides the existence of a solution to (4.3) for h k = 1. Let u1, W1 and u2, W2 be two solutions to (4.3) with k =1. Denote e = u1 u2 and E = W1 W2. One sets vh = e and Zh = E in (4.3) and readily gets Proof. First we prove that D is non-singular. To this end, consider the matrix factorization 2 2lkDek þ geðu1; E; EÞþðgeðu1; W2; EÞgeðu2; W2; EÞÞ ¼ 0: ð4:5Þ "# I 0 A 0 IA1C Thus (4.5) and (3.14) yields D ¼ T 1 ; C A I 0 RW 0 I 2 2 1 6 2lkDek þ ekEk kW2kL kDekkEk 0: 1 1 T with Schur complement matrix RW ¼ðA þ CN C Þ. Thus D is

Now the a priori bound (4.2) and the smallness assumption yield the non-singular if N and RW are non-singular. Since e >0,N is invert- uniqueness result. The standard argument [23] shows the existence ible. The non-singularity of RW follows from the observation that and uniqueness of the pressure ph as a Lagrange multiplier. h 2 Even though the equivalence between primitive variables and In 3D we have 3 velocity components and 6 different components of the tensor mixed formulation at the discrete level does not necessarily hold, W. 3 If we impose u = 0 on the entire boundary oX, then B is one-rank deﬁcient having in Section 6 we will consider a particular selection of ﬁnite element hydrostatic pressure mode in its kernel. This will however not alter our further subspaces that ensures this equivalence to hold. considerations. A. Aposporidis et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2434–2446 2441

Fig. 5.1. Uzawa scheme for the Lions–Glowinski augmented formulation.

A and N are symmetric positive definite. By the same argument, A is The convergence can be tested using different stopping criteria. 1 T ðkþ1Þ ðkÞ non-singular if D and the Schur complement BD B are non-sin- Here we consider the difference between sh and sh , that basi- ðkÞ ðkÞ gular. For the pressure Schur complement of A we get cally means that we test the residual Duh ch . One of the strong aspects of this scheme is that it does not require a regularization 1 T 1 T BD B ¼BRW B ; parameter e > 0 and the method is proven to be convergent. How- ever, no convergence rate has been rigorously established and the which is a symmetric positive definite matrix, so the proposition is numerical evidence suggests that convergence can be very slow. proven. h We include some numerical results with this approach in the next For e = 0, the previous result does not hold and the non-singu- section. larity of the system is not guaranteed. A necessary condition in this case is that Ker(C) \ Ker(N)=;. The question of the selection of finite dimensional spaces forcing the well posedness of the discrete 6. Numerical results problem in this case is open. In this section we present results of several numerical experiments illustrating the performance of the approach introduced in 5. The Lions–Glowinski augmented formulation the paper. We compare the results with the primitive variables formulation and show an advantage in using the mixed formulation, In [24] an alternative variational formulation is considered. To especially for small values of e. A few experiments with the mixed underline differences and similarities with our approach we briefly formulation are done for the non-regularized case (e = 0). In this recall the method from [24]. The solution of the Bingham problem case its performance is compared with the ALG algorithm for the (1.1) u, s together with c is the saddle point of the following Lions-Glowinski augmented formulation. We will also consider Lagrangian: the Newton continuation method and give numerical evidence of Z Z Z its effectiveness. Finally, we will show some results for domains 2 with a less trivial geometry. Lðu; c; sÞ¼l jcj dx þ ss jcjdx þ ðDðuÞcÞ : sdx XZ X Z X 2 6.1. Primal vs mixed, quadrilateral elements þ k jDðuÞcj dx f udx: X X In this first set of experiments we use the IFISS package in Here kP0 is an auxiliary augmentation parameter. Then MATLAB [17]. Problem (1.5) is discretized using Q2–Q1 finite elements for velocity and pressure, respectively, and the mixed Lðu; c; sÞ¼ min max Lðv; r; nÞ: ð5:1Þ 2 2 variable W is discretized in Q1. Starting with the zero vector as r2Ls ;v2V n2Ls an initial guess, we perform Picard iterations until krkk1 6 106, kr0k1 Note that the auxiliary variable c is introduced in a way that c = Du i.e. until the initial residual drops by six orders of magnitude. in a weak sense. This is in contrast to our approach where jDuj1Du Unless stated otherwise, l = 1 and f = 0. MATLAB’s backslash is treated as the auxiliary variable W. operator serves as the linear solver. Note that when e gets smaller The variational saddle point problem (5.1) can be equivalently the linear systems are expected to become increasingly ill- written as an integral inequality problem, see Duvaut and Lions conditioned (for both primitive variables and mixed form) and [15]. For e > 0, however, one can also look for the equivalent inte- ad hoc preconditioned iterative method should be developed to gral equality formulation. In our notation it would read: find u, p solve them efficiently for mid-size and larger problems. and c such that In practice, another good initial guess is given by the solution of a Stokes (Newtonian) problem. 1 kaðu; vÞþbðp; vÞþ2 cðs 2kc; vÞ¼ðf ; vÞ; bðq; uÞ¼0; 6.1.1. An Analytical Test Case cðw; uÞðc; wÞ¼0; One of the few available analytical solutions to (1.1) describes the flow between two parallel plates, and in two dimensions it is s ¼ 2lc þ sscjcje: given by for any test functions v, q, w in appropriate spaces. Departing from 8 > 1 1 2 2 1 2 2y 2 ; if 0 6 y < 1 ; this formulation, different discretization schemes can be consid- < 8 ½ð ssÞ ð ss Þ 2 ss ered. We note that the penalty parameter k also plays the role of u ¼ 1 2 1 6 6 1 1 > 8 ð1 2ssÞ ; if 2 ss y 2 þ ss; :> the relaxation parameter in the iterative algorithm of Uzawa type. 1 2 2 1 ½ð1 2ssÞ ð2y 2ss 1Þ ; if þ ss < y 6 1; In [24] the iterative algorithm (further referred as ALG) for solving 8 2 (5.1) is considered, see Fig. 5.1. ð6:1Þ 2442 A. Aposporidis et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2434–2446

Table 6.1 6 Number of Picard iterations required for reducing the residual of a factor 10 in the analytical test case for ss = 0.3 and different h and e in both primitive variables and mixed formulation. The last column shows results for the non-regularized case (that is not working in the primitive variables formulation).

h;ne? 101 102 103 104 105 101 102 103 104 105 0 Primitive variables Mixed variables 1 / 16 7 20 15 30 38 5 12 17 20 23 19 1 / 32 7 23 60 81 83 4 12 19 24 25 24 1/647 278995884 914161622 1 / 128 6 23 61 134 192 3 8 11 12 12 13

−1 ÈÉ 10 1 6 6 1 u2 0 and p = x. The rigid region y 2 Xj 2 ss y 2 þ ss is the kernel moving at a constant velocity. In our experiment, we

choose ss = 0.3. We impose Dirichlet boundary conditions on the domain X = [0,1]2 according to (6.1). The number of Picard iterations for both primitive variables and mixed formulations can be 10−2 seen in Table 6.1. The comparison clearly outlines the advantage ε=10−1 of using the mixed formulation. In particular, we noticed that even A || −2 for the non-regularized case the mixed formulation is convergent

e ε=10 || and gets accurate results. ε −3 =10 Fig. 6.1 shows the numerical error in the discrete energy norm 10−3 −4 ε=10 for different choices of e and h (obtained with the mixed formula- ε=10−5 tion). For the analytical velocity solution u from (6.1) it holds 2 u 2 H2, but u R H3. This limits the guaranteed order of convergence 2 O(h ) O(h) to O(h) in the energy norm. Large values of e with small values of h 1 −4 clearly prevent the optimal convergence rate of the finite element 10 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 method. For smaller values of e the observed order of convergence h is between 1 and 2. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T Fig. 6.2, top left, shows the computed pressure field. The pres- Fig. 6.1. Numerical error ku uexkA ¼ ðu uexÞ Aðu uexÞ for different sizes of the mesh and different e. The matrix A is a finite element stiffness matrix. The thin sure field indicates a numerical error in the area around the rigid solid line is a reference line for O(h2), the thin dotted one is for O(h). region. This is expected, since in the model the stress tensor is un-

1 5 Fig. 6.2. Top, left: pressure field of the analytical test case computed with ss = 0.3, h ¼ 128 and e =10 . Top, right: velocity error for ss = 0.1. Bottom, velocity error for ss = 0.3 and ss = 0.4. A. Aposporidis et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2434–2446 2443 der-determined in the rigid region. The other subplots of the secutive computation of the stress. This is consistent with the 1 Figure illustrate the velocity error for different values of ss, common stopping criteria for the ALG. We used ss =2, h ¼ 32. pointing out the presence of error spikes in the neighborhood of The convergence rate of the mixed formulation is better than the rigid region. Note however that the maximum error is of the the one of the Lions-Glowinski formulation measured in terms order of 105. of ks(k+1) s(k)k (notice however that Picard residual indicates the convergence to the same solution in less iterations). In gen- 6.1.2. The lid-driven cavity eral the identification of the optimal parameter k is not easy We perform two different numerical simulations, both of them whilst the solver for the new mixed formulation is parameter in the unit square domain X = [0,1]2. For the first case, we solve free. (1.4) and (1.5) imposing Dirichlet boundary conditions by Monitoring the difference kc(k+1) c(k)k we have obtained very T ujy=1 = (1,0) and u = 0 everywhere else. Table 6.2 shows the similar results both for the convergence with different values of k number of iterations of the Picard method described in Section 3 and the comparison with the mixed formulation (actually, with for different sizes of the mesh and different values of ss and e. an even more evident better convergence rate for the mixed Again, the comparison between the mixed formulation and the ori- formulation), so we did not report the results here. ginal one demonstrates the effectiveness of the former; for the In the second experiment, we solve the unsteady Bingham primitive variables formulation the method does not converge problem taking into account the effect of the inertia terms, i.e. within 500 iterations in some cases (denoted by ). including the nonlinear convective term. We set the density q =1 We compare also the numerical results of the mixed formula- for simplicity. Unsteady problem in the mixed form reads tion with e = 0 to the augmented formulation by Lions–Glowinski 8 > @u described in Section 5. The latter involves an additional parame- <> þðu rÞu divW þ rp ¼ f; @t ter k.InFig. 6.3 on the left we compare the dynamics of the 6:2 > r u ¼ 0; ð Þ reduction of the difference ks(k+1) s(k)k along the iterations for :> several values of k. The best choice for parameter k is k = 0.01. WjDujDu ¼ 0: Fig. 6.3 on the right illustrates the dynamics of ks(k+1) s(k)k T We impose Dirichlet boundary conditions by ujy=1 = (10,0) and along 200 iterations of the non-regularized version of (3.6), i.e. homogeneous Dirichlet boundary conditions everywhere else. We e = 0 and the Lions-Glowinski method with k = 0.01. Note that 1 5 choose Dt = 0.1 as the time step, h ¼ 128, e =10 and ss = 2 and in the previous results we used the Picard residual for the stop- l = 0.1. We solve the Stokes problem and the Navier-Stokes one. ping criterion, here we check the difference between two con- In Fig. 6.4 we show the Stokes and the Navier-Stokes solutions with

Table 6.2 Number of Picard iterations required for reducing the residual by 106 for the primitive variables formulation (left) and the mixed one (right) for the lid-driven cavity. For h = 1/128 and in many cases for h = 1/64 the iterations in primitive variables do not converge (-). The non-regularized case e = 0 is included in the last column for the mixed formulation.

h;ne? 101 102 103 104 105 101 102 103 104 105 0 Primal Dual

ss =2: 1 / 16 22 49 51 51 51 11 21 26 27 27 21 1 / 32 99 173 224 224 224 10 15 17 17 17 23 1/64213––––81212121215 1/128–––––7999911

ss =5: 1 / 16 18 37 48 51 51 17 31 37 37 38 27 1 / 32 66 94 269 267 266 14 22 23 24 24 32 1/64128––––121718181822 1/128–––––101314141415

4 2 10 10 λ=1 ALG (Formulation (5.1)) λ=0.1 Picard (Formulation (3.6)) λ=0.01 102 100 λ=0.001

0 −2

10 || 10 || k k τ τ − − k+1 k+1 τ τ −2 −4 || || 10 10

10−4 10−6

−6 −8 10 10 0 50 100 150 200 0 50 100 150 200 Number of Iterations Number of Iterations

k+1 k 1 Fig. 6.3. Reduction of the difference ks s k for the ﬁrst 200 iterations when solving the lid-driven cavity problem with ss = 2 and h ¼ 32. Left: ALG with different choices of k. No convergence occurs for the case of k = 0.001. Right: Comparison between Picard iterations (3.6) with e = 0 and formulation (5.1) with the algorithm ALG and k = 0.01. 2444 A. Aposporidis et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2434–2446

1 5 Fig. 6.4. Streamlines of the lid-driven cavity ﬂow (ss =2,h ¼ 128 and e =10 ). Left: Solution of the Stokes problem. Center and right: solution of the unsteady Navier–Stokes- T type problem computed with ujy=1 = (10,0) , l = 0.1 (center) and l = 0.01 (right).

u p W

Fig. 6.5. Degrees of freedom for the P1isoP2–P1–P0isoP2 elements used for the mixed formulation. are the degrees of freedom for the velocity components, s for the pressure, s for the auxiliary tensor W.

points of the edges of the original triangulation), Wh consists of piecewise constant functions with respect to the same reﬁned triangulation (by analogy we will denote this element by P0isoP2), cf. Fig. 6.5. Further in the tables, h denotes the size of the (larger) pressure element. Despite of a low approximation order, this choice of

ﬁnite element spaces has several advantages: (i) Vh and Qh form an LBB stable pair, (ii) it holds DðVhÞWh, (iii) for any uh 2 Vh and Wh 2 Wh it holds jDuhjeWh 2 Wh. Therefore (4.1) implies jDuhjeWh = Duh to be valid in a usual pointwise sense. Thus P1isoP2–P1–P0isoP2 is a (somewhat exceptional) example of a stable FE triple which provides the the equivalence of the discrete primitive variables and mixed formulations, similar to their continuous counterparts. Thus, the difference between the two formulations lies entirely in the performance of the Newton type solvers, while the resulting discrete solution accuracy is the same. We also

Table 6.3 Fig. 6.6. Pseudocode for the continuation algorithm with the Newton method. Total number of the Newton iterations within the continuation algorithm for the lid- newton(u,p,W;f;ecurr) stands for one iteration of the Newton method; ‘‘suc- driven cavity problem with etarget =1e 5, ss = 2, and varying h. cess’’ == ‘‘true’’ if the new residual is less than ecurr. h 1 1 1 1 32 64 128 256 l = 0.1 and l = 0.01. The nonlinear term is handled with a lineariza- Primal form 95 104 107 Augmented form 14 15 15 15 tion through a semi-implicit time advancing scheme. Effectiveness wmin 0.08070 0.08182 0.08231 0.08254 of the mixed formulation is conﬁrmed also in this case. Plots of the solution in Fig. 6.4 show the equally distributed velocity streamlines at t = 1 after the solutions reached steady states.

6.2. Newton method with continuation, triangular elements Table 6.4 Total number of the Newton iterations within the continuation algorithm for the lid- driven cavity problem with e =1e 5 and varying s ; h ¼ 1 . In this section, we show results of several numerical experi- target s 128 ments with the Newton method enhanced by a continuation algo- ss 2510 rithm. We run tests for both primitive variables and mixed form of the discrete problem. These numerical results were produced using Primal form 107 123 140 the different triple of ﬁnite element pairs: P1isoP2-P1 for velocity Augmented form 14 29 46 wmin 0.08231 0.06874 0.05696 and pressure (Vh consists of continuous piecewise linear functions yw 0.805 0.836 0.861 with respect to the triangulation built by connecting the middle A. Aposporidis et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2434–2446 2445

τ =2 τ =5 τ =10 s s s 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6 y y y

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x x

1 Fig. 6.7. Velocity streamlines (blue) and predicted yield surfaces (brown) of the lid-driven cavity ﬂow computed for different values of the stress yield ss, with h ¼ 64. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

values the linear systems become increasingly ill-conditioned 2 e 4 (for both primitive variables and mixed form) and special precon- 3.6 1 3.2 ditioned iterative method should be developed to solve them efﬁ- 2.8 0 2.4 ciently. Lacking such solvers in our current implementation 2 5 1.6 prevented us from decreasing etarget in this test below 10 . −1 1.2 0.8 Tables 6.3 and 6.4 show the total number of Newton iterations 0.4 within the continuation algorithm (the total number includes also −2 0 −2.5 those iterations which were not accepted, i.e. ended with ‘‘suc- 0 10 cess’’ = ’’false’’). The results are shown for the problem in both primitive variables and the mixed form for different values of mesh size and stress yield. We also show the minimum of the stream function and the position of the main vortex center for the computed solutions. Fig. 6.7 presents the equally distributed velocity 1 streamlines and the isoline of ð2l þ ssjDuje ÞjDuj¼ss, the latter may be interpreted as a reasonable approximation of the yield surfaces [35]. Comparing these statistics to those found in [29,30] we assure that sufﬁciently accurate solutions are computed.

Fig. 6.8. Plot of jDujfor a nontrivial domains with a circular cavity attached to a 6.3. A 2D computation on non-rectangular domains rectangular pipe (top) and an occluded pipe (bottom). Results obtained with the mixed non-regularized formulation. We finally present two simple test cases carried out in non- rectangular geometries, showing that the mixed formulation even remark that due to inherent irregularity of the visco-plastic solu- with no regularization is a viable approach for more realistic prob- tions4 it may not be beneficial to use high order elements. lems. Simulations are carried out with the code FreeFem++ A pseudocode of the continuation algorithm we used is shown (version 3.9), P1 bubble finite elements for the velocity, P1 for in Fig. 6.6. Newton iterations inside the continuation method are the pressure and each component of the tensor W. A streamline stopped (with ‘‘success’’ == ‘‘true’’) once the L2 norm of the nonlin- upwind Petrov–Galerkin stabilization has been added for the treat- ear residual is less than e, where e is a current value of the regular- ment of the convective term (linearized with a Picard approach). In the first case we simulated a 2D circular cavity problem attached to ization parameter. Numerical experiments show that finding unew, a 2D rectangular channel. In the second case we considered a rect- pnew, Wnew with better accuracy does not lead to a better convergence of the subsequent Newton step. If during the newton(...) angular channel featuring an obstacle represented by a sinusoidal step the residual exceeds twice the initial residual (it was found bump. We tested several values of the parameters, both the phys- beneficial to allow a small increase of residual on the first step of ical and the numerical ones. In particular, in Fig. 6.8 – left – we the Newton method), then the Newton iterations are terminated show the computed shear rate for the first geometry with s =1, with ‘‘success’’ == ‘‘false’’. The target e in all our tests is 105.On l = 0.05 and an incoming velocity profile with peak uM =2. In every step of the Newton method a linear system of algebraic Fig. 6.8 – right – we present results for the second geometry with equations was solved approximately by a preconditioned GMRES s =1,l = 0.1, s = 2 and uM = 10. The non-regularized mixed formu- method. For these inner linear iterations the stopping criteria lation gets convergent results also in these cases. was the reduction of the initial residual by a factor of 0.1e. Increas- ing the inner tolerance did not lead to a notable reduction of the 7. Conclusion total number of nonlinear iterations. We note again that for small In this paper we introduced a new formulation for the regularized Bingham flow equations. This formulation has the advan- 4 The velocity gradient is expected to have a kink on the yield surface, as clearly seen from the analytical example in (6.1), thus u R H3. Of course, introducing e >0 tage of being numerically more robust when the regularization smears the irregularity. parameter e gets smaller. We proved well-posedness results for 2446 A. Aposporidis et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2434–2446 the weak form of the problem and discussed algebraic properties [18] I.A. Frigaard, C. Nouar, On the usage of viscosity regularization methods for of the discretized equations. We have performed several numer- viscoplastic fluid flow computation, J. Non-Newton. Fluid Mech. 127 (2005) 1–26. ical experiments using finite element methods. These experi- [19] I.A. Frigaard, G. Ngwa, O. Scherzer, On effective stopping time selection for ments show that the number of nonlinear iterations is visco-plastic nonlinear BV diffusion filters used in image denoising, SIAM significantly reduced for the new formulation compared to the J. Appl. Math. 63 (6) (2003) 1911–1934. [20] I.A. Frigaard, O. Scherzer, Herschel–Bulkley diffusion filtering: non-Newtonian classical formulation in primitive variables. The mixed formula- fluid mechanics in image processing, Z. Angew. Math. Mech. 86 (6) (2006) tion does not become singular in the plug region, so that, even 474–494. if the theory of this paper covers only the regularized case, [23] V. Girault, P.A. Raviart, Finite element methods for Navier–Stokes equations: theory and algorithms, Springer Series in Computational Mathematics, vol. 5, numerical experiments also demonstrate good performances for Springer, Verlag, 1986. the non-regularized case. For the non-regularized case the mixed [24] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, formulation was found to compare favorably to the classical Springer, Verlag, New York, NY, 1984. [25] P.P. Grinevich, M.A. Olshanski, An iterative method for the stokes-type Uzawa method based on the Lions-Glowinski approach. Further problem with variable viscosity, SIAM J. Sci. Comput. 31 (5) (2009) 3959–3978. we plan to validate and use the new formulation for more [26] J. Hron, A. Ouazzi, S. Turek, A Computational Comparison of two FEM Solvers practical 3D problems. Validation will include the accurate For Nonlinear Incompressible Flow, Challenges in Scientific Computing, computation of the plug region in problems of practical interest Springer, 2004. [27] E.A. Muravleva, M.A. Olshanskii, Two finite-difference schemes for the (see [35]). calculation of Bingham fluid flows in a cavity, Russ. J. Numer. Anal. Math. Model. 23 (6) (2008) 615–634. References [29] E. Mitsoulis, Th. Zisis, Flow of Bingham plastics in a lid-driven cavity, J. Non- Newton. Fluid Mech. 101 (2001) 173–180. [30] M.A. Olshanskii, Analysis of semi-staggered finite-difference method with [2] I.V. Basov, V.V. Shelukhin, Nonhomogeneous incompressible Bingham application to Bingham flows, Comput. Methods Appl. Mech. Eng. 198 (2009) viscoplastic as a limit of nonlinear fluids, J. Non-Newton. Fluid Mech. 142 975–985. (2007) 95–103. [31] H. Schmitt, Numerical simulation of Bingham fluid flow using prox- [4] M. Bercovier, M. Engelman, A finite element method for incompressible non- regularization, J. Optim. Theory Appl. 106 (3) (2000) 603–626. Newtonian flows, J. Comput. Phys. 36 (1980) 313–326. [32] T.C. Papanastasiou, Flows of materials with yield, J. Rheol. 31 (1987) 385–404. [5] L.C. Berselli, L. Diening, M. Ruzicka, Existence of Strong Solutions for [34] N. Roquet, P. Saramito, An adaptive finite element method for Bingham fluid incompressible fluids with shear dependent viscosities, J. Math. Fluid Mech. flows around a cylinder, Comput. Methods Appl. Mech. Eng. 192 (31-32) 12 (1) (2010) 101–132. (2003) 3317–3341. [6] R. Byron-Bird, G.C. Dai, B.J. Yarusso, The Rheology and Flow of Viscoplastic [35] A. Putz, I.A. Frigaard, D.M. Martinez, On the Lubrication Paradox and the use of Materials, Rev. Chem. Eng. 1 (1) (1983) 2–70. Regularisation Methods for Lubrication Flows, Journal of Non-Newtonian Fluid [7] J. Cea, R. Glowinski, Méthodes numériques pour lécoulement laminaire dún Mechanics, Elsevier, 2009. fluide rigide visco-plastique incompressible, Int. J. Comput. Math. Sec. B 3 [36] M. Sahin, H.J. Wilson, A semi-staggered dilation-free finite volume method for (1972) 225–255. the numerical solution of viscoelastic fluid flows on all-hexahedral elements, [11] T. Chan, G. Golub, P. Mulet, A nonlinear primal-dual method for total variation- J. Non-Newton. Fluid Mech. 147 (2007) 79–91. based image restoration, SIAM J. Sci. Comput. 20 (6) (1999) 1964–1977. [37] A.M. Robertson, A. Sequeira, V. Kameneva, in: G.P. Galdi, R. Rannacher, A.M. [12] E.J. Dean, R. Glowinski, Operator-splitting methods for the simulation of Robertson, S. Turek (Eds.), Hemorheology in Hemodynamical Flows, Bingham visco-plastic flow, Chin. Ann. Math. 23 (2002) 187–204. Birkhauser, Basel, 2008. [13] E.J. Dean, R. Glowinski, G. Guidoboni, On the numerical simulation of Bingham [38] G. Vinay, A. Wachs, J.F. Agassant, Numerical simulation of weakly visco-plastic flow: old and new results, J. Non-Newton. Fluid Mech. 142 (2007) compressible Bingham flows the restart of pipeline flows of waxy crude oils, 36–62. J. Non-Newton. Fluid Mech. 136 (2–3) (2006) 93–105. [15] G. Duvaut, J.L. Lions, Inequalities in Mechanics and Physics, Springer, 1976. [39] G. Vinay, A. Wachs, I. Frigaard, Start-up transients and efficient computation of [16] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. isothermal waxy crude oil flows, J. Non-Newton. Fluid Mech. 143 (2–3) (2007) 19, Springer, 1998. 141–156. [17] H. Elman, D. Silvester, A. Wathen, Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics, Oxford University Press, 2005.