Uzawa method on semi-staggered grids for unsteady Bingham ...

Russ. J. Numer. Anal. Math. Modelling, Vol. 24, No. 6, pp. 543–563 (2009) 

DOI 10.1515/ RJNAMM.2009.034 

cde Gruyter 2009 

<strong>Uzawa</strong> <strong>method</strong> on semi-staggered grids for unsteady 

Bingham media flows 

L. V. MURAVLEVA£and E. A. MURAVLEVA † 

Abstract — A finite difference scheme on semi-staggered grids is used for numerical simulation of 

unsteady flows of an incompressible viscoplastic Bingham medium. The Duvaut–Lions variational 

inequality is considered as a mathematical model. The convergence of the <strong>Uzawa</strong> <strong>method</strong> is proved. 

The efficiency of the <strong>method</strong> is demonstrated on a model problem with a known exact solution (plane 

Poiseuille flow) and on the unsteady driven cavity problem. 

1. Statement of the problem 

There exists a number of materials behaving as a viscoplastic medium (Bingham 

medium), namely, i.e., the medium behaves as a rigid body below some limit stress 

value and above this level it behaves as an incompressible viscous fluid. A variational 

statement for the flows of viscoplastic media was first proposed by Il’yushin 

[11]. In [16] the existence and uniqueness theorems were proved for the problem 

of the flow in a pipe and a qualitative study of flow specifics was performed. 

Monograph [8] contains strict mathematical analysis of variational inequalities corresponding 

to this model. 

Let Ω be a bounded domain in R dd23and Γ be a locally Lipschitz boundary 

of the domain Ω. A flow of an incompressible viscoplastic medium during the 

σ Ω¢´0Tµ 

time interval´0Tµis described by the following system of equations and constitutive 

relations: 

ρ∂v 

∂t·´v¡∇µv∇¡σ·f∇¡v0 s D´vµ Γ¢´0Tµ 

in (1.1) 

pI·ττ2µD´vµ·σ s 

D´vµD´vµ0 

(1.2) 

τσ D´vµ0 

v´0µv 0 in Ω∇¡v 00v0 on (1.3) 

£Moscow State University, Faculty of Mechanics and Mathematics, Moscow 119991, Russia 

† Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow 119333, Russia 

The work was partially supported by the Russian Foundation for Basic Research (08–01–00353, 

09–01–00565).

544 L. V. Muravleva and E. A. Muravleva 

Equations (1.1)–(1.3) use the standard notations: ρµσ s are positive constants, 

namely the density, the viscosity coefficient, and the yield stress of the Bingham 

medium, respectively (sometimes we use τ s as shear yield stress, σ sτ sÔ2); v is the 

unknown velocity field, f is the given field of external forces; σ is the stress tensor, 

τ is the deviator of the stress tensor, and p is the pressure, D´vµ∇v·´∇vµT℄2 

is the strain rate tensor with the normD´vµ 

0¨xt¾Ω¢´0Tµ 

∑ d i1 ∑ d j1 D i j´vµD j´vµ¡12 i . 

System (1.1)–(1.3) holds in the domain of motion (i.e.,D´vµ0) and, generally 

speaking, has no meaning in the rigid zone Ω 

typical peculiarity in problems of viscoplastic media flows 

D´vµ´xtµ0©. The 

is the necessity to construct solutions in domains with unknown boundaries. This 

causes significant difficulties in construction of efficient <strong>method</strong>s for their study. 

The main difficulty in numerical simulation of the flow of a viscoplastic medium is 

related to nondifferentiability of constitutive relations. One of the most successful 

techniques to overcome the mentioned difficulties is using the theory of variational 

inequalities [8, 10, 12]. The finite element <strong>method</strong> is traditionally employed as the 

discretization technique [7] in numerical simulation of Bingham media. In this paper 

we apply the finite-difference scheme from [19–23] for determination of an 

approximate solution. We consider the numerical solution of the unsteady driven 

cavity problem for a Bingham medium as a model example. The obtained results 

are compared to those known from the literature. 

2. Variational inequality 

Duvaut and Lions [8] proposed the problem on the solvability of following variational 

inequality (2.1) (with conditions (2.2),(2.3)) as a strict formulation of problem 

(1.1)–(1.3) of a viscoplastic flow: determine v´tµ¾´H0´Ωµµd 1 so that for each 

t¾´0Tµthe following inequality holds: 

∂v 

ρΩ ∂t¡ú v´tµµdx·ρΩ´v´tµ¡∇µv´tµ¡ú v´tµµdx 

D´v´tµµµdx 

(2.1) 

sΩ´Dúµ 1´Ωµµd∇¡u0 ·2µΩ 

D´v´tµµ:Dú v´tµµdx·σ Ω f´tµ¡ú v´tµµdxu¾U ∇¡v´tµ0 in Ωv´0µv 0 in Ωv´tµ0 Uu¾´H (2.3) 

i 

on 

i 

Γ (2.2) 

where A : B∑ d i1 ∑ d j1 a i jb i j for all tensors Aá jµand B´b jµof the second 

order. Inequality (2.1) ‘automatically’ includes the problem on a ‘free boundary’. 

In our case it is the surface separating the domain in which the flow is described 

by equation (1.1) from the domain where the medium moves as a rigid body. The 

following theorem on multipliers was also proved in [8].

∂t·´v¡∇µv 

<strong>Uzawa</strong> <strong>method</strong> on semi-staggered grids 545 

Theorem 2.1 [8]. Let v be a solution to problem (2.1)–(2.3). Then there exists 

a tensor-valued function λλ i j1ijd Ω¢´0Tµ Ω¢´0Tµ 

such that 

Γ¢´0Tµ 

λ¾´L ∞´Ωµµd¢dλλ Tλ1 

λ : D´vµD´vµa.e. in (2.4) 

ρ∂v 

µv σ s ∇¡λ·∇pf in (2.5) 

∇¡v0 in Ω¢´0Tµv´0µv 0 in Ωv0 on (2.6) 

If, on the contrary, the triplevpλsatisfies relations (2.4)–(2.6), then v is the 

solution to problem (2.1)–(2.3). 

3. Discretization of problem (2.1)–(2.3) in time 

Suppose the flow is slow, i.e., we can neglect the convective term (the Stokes approximation). 

Following [7], we use the well-known backward Euler scheme for 

time discretization of problem (2.1)–(2.3). Lett0 be a constant time step. Assume 

v 0v 0 

kµdxu¾U 

then for k1 we compute v k from v k 1 as the solution to the following inequality 

(here and further f kf´ktµ) 

ρtΩ´v k v k 1µ¡ú v kµdx·2µΩ D´v kµ:Dú v k¡ú 

f v 

We introduce αρtff k 1 , in what follows 

sΩ´Dúµ 

we omit the subscript k. 

Then in the new notations we have 

αΩ 

v¡ú D´vµ:Dú vµdx·σ vµdx·2µΩ 

D´vµµdx 

(3.1) 

sΩ´Dúµ ·σ k·αv 

Ω f¡ú vµdxu¾U 

D´v kµµdxΩ 

4. <strong>Uzawa</strong>’s algorithm 

Numerical <strong>method</strong>s for solving variational inequality (3.1) are based on the following 

approach. Introduce the functional 

dx·σ sΩDúµdx f¡udx (4.1) 

Ω 

Júµα 

2Ωu2 dx·µΩDúµ2

Júµ 


The functional Júµis strictly convex, but is not differentiable because of the term 

σ sÊΩDúµdx. The solution v of problem (3.1) is the minimiser of the functional J 

on U [10]: 

vargmin 

(4.2) 

u¾U Taking into account inequality (2.4), for the integral of the contraction λ : Dúµ, 

0´Ωµµd 1 

λ : D´vµdxΩD´vµdx 

Ω 

λ : 

Ω 

DúµdxΩλDúµdxΩDúµdxu¾´H 

we get the following relation: 

ΩD´vµdxsup λ : D´vµdx (4.3) 

λ¾ΛΩ 

where Λλλ¾´L∞´Ωµµd¢dλλ Tλ1a.e. on Ω. Further, substituting 

expression (4.3) into the functional, we have 

s sup 

f¡udx 

dx·σ λ : Dúµdx 

λ¾ΛΩ 

Ω 

or 

dx·σ λ : Dúµdx sΩ 

f¡udx 

Ω 

Introduce the Lagrange functionalÄ:´H 1´Ωµµd¢´L 2´Ωµµd¢dÊcorresponding 

to (4.1) as 

dx·σ η : Dúµdx f¡udx(4.4) 

sΩ 

Ω 

inf 

u¾Uα 

2Ωu2 dx·µΩDúµ2 

inf sup 

u¾U λ¾Λα 


Äú;ηµα 


In accordance with the minimax theorem [9, 10], the pair´vλµis a saddle point of 

the Lagrange functionalÄú;ηµon U¢Λ: 

´vλµ¾U B¢ΛÄ´v;ηµÄ´v;λµÄú;λµúηµ¾U¢Λ (4.5) 

and the first component of the pair v is uniquely determined and it is the solution to 

optimization problem (4.2). The first inequality of (4.5) is equivalent [9, 26] to the 

equation 

Ω¢´0Tµ λ´tµP Λ´λ´tµ·rσ s D´v´tµµµr0a.e. on 

(4.6)

Λ´qµ´xµ´q´xµ 


where P Λ :´L∞´Ωµµd¢dΛ is the operator of orthogonal projection defined by the 

following expression: 

2µ∇¡D´vµ 

q´xµ1 

D´vµµ 

P on 

q´xµq´xµq´xµ1a.e. Ωq¾´L This remark plays an important role in computations. Due to Theorem 2.1, the pair 

´vλµsatisfies the following relations: 

αv σ s ∇¡λ·∇pf∇¡v0 in Ωv0 on Γ (4.8) 

λP Λ´λ·rσ s (4.9) 

Based on (4.8)–(4.9), nµ we construct the following algorithm, which is the <strong>Uzawa</strong> 

algorithm applied to the saddle-point problem withÄú;ηµ: 

λ 0¾Λ is given (arbitrarily) (4.10) 

for n0, assuming that λ n´¾Λµis nµµ known, we compute v n as an element of U that 

is the saddle point of the LagrangianÄú;ηµ, i.e., as the solution to the problem 

αv n 2µ∇¡D´v σ s ∇¡λ n·∇p nf∇¡v0 in Ωv0 on Γ (4.11) 

after that, the new approximation λ n·1 is determined as 

λ n·1P Λ´λ n·rσ s D´v (4.12) 

The following remark presents several useful formulae we need further in the 

proof of the theorem. 

∞´Ωµµd¢d (4.7) 

Remark 4.1. Considering the velocity gradient ∇v as a second-order tensor, represent 

it in the form of a sum of the symmetric strain rate tensor D´vµand the antisymmetric 

rotation velocity tensor w´vµ1 

2¢∇v ´∇vµT£with the components 

´w i j1 

2´∇ i v j ∇ j v iµµ. Then, taking into account that the contraction of the symmetric 

and antisymmetric tensors is equal to zero, we get the following relations: 

D´vµ:w´vµ0λ : w´vµ0´λλ j´vµµ 

In addition, for D´vµ:∇v and λ : ∇v we have 

i 

d 

∑ D i j´vµD j´vµD´vµ2 

D´vµ 

i 

ij1 

: 

D´vµ:∇vd ∑ D i j´vµ∇ i v jd ∑ D i j´vµ´D ij1 

ij1 

λ : ∇vd ∑ λ i j´D i j´vµ·w ij1 

j´vµµd i 

Tµ 

i j´vµ·w 

∑ λ i j D i j´vµλ ij1


Prove the theorem of convergence of <strong>Uzawa</strong>’s <strong>method</strong> for problem (2.1)–(2.3). 

Theorem 4.1. Suppose the condition 

0r4µσ s 2 (4.13) 

holds. Then for the sequencevnn0 generated by algorithm (4.10)–(4.12) we have 

lim vnv in´H 0´Ωµµd 1 

n·∞ where v is the solution nµ n 

to (4.8)–(4.9). 

Proof. Letvλbe the solution to (4.8)–(4.9). Assuming v nvn vp p n p and λ nλ n λ and taking into account that the projection operator P Λ 

satisfies the relationP Λ x P Λ 2´Ωµµd¢d yx y, as a result of subtraction of (4.8)–(4.9) 

from (4.11)–(4.12), we get that for all n0 we have 

αv n 2µ∇¡D´v σ s ∇¡λ n·∇pn0∇¡vn0 in Ωvn0 on Γ (4.14) 

n·1´L λ 2´Ωµµd¢dλ n·rσ s D´vnµ´L (4.15) 

Inequality (4.15) implies 

λ n·12´L2´Ωµµd¢dλ n2´L2´Ωµµd¢d·2rσ nD´v s´λ nµµ·r 2 σsD´v 2 nµ2´L 2´Ωµµd¢d 

which gives 

nµ nµµ 

nµ 

λ n2´L r 2 σsD´v 2 nµ2´L 2rσ s´λ nD´v 2´Ωµµd¢d 2´Ωµµd¢d (4.16) 

Multiply (4.14) by v n : 

´αv nv 2µ´∇¡D´v nµv σ nv s´∇¡λ nµ·´∇p nv nµ0 

integrate by parts, nµ and take into nµ nµ account Remark 

nµµ nµµ 

4.1: 

αv n2´L 2´Ωµµ2·2µ´D´v nµD´v nµµ·σ s´λ nD´v nµµ0 

because of 

´∇¡D´v nµv ´D´v nµ∇v ´D´v nµD´v D´v nµ2´L 2´Ωµµd¢d 

´∇¡λ nv ´λ n∇v nD´v ´λ 

For all v such that v¾´H 0´Ωµµd∇¡v0 1 we have 2D´vµ2´L because of ∇v2´L2´Ωµµd¢d 2´Ωµµd¢d λ n·12´L 

2´Ωµµd¢d

<strong>Uzawa</strong> <strong>method</strong> on semi-staggered grids 

jµ 

549 

d 

2´D´vµD´vµµ2´D´vµ∇vµ2 ∑ 

jµd i j∇ i v ∑ 

jµ 

i v j·∇ j v i∇ i v 

ij1´D ij1´∇ d 

∑ i ∇ j v iv ij1´∇ 2´Ωµµd¢d 2´Ωµµd¢d 

j∇v2´L Thus, 

nµµ 2rαv n2´L 2´Ωµµd·4rµD´v nµ2´L 2rσ nD´v s´λ (4.17) 

Taking into account relations (4.16) and (4.17), we get 

λ n2´L n·12´L λ 2´Ωµµd¢d 2´Ωµµd¢d 

r 2 σsD´v 2 nµ2´L 2´Ωµµd¢d·2rαv n2´L 2´Ωµµd·4rµD´v nµ2´L 2´Ωµµd¢d 2r´D´v nµ2´L 2´Ωµµd¢d´2µ rσs2µ·αv 2 n2´L 2´Ωµµdµ 2´Ωµµdµ 

2´Ωµµd r´∇´v nµ2´L 2´Ωµµd¢d´2µ rσs2µ·2αv 2 n2´L α 

rσ 

r∇´v nµ2´L s 

2 rσ 2 

µ2 

2µ·α2 

s s 

2µ·rσ 2 

2µv 

n2´L 2´Ωµµd¢d rσs 

2 r2 

2µ´µ∇´v nµ2´L 2´Ωµµd¢d·αv n2´L (4.18) 

Condition (4.13) implies r´2 rσs2µµ0 2 (where, as we have seen, 

ρ∆t). Relations (4.18) imply that the sequence λ n2´L2´Ωµµd¢d¡n0 decreases and 

hence converges to some limit. The latter means 

lim 

(4.19) 

n·∞ Combining (4.18) and (4.19), we get lim n·∞ v n0 in´H1´Ωµµd , namely, the convergence 

ofvnn0 to v in´H0´Ωµµd 1 . 

Remark 4.2. Estimate (4.13) determines the interval of admissible values of the 

parameter r providing the convergence of <strong>Uzawa</strong>’s algorithm (projection <strong>method</strong>) 

to the solution. But in the proof of Theorem 4.1 we neglect the term αv n2´L 2´Ωµµd 

d 

∑ 

ij1´∇ i v j∇ i v jµ·d 

∑ 

ij1´∇ j v i∇ i v jµ∇v2 

∇v2 d 

∑ 

j1∇ jd 

∑ 

i1∇ i v iv 

λ n2´L2´Ωµµd¢d 2´Ωµµd¢d¡0 λ n·12´L 

in formula (4.18). Estimate (4.13) allows us to choose a value of r both for steady 

and unsteady problems. For evolutional problems the range of admissible values of


the parameter r may be extended a little. According to the Poincaré inequality, the 

following relation holds: 

γ minv2´L 2´Ωµµd∇v2´L 2´Ωµµd¢d 

where γ min 

2´Ωµµdµ 2´Ωµµd¡ 

is the least eigenvalue of the Laplace operator with the homogeneous 

Dirichlet boundary conditions in the domain Ω. Replacing in (4.18) the value 

∇´v nµ2´L by a smaller expression containing onlyvn2´L2´Ωµµd¢d, we get 

2´Ωµµd¢d r´∇´v nµ2´L 2´Ωµµd¢d´2µ rσs2µ·2αv 2 n2´L r γ minv n2´L2´Ωµµd¢d´2µ rσs2µ·2αv 2 n2´L rv n2´L2´Ωµµd¢d2µ rσs 

2·2α 

2 

γ minγ 

Thus, the extended interval of admissible values for the parameter 

min 

r has the form 

0r4σ 2 

s´µ·αγ minµ 

5. Finite-difference scheme 

The <strong>Uzawa</strong> <strong>method</strong> described in Section 4 is applicable to the finite-difference approximation 

of variational inequality (3.1). Further we restrict ourselves to the case 

of a rectangular domain Ω. Since the <strong>Uzawa</strong> <strong>method</strong> is justified here in its functional 

form, its application to a finite-dimensional problem cannot be assumed completely 

justified. This gap will be filled in one of the forthcoming papers. As it was 

pointed out in Section 1, most papers concerning numerical simulation of a Bingham 

medium use the finite element <strong>method</strong> for discretization [14]. In [19–22] some 

difference schemes were proposed, and one of those schemes is used 

2in this paper. 

Let Ω´01µ2 1 

1 

and assume h 1N 1 

and h 2N 2 

for given natural N 1N 2 

1 

. Define 

the following grid domains: 

¯Ω 1x i jíh 1jh 2µi0N 1j0N Ω 2x i j´í·12µh 1´j·12µh 2µi0N 1 1j0N 2 

Define the spaces of the grid components of the velocity and pressure functions 

taking real values: 

V 0 hv ijú ijv ijµú´x P hp j :p´x i jµx 

i 

i jµv´x i 

i jµµx 

j¾Ω 2∑ 

j0 

i 

p i 

ij 

j¾¯Ω 1v 0jv iN 2v N1jv i00 

By V h we denote the spaces of the grid vector functions determined only at the internal 

points of Ω 1 . All components of the strain rate tensor (D h´v hµD h´v hµij)

<strong>Uzawa</strong> <strong>method</strong> on semi-staggered grids 

2 

551 

and the stress tensor (λ hλ hij) are determined on the grid Ω 2 . The corresponding 

space of real grid tensor functions is denoted by Q h : 

Q hq hq hq 11 

i j¾Ω For vector grid functions of the form v húvµT determined on ¯Ω 1 we define the 

scalar product by the formula 

1v 1µTv hú 2µT 2v 

hq 12 

hq 21 

hq 22 

h;´q hµijq h´x i jµx 

ú hv hµh∑ 

x i j¾¯Ω 1ú 1 ij u2 ij·v 1 ij v2 ijµh 1 h 2u hú 

for scalar functions determined on Ω 2 we use the formula 

´p hq hµh∑ 

x i j¾Ω 2 

p ijq ijh 1 h 2 

for tensor functions determined on Ω 2 we use the formula 

´q hλ hµh∑ 

x i j¾Ω 2´q 11 

ijλij·q 11 12 



ijλ 22 

2 ijµh 1 

h 

h 

Further in the text we use norms for vector, scalar, and tensor grid functions. All 

those norms are calculated based on the introduced scalar products: 

v hh´v hp hh´p hp hλ hh´λ hλ 

hv hµ12 

hµ12 

hµ12 

Define the discrete analogue of the differential divergence operator ∇ h¡:V 0 hP h : 

´∇ i·1j·1 u ij·1·u i·1j u ij 

h¡v hµiju ·v i·1j·1 v i·1j·v ij·1 v ij 

2h 1 

2h 2 

(5.1) 

and the strain rate tensor D h : V 0 hQ h : 

i·1j·1 u ij·1·u i·1j u ij 

2h 1 

hµµijv ´D 22 

i·1j·1 v i·1j·v ij·1 v ij 

h´v 2h 2 

 

hµµiju ´D 12 

h´v hµµij´D 21 

i·1j·1 u i·1j·u ij·1 u ij 

h´v 4h 2 

·v i·1j·1 v ij·1·v i·1j v ij 

4h 1 

A direct check verifies that the difference scheme obtained here approximates the 

original problem with the orderÇ´h2µfor h1 

smooth functions. Define a closed convex 

set Λ h and spaces Q h by 

Λ hλ hλ h¾Q hλ hλ T hλ 

´D 11 

h´v hµµiju


λ hij´´λ hµ2 11 

ij·2´λ hµ2 12 

ij·´λ hµ2 22 

ijµ12 is the norm of a tensor function at a point 

x i j¾Ω 2 

h¡ 

. 

Consider the discrete analogue of the LagrangianÄú;ηµ(4.4): 

Äh´v hλ hµh 1 h 2α 

2 

∑ú hµ2·´v hµ2¡·µ 

h¡ hµ2¡ 

∑´D 11 

hµ2·´D 22 

hµ2·2´D 12 

M i j¾Ω 1 M i j¾Ω 2 

·σ s ∑ D 11 

h λh·D 11 22 

h λh·2D 22 12 

h λ 12 

∑ fh 1 u h·f h 2 v 

M i j¾Ω 2 M i j¾Ω 1 

In order to find the saddle point of the discrete LagrangianÄh´v hλ hµ, we apply 

the discrete version of algorithm (4.10)–(4.12) in the following form: 

λ 0 h¾Λ h is given arbitrarily 

given λ n h´¾Λ hµ, n0, we compute v n n·1 

h 

, after that the new approximation λ h 

determined by the following steps. 

(5.2) 

is 

h0 

Step 1. Solve´αv n h 

µhv n·∇ h p n hσ s ∇ h¡λ n h·f h 

∇ h¡vn Step 2. Calculate 

(5.3) 

then repeat from Step 1. 

hµ´ hµ 

For the discretizations´∇ h¡v hµand´D h´v hµµintroduced here and under the 

condition ∇ h¡v h0, the following equality holds: 

2´D h´v hµD h´v hµµ´∇ h v h∇ h v hv hv 

(5.4) 

In this case we obtain the following discrete analogues of the gradient operator 

∇ h : P hV h : 

hµijp 

´∇ ij p i 1j·p ij 1 p i p 1j 1 ij p ij 1·p i 1j p i 1j 1 

h p 

2h 1 2h 2 

(5.5) 

and the Laplace operator ∆ h : V 0 hV h 

1µ 

: 

´∆ h v hµij1 

4h 2 i·1j·1 2v ij·1·v i 1j·1 

1´v ·2v i·1j 4v ij·2v i 1j·v i·1j 1 2v ij 1·v i 

1µ 

1j ·1 

i·1j·1·2v ij·1·v 4h 2 i 1j·1 

2´v 2v i·1j 4v ij 2v i 1j·v i·1j 1·2v ij 1·v i 1j 

Ifλ n·1 

h λ n hε, 

λ n·1 

hP Λh´λ n h·rσ s D h´v n hµµ


In the case h 1h 2h for ∆ h we obtain the so-called ‘shift’ approximation: 

´∆ i 1j 1·v i·1j 1·v i·1j·1·v i 1j·1 4v ij 

h v hµijv 2h 2 (5.6) 

The choice of the parameter r in algorithm (5.2), (5.3) is performed in formal accordance 

with Remark 4.2, i.e., we suppose∇ h v n hh converges to∇ h v hh for all r 

from the interval 

0r4 

σsµ·α 

2 γ min 

Before proceeding to numerical experiments, let us discuss the algorithm and 

the details of its implementation. The second step performs pointwise calculations 

and causes no difficulties. The first step consists in the solution of the generalized 

Stokes problem with a constant matrix of the system and a variable right-hand 

side. The first iterative <strong>method</strong>s for solution of the stationary Stokes problem in 

the ‘pressure-velocity’ variables were the Arrow–Hurwitz and <strong>Uzawa</strong> <strong>method</strong>s. In 

spite of the fact that more than fifty years have passed since the creation of these 

<strong>method</strong>s, they remain to be the basis for the development of new iterative <strong>method</strong>s 

[1, 5]. Consider the system of grid equations 

h0 

for the stationary Stokes problem 

∆ h v h·∇ h p hF h 

∇ h¡v If the system is not degenerate, then applying the <strong>Uzawa</strong> <strong>method</strong>, we first solve the 

equation for pressure S h p hϕ, 

1 

S h p h∇ h¡∆ h 

hµ ∇ 1 

hp h∇ h¡∆ hµ 

h 

F hϕ 

and then calculate the velocity v h , 

v h´∆ 1´F h ∇ h p 

In the case when S h is degenerate, we require the fulfillment of the condition 

ϕker∇ h . Then the problem is solvable and the normal solution should be orthogonal 

to the kernel of the discrete gradient operator. 

It is well known that for the Stokes problem the difference scheme with operators 

(5.1), (5.5), (5.6) constructed here leads to a degenerate matrix S h . In order 

to ascertain that, note that even in the two-dimensional case the discrete gradient 

operator ∇ h (5.5) has (in contrast to the continuous case) a nontrivial kernel of the 

form 

Ker´∇ hµspan´p 1p 2µp 

ij´ 1 1µi·j 

ij´ 1p 2 1µi·j·1 

1 (5.7) 

in the space P h . Therefore, the operator S h has a nontrivial kernel on P h . In this case 

one of the standard <strong>method</strong>s for solving system (5.2) is the solution of the equation 

S h p hϕ in the subspace of the space P h orthogonal to Ker´∇ hµ(5.7). However, in


the three-dimensional case the dimension of the kernel increases with a decreasing 

h [18], and such approach becomes not very efficient. Several authors [25] used 

staggered grids for calculation of the flow of a viscous incompressible fluid and in 

this case a special additional term was included into the equation for the pressure. 

This term is in some sense equivalent to the addition of a biharmonic operator into 

the continuity equation. 

In order to stabilize the scheme on semi-staggered grids, we apply the approach 

proposed recently in [3] for stabilization of LBB-unstable elements Q 1 Q 0 possessing 

an explicit analogy with this difference scheme. This approach consists in 

supplementing the Lagrangian of the Stokes problem by the term 

ÊΩ´p dx Πpµ2 

where Π is some operator of local interpolation from P h 

yµ12Ó 

into the space of piecewisebilinear 

continuous functions. 

Define the operator Π h : P hR h , where R h is the space of grid functions determined 

on ¯Ω 1 . Let x i j¾¯Ω 1 . Denote 

ω´x i jµÒx 

2´h 2 x·h 2 

kl¾Ω 2x 

j1 

kl x i 

1 

∑ 

x kl¾ω´x i jµp kl 

jµ i 

then 

´Π h p hµi jω´x h 

The operator Π h may be considered as the grid ‘interpolation’ of the function given 

on the grid Ω 2 by a function determined on the grid Ω 1 . Let Π£h 

be the operator 

conjugate to the Euclidean scalar product. Assume G hÍ h Π£h Π hµG h : P 

P h . Taking into account the stabilizing term, the continuity equation ∇¡v0 is 

approximated on the grid Ω 2 in the following way: ∇ h¡v h·G h p h0If the grid is 

uniform, then G hh2 ∆h4where p ∆ p h 

is the approximation of the Laplace operator 

with the Neumann boundary conditions having a nine-point stencil (in the twodimensional 

case): 

16½ ¼1 

16 

g 

Cv pf 

B T 

1 

8 

1 3 

8 4 

1 

16 

1 

8 

Thus, the stabilizing term has the second order in h on smooth solutions. The system 

of algebraic equations for the solution of Stokes problem (5.2) takes the following 

form for the constructed scheme: 

A B 

The matrix is sparse and has a block structure. The mathematical analysis of this 

scheme is the subject of a separate paper [23]. Note that the proposed stabilization 

is applicable to the three-dimensional case too. 

1 

16 

1 

8 

1

6. Numerical experiments 


The first step of algorithm (5.2) consists in the solution of the Stokes problem. 

Therefore, at first we present the results of numerical experiments for known analytic 

solutions in the square´01µ2 . Taking into account that the kernel of the discrete 

gradient operator is nontrivial, we compare the two <strong>method</strong>s described above, 

namely, determination of the solution on the subspace orthogonal to the kernel and 

the use of the scheme with the stabilizing term. 

6.1. Trigonometric polynomial (viscous fluid) 

Let us consider the solution: 

2πy 2πyµ 

u1 

4π 2´1 cos2πxµsin v1 

4π 2 sin2πx´1 cos (6.1) 

p1 

π sin2πxsin 

The calculation results are presented in Tables 1 and 2. Table 1 corresponds to the 

orthogonalization <strong>method</strong>, Table 2 corresponds to the stabilized scheme. The tables 

present the values of the error norms for the velocity e hvΩ 1 

v h and pressure 

r hpΩ 2 

p h in the grid analogue of the L 2 -norm. The last column shows the number 

of iterations in the <strong>Uzawa</strong>–conjugate gradient <strong>method</strong> necessary for decreasing 

the residual norm up to ε10 8 (the left column) and ε10 10 (the right column). 

In the hope to obtain a more precise solution, the stop criterion of the conjugate gradient 

<strong>method</strong> was changed up the the value ε10 10 . However, this has increased 

the number of iterations, but has not improved the accuracy of the calculations. The 

data presented in the tables correspond 

1µ 

to both cases 

1µ 

(eight significant digits coincide). 

Thus, the accuracy of 10 10 is excessive. 

1µ 

6.2. The ‘vortex’ in the cavity (viscous fluid) 

Let us consider the solution: 

cos2πéR 1x 

u1 

e R 1 1sin2πéR 2y 2 e 

e R 2 1R R 2y 

2πé R 2 

e R 1x 

1µ 

1 1µ 

1µ 

1 

cos2πé 2πé R (6.2) 

1 

pR 1 R 2 sin2πéR 1x 

e R 1 1sin2πéR 2y 

e R1x e R 2y 

e R é 2 R 1µé 1 

1µ 

vsin2πéR 1x 

e R 11 

1 

1µ 

R 2y 

1R 

e R 2 

1µ R 2


Table 1. 

Convergence of the difference solution and the number of iterations, example (6.1), orthogonalization. 

h 

e hL 2 

e hL 2 

e 2hL 2 

log 2e hL 2 

e 2hL 2 

r hL 2 

r hL 2 

r 2hL 2 

log 2r hL 2 

r 2hL 2 

#it #it 

132 24860¢10 4 40116 20042 12929¢10 3 40130 20047 11 13 

164 61969¢10 5 40029 20010 32218¢10 4 40033 20012 12 14 

1128 15481¢10 5 40007 20002 80479¢10 5 40008 20003 13 15 

1256 38695¢10 6 40003 20001 20115¢10 5 40000 20000 13 15 

1512 96730¢10 7 50288¢10 6 13 16 

Table 2. 

Convergence of the difference solution and the number of iterations, example (6.1), stabilization. 

h 

e hL 2 

e hL 2 

e 2hL 2 

log 2e hL 2 

e 2hL 2 

r hL 2 

r hL 2 

r 2hL 2 

log 2r hL 2 

r 2hL 2 

#it #it 

132 29768¢10 4 39732 19903 19277¢10 3 29239 15479 13 17 

164 74924¢10 5 39857 19948 65928¢10 4 28978 15350 13 17 

1128 18798¢10 5 39928 19974 22751¢10 4 28696 15208 13 17 

1256 47081¢10 6 39964 19987 79283¢10 5 28509 15114 14 17 

1512 11781¢10 6 27810¢10 5 14 17 

The solution (6.2) imitates a ‘vortex’ in a cavity, the center of the vortex is at the 

point with the coordinates x 01R 1 logéxp´R y 01R 2 logéxp´R 2µ·1µ2µ0512. Thus, the solution 

1µ·1µ2µ0842 

has a boundary layer 

near the right boundary of the domain (see [2]). Figure 1 illustrates the distribution 

of the velocity vector (left) and the pressure field (right). The calculation results for 

R 142985R 201 are presented in Tables 3 and 4. Table 3 contains the calculation 

results for the <strong>method</strong> of orthogonalization, Table 4 contains the results for 

the stabilized scheme. Since large gradients of velocity and pressure appear in the 

boundary layer, the number of iterations necessary for the convergence is greater 

than in the previous example. As in the first example, the solution obtained with 

ε10 8 coincides with the solution obtained for the same approximate problem 

with ε10 10 up to eight digits after the decimal point. 

The results presented in Tables 1– 4 demonstrate the second order of convergence 

for velocity and the same order for pressure using orthogonalization, 

and slightly worse using stabilization. Moreover, the number of iterations for the 

<strong>Uzawa</strong>–CG <strong>method</strong> with the stabilized scheme practically does not depend on the 

mesh size. The results of both <strong>method</strong>s are close; in the case of orthogonalization 

they are slightly better. Unfortunately, in the three-dimensional case the <strong>method</strong> of 

orthogonalization is not efficient because of the growth of the kernel dimension, but 

the stabilization <strong>method</strong> remains applicable.


1 

velocity field 

0.8 

4 

y 

0.6 

z 

2 

0 

0.4 

−2 

0.2 

−4 

1 

0.5 

0.5 

1 

0 0.2 0.4 0.6 0.8 1 

x 

Figure 1. The velocity vector (left) and pressure (right) for example (6.2). 

y 

0 

x 

Table 3. 

Convergence of the difference solution and the number of iterations, example (6.2), orthogonalization. 

h 

e hL 2 

e hL 2 

e 2hL 2 

log 2e hL 2 

e 2hL 2 

r hL 2 

r hL 2 

r 2hL 2 

log 2r hL 2 

r 2hL 2 

#it #it 

132 73731¢10 3 41343 20476 13453¢10 1 43257 21129 14 18 

164 17834¢10 3 40346 20124 31101¢10 2 40813 20290 14 19 

1128 44203¢10 4 40087 20031 76204¢10 3 40203 20073 14 19 

1256 11027¢10 4 40022 20008 18955¢10 3 40051 20018 15 19 

1512 27552¢10 5 47327¢10 4 15 19 

Table 4. 

Convergence of the difference solution and the number of iterations, example (6.2), stabilization. 

h 

e hL 2 

e hL 2 

e 2hL 2 

log 2e hL 2 

e 2hL 2 

r hL 2 

r hL 2 

r 2hL 2 

log 2r hL 2 

r 2hL 2 

#it #it 

132 82125¢10 3 40081 20029 14631¢10 1 34115 17704 20 23 

164 20490¢10 3 39559 19840 42889¢10 2 30684 16175 20 24 

1128 51796¢10 4 39663 19878 13977¢10 2 29778 15742 21 24 

1256 13059¢10 4 39808 19931 46940¢10 3 29234 15476 21 24 

1512 32805¢10 5 16057¢10 3 21 24


Figure 2. The velocity profile for example (6.3) with σ s02. 

6.3. Flow between 

v 

plates (viscoplastic medium) 

There are few analytic solutions for the models of a viscoplastic medium. Consider 

one of them, a flow 

 

between fixed 

sµ2 

plates (the plane Poiseuille flow): 

u0p 

1 

8´1 2σ ´1 2σ s 2xµ2℄0xx 1 

1 

2σ x 1xx 2 (6.3) 

8´1 1 

8´1 2σ sµ2 y·1 

2 

where 

s x 11 

σ sx 21 

(6.4) 

2 2·σ If σ s0, then we have not a viscoplastic medium, but a viscous liquid possessing 

a parabolic velocity profile. In the range 0σ s12 the velocity profile has the 

following form: the interval x¾x 1x 2℄corresponds to the rigid zone whose points 

all move at the same velocity´1 2σ sµ28, in the lateral intervals x¾0x 1µand 

x¾´x 21℄adeformed medium moves. For σ s02 the velocity profile is presented 

in Fig. 2. The greater σ s , the wider is the rigid zone and the lower is the velocity. 

For σ s12 we have v0, i.e., the whole domain forms a rigid fixed zone. 

The dependence of the number of external iterations necessary for the convergence 

of <strong>Uzawa</strong> <strong>method</strong> (5.2)–(5.3) for µ1σ s02, is presented in Table 5. 

According to estimate (4.13), the sufficient condition for the convergence of the algorithm 

is 0r4µσ s 2 . The first row presents the number of iterations in the 

calculation on a uniform grid with the mesh size 140, the second row corresponds 

to the size 164. For r4µσ s100 2 the <strong>method</strong> converges, and in this case the 

value r999 requires 1024 and 1385 iterations, respectively. For r100 (and 

1µ2℄1 

´2x 2σ s sx1 

2·σ


Table 5. 

The dependence of the number of iterations on the parameter r, example (6.3) 

with σ s02. 

h 1 10 20 30 40 50 60 70 80 90 99 99.5 99.9 

140 331 164 106 80 65 55 48 43 39 36 101 204 1024 

164 306 190 125 97 80 69 61 56 317 325 332 351 1385 

Table 6. 

The convergence of the difference solution and the number of iterations, example (6.3). 

h 

e hL 2 

e hL 2 

e 2hL 2 

log 2e hL 2 

e 2hL 2 

#it 

e hL 2 

e hL 2 

e 2hL 2 

log 2e hL 2 

e 2hL 2 

120 6079¢10 6 42289 20803 28 1668¢10 5 39311 19749 52 

140 1437¢10 6 34208 17743 48 4244¢10 6 36274 18589 98 

180 4202¢10 7 27754 14727 88 1170¢10 6 26495 14057 191 

1160 1514¢10 7 165 4416¢10 7 346 

#it 

greater) we get divergence. Thus, restriction (4.13) related to the continuous problem 

is confirmed in numerical calculations. As can be seen, for small r the convergence 

is sufficiently slow, and for an r close to the right boundary of the interval 

it also becomes slower. The best convergence is observed for the values from the 

middle of the interval. Under a fixed r, the number of iterations increases for a finer 

grid. 

Table 6 presents the values of the velocity error norm e hvΩ 1 

v h in the grid 

analogue of the L 2 -norm for example (6.3). The left half of the table corresponds to 

the yield stress σ s02, r60, the right one corresponds to σ s03, r20. The 

Stokes problem in internal iterations was solved by the orthogonalization <strong>method</strong>. 

The number of external iterations in the <strong>Uzawa</strong> <strong>method</strong> corresponds to the stopping 

criterion ε10 3 . 

6.4. Driven cavity problem (viscoplastic medium) 

There are two test problems for numerical algorithms solving viscoplastic problems: 

a flow in a pipe (the Western authors sometimes refer to it as Mosolov’s problem) 

and a flow in a cavity. The first problem is simpler; its equivalent problem for a viscous 

fluid is reduced to the solution of the Poisson equation and is of great interest 

for mechanics. Therefore, it is solved especially often [17, 19, 21, 24]. The second 

problem is the best known test in computational hydrodynamics for the Stokes 

problem. It also becomes the test for a Bingham medium [6, 7, 10, 15, 22, 27] and 

consists in the following: let Ω´01µ2 , f0, the boundary conditions be given in


‖v‖ L2 

0.2 

0.15 

5 

3 

2 

τ s 

= 1 

0.1 

0.05 

0 0.02 0.04 0.06 0.08 

t 

Figure 3. The velocity norm 

B´xµ:´0 

and stop time. 

the following way: 

B 

x¾ΓÒΓ B 

v 

16´x2 1´1 x 1µ2µ0x¾Γ where Γ Bxx´x 2µ0x 

1x 11x 21is the moving upper boundary. This 

choice of the nonzero horizontal velocity component corresponds to the so-called 

regularized driven cavity problem. 

Unsteady flows for viscoplastic media have been little studied, only one-dimensional 

problems were usually considered [4]. In recent papers [20, 21], numerical 

simulations of the unsteady flow of a Bingham medium were performed for channels 

of various cross-sections. An unsteady problem of the flow in a cavity was 

considered in [6, 7] based on the splitting <strong>method</strong> [13] with the use of FEM. In [7], 

rigid zones were obtained for the steady driven cavity problem, and the graph of the 

dependence of the velocity vector norm on time was presented for the process of the 

start-up of the flow (reaching the stationary state) and subsequent cessation of the 

flow. Let us describe these modes in more detail: 

B 

(1) Start-up of the flow: starting from the rest (v´0µ0) under the action of 

the instantaneously applied motion of the ‘cover’ of the cavity (v´tµΓv ) the 

medium accelerates in the course of the time interval´0t 1℄, and the flow actually 

reaches the steady state; 

(2) Cessation of the flow: from the state reached by the time moment tt 1 the 

medium comes to a halt because the motion of the cover is ‘blocked’: for tt 1 we 

have v B´xµ0. The important qualitative peculiarity of the problems concerning 

the unsteady flows of viscoplastic media is the finiteness of the damping time of the 

motion in the absence of external forces [6]. This is a fundamental distinction from 

the corresponding flow of a viscous liquid, which damps exponentially in infinitely 

long time.


(a) 

(b) 

(c) 

(d) 

s 

Figure 4. Development of rigid zones for τ s1: (a) t0, (b) t0004, (c) t0008, (d) t001. 

The graphs in Fig. 3 illustrate the dependence of the velocity vector normv hh 

on time. The results are presented for τ s1´t 1006µτ s2´t 10055µτ 3´t 1005µτ s5´t 10045µ. It is easy to see that the medium stops in a finite 

time interval. The curve for τ s1 is identical to Fig. 11 from [7]. 

It was noted in a classic monograph (see [10]) that ‘naturally, the problem on determination 

of rigid zones is of most interest in numerical experiments’. Figures 4a– 

4d illustrate the evolution of rigid zones in time for the stopping problem. The results 

are presented for µ1ρ1τ s1´σ sÔ2µ. The results were obtained on 

a grid with the mesh size h1128 and the convergence criterion ε10 4 . Rigid 

zones were determined as isolinesτ hσ s (the von Mises criterion is applied to the 

discrete field τ h ). Figure 4a corresponds to the steady flow of a viscoplastic medium 

in a cavity and it is in a good agreement with the previous papers [6, 7, 15, 22, 27]. 

Three rigid zones are present in this flow at the initial moment corresponding to the 

steady state. One of them is located near the center of the vortex and two others lie 

in the bottom part of the cavity. The bottom rigid zones gradually merge (Fig. 4b) 

and increase in size (Fig. 4c). The central rigid zone sinks and also increases. In the 

course of time (sufficiently soon) three additional rigid zone appear in the upper corners 

and above the central rigid zone. Some symmetry is quite natural here, because


the upper boundary of the domain becomes fixed. Further increase and merging of 

rigid zones progresses. Figure 4d illustrates the situation shortly before the complete 

stop of the flow: only a thin band of the moving medium remains between the outer 

(adjacent to the boundary) and the inner rigid zones. The appearance of additional 

rigid zones (beside those existing in the steady state) and their further merging is 

the typical peculiarity of unsteady flows of viscoplastic media. A similar pattern 

was observed when the flow in a channel was stopped [21]. 

7. Conclusion 

In this paper we justify at the functional level the application of the <strong>Uzawa</strong> <strong>method</strong> 

for the solution of unsteady viscoplastic problems (without convective terms). Estimates 

of the convergence interval are obtained for the iterative parameter of the 

<strong>Uzawa</strong> <strong>method</strong>. In this case the restriction relating to the continuous problem practically 

coincides with the convergence interval obtained numerically in the finitedimensional 

problem. The <strong>method</strong> has been implemented in a rectangular domain 

by a difference scheme on semi-staggered grids. Numerical experiments confirm 

the efficiency of the proposed approach. The evolution of rigid zones for a plane 

unsteady viscoplastic problem has been obtained (to best of our knowledge) for the 

first time and is of great interest for applied problems. In future we expect to present 

a more detailed justification of the application of difference approximations and the 

convergence of the algorithm in the finite-dimensional case, and also to extend the 

results to the three-dimensional unsteady case. 

Acknowledgement 

The authors are sincerely grateful to V. I. Agoshkov for useful comments. 

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