Dual-Mixed Finite Element Approximation of Stokes and Nonlinear Stokes Problems Using Trace-free Velocity Gradients

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1 Dual-Mixed Finite Element Approximation of Stokes and Nonlinear Stokes Problems Using Trace-free Velocity Gradients Jason S. Howell a,1 a Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, , USA. Abstract In this work a finite element method for a dual-mixed approximation of Stokes and nonlinear Stokes problems is studied. The dual-mixed structure, which yields a twofold saddle point problem, arises in a formulation of this problem through the introduction of unknown variables with relevant physical meaning. The method approximates the velocity, its gradient, and the total stress tensor, but avoids the explicit computation of the pressure, which can be recovered through a simple post-processing technique. This method improves an existing approach for these problems and uses Raviart-Thomas elements and discontinuous piecewise polynomials for approximating the unknowns. Existence, uniqueness, and error results for the method are given, and different approaches for solving the linear systems that arise in computations are discussed. Key words: Stokes problem, nonlinear Stokes problem, dual-mixed method, finite element method, twofold saddle point problem, Raviart-Thomas, pseudostress. 1. Introduction In this article a dual-mixed formulation and corresponding finite element approximation of steady Stokes and nonlinear Stokes problems is studied. The nonlinear Stokes problem arises in modeling flows of, for example, biological fluids, lubricants, paints, polymeric fluids, where the fluid viscosity is assumed to be a nonlinear function of the fluid s velocity gradient tensor. A nonlinear Stokes problem is given by: Find (u, p) such that (ν( u ) u) + p = f in, (1.1) u = 0 in, (1.2) u = u Γ on Γ, (1.3) where is a bounded open subset of R n with Lipschitz continuous boundary Γ. The fluid velocity is denoted by u, and u := ( u) i j = u i / x j is the tensor gradient of u. Here and throughout the paper the following notation is used: for tensors σ = (σ i j ), τ = (τ i j ), σ : τ = i, j σ i j τ i j, σ 2 = σ : σ. The pressure is denoted by p, and f describes the external forces on the fluid. The function ν describes the nonlinear kinematic viscosity of the fluid. addresses: howell4@andrew.cmu.edu (Jason S. Howell) 1 This material is based upon work supported by the Center for Nonlinear Analysis (CNA) under the National Science Foundation Grant No. DMS February 10, 2009

2 Some classical examples of ν are given by: Power Law: ν( d(u) ) = ν 0 d(u) r 2, ν 0 > 0, 1 < r < 2, (1.4) where d(u) = 1 2 ( u + ut ) denotes the fluid deformation tensor. The power law model has been used to model the viscosity of many polymeric solutions and melts over a considerable range of shear rates [1]. Ladyzhenskaya Law[2]: ν( u ) = (ν 0 + ν 1 u ) r 2, ν 0 0, ν 1 > 0, r > 1, (1.5) which has been used in modeling fluids with large stresses. Carreau Law: ν( d(u) ) = ν 0 ( 1 + d(u) 2 ) (r 2)/2, ν0 > 0, r 1, (1.6) used in modeling visco-plastic flows and creeping flow of metals. The traditional linear Stokes problem is recovered from (1.1) (1.3) with the Ladyzhenskaya law if ν 0 > 0 and ν 1 = 0. In this case, the parameter r is taken to be 2. General descriptions of the conservation of momentum equation (1.1) are often written in terms of the tensor σ = ν( u ) u: σ + p = f in. (1.7) The work in this paper is based on the approximation methods presented in [3] by Gatica, González, and Meddahi, and [4] by Ervin, Howell, and Stanculescu. Their approach extends the investigations of [5, 6]. Other works on finite element approximation of nonlinear Stokes problems, including a priori and a posteriori error estimation, include [7, 8, 9]. Investigations by Gatica in [10] and Gatica, Heuer, and Meddahi in [11] provided a general theory for solvability and Galerkin approximations of a class of nonlinear twofold saddle point problems posed in Hilbert spaces. In [3], Gatica, González, and Meddahi reformulated the modeling equations for a nonlinear Stokes flow as a twofold saddle point problem, using the tensor ψ in place of σ (ψ = σ pi) and introducing an additional variable for u. In doing so, their formulation used the constitutive equation for σ as a function of u and reduced the regularity requirement for the velocity. Advantages of this approach include: (i) more flexibility in choosing the approximating finite element space for u, (ii) Dirichlet boundary conditions for u become natural boundary conditions and are easily incorporated into the variational formulations, (iii) avoids the assumption of expressing u was a function of σ. A disadvantage in this formulation is that additional unknowns are introduced. The analysis of this approach was studied in a Hilbert space setting in [3]. In [4], Ervin, Howell, and Stanculescu extended the results in [3] to include the setting where the functional (Sobolev) spaces involved were defined based on the nonlinearity found in (1.7), as well as showing that the method could be extended to higher-order approximations. A priori error estimates that were dependent on the value of r were derived. The purpose of this work is to reformulate the variational problem that arises from the dualmixed approach in [3] and [4]. This is accomplished by restricting the function space for the velocity gradient to that of trace-free tensors. In this way, the pressure unknown is eliminated from the variational formulation, resulting in a smaller system of equations to be solved. Upon computing a finite element approximation to the variational solution, an accurate approximation to the pressure can be recovered via a postprocessing calculation. 2

3 There are also several interesting aspects of the linear systems that arise in this approximation method. While the inclusion of a scalar Lagrange multiplier will often spoil the natural banded structure of a system arising from finite element discretization, a bordering algorithm can be applied to preserve the desired banded structure. These twofold saddle point problems also yield linear systems that can be structured to satisfy the requirements of an Uzawa algorithm through the use of a penalty method or fixed-point iteration. In the lowest-order case, this approach results in an algorithm for which the block structure of the approximating system can be exploited on any computational mesh. The difficulty in using symmetric tensor finite elements in Stokes and elasticity problems is well-documented [12, 13]. In [14], the pseudostress was introduced as an additional variable and used in the modeling equations instead of the symmetric stress tensor. This approach allows for the use of nonsymmetric tensor finite elements and the symmetric stress can be computed easily via a postprocessing calculation. Least-squares finite element methods for the steady incompressible Stokes problem have been studied [15, 16]. Many of these methods are based on twoor three-field first-order partial differential systems: (i) velocity, vorticity, and pressure [15, 16], (ii) velocity, pressure, and stress [17], (iii) velocity, velocity gradient, and pressure [18], (iv) velocity, velocity gradient, and pressure with additional constraints [18], (v) constrained velocity gradient and pressure [19], and velocity and stress [14]. If pure Dirichlet boundary conditions are assumed, then a two-field formulation in velocity and pseudostress [14, 20] can be described. The method presented here represents an addition to the list of first-order approximation schemes for the incompressible Stokes problem in three variables as it approximates the velocity, velocity gradient, and pseudostress, with a constraint on the mean trace of the pseudostress (this constraint effectively forces the pressure to have zero mean in ). Application of leastsquares methods and two-variable formulations of nonlinear Stokes problems will be discussed subsequent work. A description of the notation used in this paper, the mathematical problem, and the original dual-mixed variational formulation is given in Section 2. The modified variational formulation is presented in Section 3, along with results that establish the solvability of the continuous problem. The discrete form of the modified problem and its solvability and error estimation is described in Section 4. In Section 5, the structure of linear systems arising in the computation of solutions to the problem is discussed, and an implementation of the Uzawa algorithm is presented. Numerical results are given in Section Mathematical Setting In this section, as described in [4], the notation, assumptions, and dual-mixed variational formulation of the nonlinear Stokes problem (1.1) (1.3) are presented. Throughout the remainder of this paper the case where 1 < r 2 is considered, with r denoting the unitary conjugate of r, satisfying 1/r + 1/r = 1. Used in the analysis below are the following function spaces and norms: T := (L r ()) n n = { τ = (τ i j ) τ i j L r (), i, j = 1,..., n }, with norm τ T := ( τ r d ) 1/r ; T := ( L r () ) n n and T := { τ T τ ( L r () ) n }, 3

4 with norm τ T := ( ( τ r + τ r ) d ) 1/r ; U := (L r ()) n, and P := L r (). Norms will be denoted by either X for a given function space X or by m,r, for the Sobolev space W m,r (). The infinity norm will be denoted by. For a Banach space X, X denotes its dual space with associated norm X. Note that T = T, and ( T ) = T Derivation of the Variational Formulation Motivated by (1.4) (1.6), assume that the extra stress tensor is a function of the velocity gradient, i.e. σ := g( u) = ν( u ) u. (2.1) Specifically, assume that if 1 < r < 2, A1: g : T T is a bounded, continuous, strictly monotone operator [21]; and that there exist constants Ĉ 1 and Ĉ 2 such that, for s, t, w T, A2: (g(s) g(t)) : (s t) d Ĉ 1 g(s) g(t) s t d + s 2 r T s t 2 T + t 2 r T, (2.2) 2 r s t ( ) 1 r r A3: (g(s) g(t)) : w d Ĉ 2 s + t g(s) g(t) s t d w T, (2.3) with the convention that g(s) = 0 if s = 0 and s(x) t(x) /( s(x) + t(x) ) = 0 if s(x) = t(x) = 0. Properties A1 A3 have been established for power law and Carreau law fluids [22]. (For the case of a power law fluid monotonicity is also shown in [23, 24].) For Ladyzhenskaya law fluids, the analysis in [23] is easily extended to show that A1 A3 hold. For ease of notation, let E(s, t) = s t s + t 2 r r, (2.4) and note that E(s, t) 1 for all s, t T. It is also shown in [23] that there is a constant Ĉ 3 > 0 such that, for all s, t T, g(s) g(t) s t d Ĉ 3 (g(s) g(t)) : (s t) d. (2.5) For the traditional Stokes problem r = 2, the linear operator g(s) = ν 0 s is clearly continuous and coercive. Remark 2.1. From (1.2) it follows that u Γ must satisfy the compatibility condition u Γ n dγ = 0, where n denotes the outward pointing unit normal vector to. Γ The dual-mixed formulation is obtained by first introducing the variables φ and ψ and using the characterization of the momentum equation (1.7): φ := u, (2.6) ψ := σ pi, the total stress tensor, (2.7) = g(φ) pi, using (2.1). (2.8) 4

5 With the definition of ψ a variational form for (1.1) can be written as v ψ d = v f d, for v T. (2.9) Note that from the definition of φ, for τ T, 0 = φ : τ d + u : τ d = φ : τ d + (τ n) u Γ dγ Γ u τ d (2.10) where the integral over Γ is the duality pairing of (W 1/r,r (Γ)) n and (W 1 1/r,r (Γ)) n with respect to the (L 2 ()) n inner product. The incompressibility condition u = 0 is equivalent to tr(φ) = 0, (2.11) where tr(φ) denotes the trace of φ. Combining (1.4), (2.10), and (2.9) a variational formulation to (1.4), (2.10), and (2.9) is: Given f ( L r () ) n, uγ ( W 1 1/r, r (Γ) ) n, determine (φ, ψ, p, u) T T P U such that g(φ) : ς d ψ : ς d p tr(ς) d = 0, ς T, (2.12) τ : φ d q tr(φ) d u τ d = (τ n) u Γ dγ, Γ (τ, q) T P, (2.13) v ψ d = v f d, v U. (2.14) Note that equations (2.12)-(2.14) do not uniquely define a solution; as adding (0, ci, c, 0) to a solution (φ, ψ, p, u), also satisfies (2.12)-(2.14) for any c R. In order to guarantee uniqueness, the method proceeds as in [25, 12, 3, 4] and impose, via a Lagrange multiplier, the constraint tr(ψ) d = 0. The variational formulation may then be restated as: Given f ( L r () ) n, u Γ ( W 1 1/r, r (Γ) ) n, determine (φ, ψ, p, u, λ) T T P U R such that g(φ) : ς d ψ : ς d p tr(ς) d = 0, ς T, (2.15) τ : φ d q tr(φ) d u τ d + λ tr(τ) d = (τ n) u Γ dγ, (τ, q) T P, (2.16) Γ v ψ d + η tr(ψ) d = v f d, (v, η) U R. (2.17) Remark 2.2. As commented in [3] and [4], the value of the Lagrange multiplier λ is 0, as can be seen from the choice of τ = I and q = 1. However, it is included in the variational formulation so that the formulation has a twofold saddle point structure. 5

6 To formally rewrite (2.15)-(2.17) as a twofold saddle point problem define the following operators: A : T T, B : T (T P), C : T P (U R). [A(φ), ς] := g(φ) : ς d, (2.18) [B(φ), (τ, q)] := τ : φ d q tr(φ) d, (2.19) [C(ψ, p), (v, η)] := v ψ d + η tr(ψ) d. (2.20) The modeling equations can then be written in the form [A(φ), ς] + [ς, B (ψ, p)] = 0, ς T, (2.21) [B(φ), (τ, q)] + [(τ, q), C (u, λ)] = (τ n) u Γ dγ, (τ, q) T P, (2.22) Γ [C(ψ, p), (v, η)] = v f d, (v, η) U R, (2.23) where B and C denote the respective adjoint operators of B and C, respectively Solvability of the Continuous and Discrete Variational Formulations Results regarding the existence and uniqueness of solutions to (2.21)-(2.23) are now reviewed. Define the kernel of the C operator Z 1 := { (τ, q) T P : [C(τ, q), (v, η)] = 0, (v, η) U R}, } = {(τ, q) T P : τ = 0 in, and tr(τ) d = 0. (2.24) The following result provides sufficient conditions for the solvability of the variational problem when 1 < r < 2. For the case r = 2, the reader is referred to [3]. Theorem 2.1. Assume that: (i) A defines a bounded, continuous, strictly monotone operator on a reflexive Banach space. (ii) There exists a constant β B > 0 such that (iii) There exists a constant β C > 0 such that inf [B(φ), (τ, q)] inf sup β B. (2.25) (τ,q) Z 1 φ φ T T (τ, q) T P sup (u,λ) U R (τ,q) T P [C(τ, q), (u, λ)] (τ, q) T P (u, λ) U R β C. (2.26) Then, for f ( L r () ) n and uγ ( W 1 1/r, r (Γ) ) n, there exists a unique solution (φ, ψ, p, u, λ) T T P U R satisfying (2.21) (2.23). In addition, the solution satisfies φ T + u U + λ C ( u Γ 1 1/r,r,Γ + f r /r 0,r,), (2.27) ψ T + p P C ( u Γ 1/r 1 1/r,r,Γ + f 0,r, + f 1/r 0,r,), (2.28) for some constant C > 0. 6

7 The proof of Theorem 2.1 for the general Sobolev case (1 < r < 2) is given in [4], and proof of the Hilbert space case is given in [10]. The inf-sup conditions (2.25) and (2.26) required by Theorem 2.1 for the variational problem (2.21) (2.23) are shown to hold in [4] and [3]. The existence, uniqueness, and abstract error estimates for the corresponding discrete formulation to (2.21) (2.23) are also shown in [4] and [3], and a finite element approximation using Raviart-Thomas and discontinuous piecewise polynomial elements is presented in both cases. The finite element approximation discussed in this work is very similar, thus further discussion of the existing variational approach will be not be discussed here, the reader should refer to [4] and [3] for further information. 3. The Modified Variational Formulation In this section a modified version of (2.21) (2.23) is discussed. Define the subspace T 0 T of trace-free tensors in (L r ()) n n by n T 0 = {ς T tr(ς) = 0} = ς T ς ii = 0. If φ T 0, then the requirement tr(φ) = 0 that arises from the incompressibility condition (2.11) is enforced in a strong sense. In addition, if (2.15) is tested only against functions ς from T 0, then p tr(ς) d = 0 for all ς T 0, which effectively removes the pressure p from the full variational formulation. Thus the modified variational problem is given by: Given f ( L r () ) n, u Γ ( W 1 1/r, r (Γ) ) n, determine (φ, ψ, u, λ) T 0 T U R such that g(φ) : ς d ψ : ς d = 0, ς T 0, (3.1) τ : φ d u τ d + λ tr(τ) d = (τ n) u Γ dγ, τ T, (3.2) Γ v ψ d + η tr(ψ) d = v f d, (v, η) U R. (3.3) Define the modified functionals à : T 0 T, B : T 0 (T ), C : T (U R). i=1 by [Ã(φ), ς] := g(φ) : ς d, (3.4) [ B(φ), τ] := τ : φ d, (3.5) [ C(ψ), (v, η)] := v ψ d + η tr(ψ) d. (3.6) 7

8 Thus (3.1) (3.3) can be written as the twofold saddle point problem [Ã(φ), ς] + [ς, B (ψ)] = 0, ς T 0, (3.7) [ B(φ), τ] + [τ, C (u, λ)] = (τ n) u Γ dγ, τ T, (3.8) Γ [ C(ψ), (v, η)] = v f d, (v, η) U R. (3.9) With the definition of the kernel Z 1 of C written without p, the analogue to Theorem 2.1 for this variational problem is stated as: Theorem 3.1. Assume that: (i) Ã defines a bounded, continuous, strictly monotone operator on a reflexive Banach space. (ii) There exists a constant β B > 0 such that [ B(φ), τ] inf sup φ φ T 0 T τ T τ Z 1 β B. (3.10) (iii) There exists a constant β C > 0 such that inf sup (u,λ) U R τ T [ C(τ), (u, λ)] β C. (3.11) (u, λ) U R τ T Then, for f ( L r () ) n and uγ ( W 1 1/r, r (Γ) ) n, there exists a unique solution (φ, ψ, u, λ) T 0 T U R satisfying (3.7) (3.9). In addition, the solution satisfies for some constant C > 0. φ T + u U + λ C ( u Γ 1 1/r,r,Γ + f r /r 0,r,), (3.12) ψ T C ( u Γ 1/r 1 1/r,r,Γ + f 0,r, + f 1/r 0,r,), (3.13) Proof. The proof is nearly identical to that of Theorem 3.1 of [4] and is omitted for brevity The Inf-Sup Conditions for B and C In this section the conditions (3.10) and (3.11) are shown to hold. Note that, from the definition of C, Z 1 := { τ T [ C(τ), (v, η)] = 0, (v, η) U R }, { } = τ T τ = 0 in, and tr(τ) d = 0. (3.14) To show the inf-sup condition for B, a preliminary result is required. This is Lemma 3.1 of [25] when r = r = 2 and Lemma 3.1 of [4] when 1 < r < 2, the corresponding proofs can be found in those works. 8

9 Lemma 3.1. For τ T satisfying tr(τ) d = 0, let τ0 = τ 1 ntr(τ)i. Then, there exists C, depending only, such that Lemma 3.2. There exists a constant β B > 0 such that τ L r C ( τ 0 L r + τ W 1,r ). (3.15) [ B(φ), τ] inf sup φ φ T 0 T τ T τ Z 1 β B. Proof. The proof is similar to those found in [4] and [3]. Let τ 0 = τ 1 n tr(τ)i, and φ = τ0 r /r 1 τ 0 / τ 0 r 1 T. (3.16) Then φ T 0, and φ T = 1. Then, [ B(φ), τ] τ 0 r /r 1 = τ : τ 0 d φ T τ 0 r 1 T 1 = τ 0 r /r 1 τ 0 : τ 0 d, (as τ : τ 0 = τ 0 : τ 0 ) τ 0 r 1 T 1 = τ 0 r τ 0 r 1 T T = τ 0 T 1 C τ T = 1 C τ T, (3.17) as τ Z 1 (see (3.15)). To show the inf-sup condition for C, two lemmas are necessary. Lemma 3.3. Let 0 T := { τ T : tr(τ) d = 0}. Then, there exists C > 0 such that for any u U u ˆτ d u τ d sup C sup. (3.18) ˆτ T τ T ˆτ 0 T ˆτ0 τ T τ0 Proof. See [4] for the case 1 < r < 2 and [3] for the case when r = 2. Lemma 3.4. Given w (L r ()) 2, there exists τ T such that τ = w in, and τ T C w L r (). (3.19) Proof. The proof is a straightforward extension of the results in [26], pg Lemma 3.5. There exists a constant β C > 0 such that inf sup (u,λ) U R τ T [ C(τ), (u, λ)] β C. (3.20) (u, λ) U R τ T Proof. See [4] for the case 1 < r < 2 and [3] for the case when r = 2. Lemmas 3.2 and 3.5, together with the fact that à : To (To) represents a bounded, continuous, strictly monotone operator on a reflexive Banach space, imply through Theorem 3.1 that there is a unique solution (φ, ψ, u, λ) T 0 T U R satisfying (3.7) (3.9). 9

10 4. The Discrete Variational Problem and Finite Element Approximation Let R n be a polygonal domain and let T h be a triangulation of into triangles (n = 2) or tetrahedrals (n = 3). Thus = K, K T h, and assume that there exist constants γ 1, γ 2 such that γ 1 h h K γ 2 ρ K (4.1) where h K is the diameter of triangle (tetrahedral) K, ρ K is the diameter of the greatest ball (sphere) included in K, and h = max K Th h K. Define the finite-dimensional subspaces T 0 h T 0, T, h T, and U h U. Then the discrete formulation of (3.1)-(3.3) is defined as: [Ã(φ h ), ς h ] + [ς h, B (ψ h )] = 0, ς h T 0 h, (4.2) [ B(φ h ), τ h ] + [τ h, C (u h, λ h )] = (τ h n) u Γ dγ, [ C(ψ h ), (v h, η h )] = The corresponding discrete kernel of C is defined similarly. Let Γ τ h T,h, (4.3) v h f d, (v h, η h ) U h R. (4.4) Z 1h := { τ h T,h [ C(τ h ), (v h, η h )] = 0, (v h, η h ) U h R } Existence, Uniqueness, and A Priori Estimates Here the existence and uniqueness of a solution to the discrete variational formulation in the case 1 < r < 2 is reviewed. Theorem 4.1. Let g satisfy (2.2) and (2.3). Let (φ, ψ, u, λ) T 0 T U R solve (3.1)-(3.3). Assume that (1) There exists a positive constant β Bh such that inf sup τ h Z 1h ς h T 0 h (2) There exists a positive constant β Ch such that inf sup (u h,λ h ) U h R τ h T,h [ B(ς h ), τ h ] ς h T τ h T β Bh. (4.5) [ C(τ h ), (u h, λ h )] β Ch. (4.6) τ h T (u h, λ h ) U R Then, for f ( L r () ) n and uγ ( W 1 1/r, r (Γ) ) n, there exists a unique solution (φ h, ψ h, u h, λ h ) T 0 h T, h U h R to the problem (4.2)-(4.4). Proof. With the assumptions as stated above, existence and uniqueness of (φ h, ψ h, u h, λ h ) T 0 h T, h U h R solving (4.2)-(4.4) follows directly from the continuous solution approach outlined in Section 3 and summarized in Theorem 3.1. It should be noted that the stability estimates shown in Theorem 3.1 carry over to the discrete case as well. The abstract a priori error estimate for 1 < r < 2 is now given. 10

11 Theorem 4.2. Let E(φ, φ h ) = φ φ h φ + φ h Assume the hypotheses of Theorem 4.1 are satisfied. Then φ φ h 2 T + (2 r)/r g(φ) g(φ h ) φ φh d { ( C inf φ ςh 2 ς h T 0 T + E(φ, φ ) h) r φ ς h r T h + inf u v h 2 U + v h U h. (4.7) inf τ h T, h ψ τ h 2 T }, (4.8) { ψ ψ h T C inf τ h T, h ψ τ h T ( + E(φ, φ h ) g(φ) g(φ h ) ) 1/r } φ φh d, (4.9) and for some constant C > 0. u u h U + λ λ h C { } φ φ h T + inf u v h U, (4.10) v h U h Proof. The proof is very similar to the proof of Theorem 4.2 in [4] and is omitted for brevity. Remark 4.1. Note that E(φ, φ h ) 1. In addition, if 1/( φ + φ h ) C for some constant C > 0, then E(φ, φ h ) min { } 1, C φ φ h (2 r)/r. Furthermore, if φ φ h φ φ h T, the estimates (4.8) (4.10) may be written as φ φ h T + ψ ψ h T + u u h U + λ λ h { C inf φ ς ς h T + inf u v h U + h T h v h U h inf τ h T, h Remark 4.2. If r = 2, then a standard proof approach shows that the error estimate φ φ h T + ψ ψ h T holds for some C > 0. C + u u h U + λ λ h ς inf φ ς h T + h T 0 h inf τ h T, h } ψ τ h T. (4.11) ψ τ h T + inf u v h U v h U h, (4.12) 11

12 4.2. Finite Element Approximation In this section the choices for the subspaces T 0 h, T, h, and U h are described, and it is shown that these choices satisfy the conditions (4.5) and (4.6). Consider R 2 and let K T h and let P k (K) be the set of all polynomials in the variables x 1, x 2 of degree less than or equal to k defined on the triangle K. Let RT k (K) be the 2-vector of Raviart-Thomas elements [27, 28] on K defined by [ ] RT k (K) = (P k (K)) 2 x1 + P x k (K). 2 For k 0, define the following discrete spaces: T 0 h := { } φ T 0 φ K (P k (K)) 2 2, K T h, { T, h := ψ T ψ = (ψ 1 ψ 2 ) T K (RT k (K)) 2, (ψ i1 ψ i2 ) T K RT k (K), i {1, 2}, K T h }, U h := { u U u K (P k (K)) 2, K T h }. Remark 4.3. There is no interelement continuity requirement on the spaces T 0 h and U h. Remark 4.4. As n = 2, for φ T 0 h, φ 11 = φ 22. Let s > 1 and let I k h : ( W 1,s () ) 2 2 T, h be the k-th order Raviart-Thomas interpolation operator [27, 12, 29], defined by, for row j = 1, 2 of τ T, (τ j I k h τ j) n ei v k ds = 0, v k P k (K), e i K, i = 1, 2, 3, K T h, e i (τ j I k h τ j) v k 1 dk = 0, v k 1 (P k 1 (K)) 2, K T h, K where n ei denotes the outer unit normal vector to edge e i of K. Then, for 0 m k + 1, and, for v U, τ I k h τ 0,r, Ch m τ m,r,, (4.13) (τ I k h τ) 0,r, Ch m τ m,r,, (4.14) v (τ I k h τ) d = 0, τ T. (4.15) In the lowest-order case, i.e., k = 0, for τ h Z 1h and τ 0 h = τ h 1 2 tr(τ h)i, φ = τ0 h r /r 1 τ 0 h τ 0 h r 1 T T 0 h. (4.16) The proof of the discrete inf-sup condition for B then follows as in the continuous case. However, for higher-order approximations, φ defined by (4.16) for (τ h, q h ) Z 1h is not a polynomial 12

13 and hence not in T h. In these cases a suitable projection of φ is required. Let Π : T 0 T 0 h denote the L 2 projection operator, defined by Π(φ ) := φ h, where φ : τ h d = φ h : τ h d τ h T 0 h. Lemma 4.1. Let φ T 0 and φ h = Πφ. Then there is a constant C > 0 such that φ h T C φ T. (4.17) Proof. See the proof of Lemma 4.4 in [30]. Lemma 4.2. For the choices of T 0 h, T, h, and U h above, there exists a positive constant β Bh such that [ B(φ inf sup h ), τ h ] β τ Bh. h Z 1h φ h T τ h T φ h T 0 h Proof. Note that for τ h Z 1h, τ h = 0 implies τ h K (P k (K)) 2 2 for all K T h. Thus τ h Z 1h and τ 0 h = τ h 1 n tr(τ h)i implies τ 0 h T 0 h. Let φ = τ 0 h r /r 1 τ 0 h / τ0 h r 1 T. Then φ T 0 h and φ T = 1. Let ς h = Πφ. From Lemma 4.1, ς h T C φ T = C. Also [ B(ς h ), τ h ] = [ B(φ ), τ h ] for all τ h Z 1h. Continuing as in (3.17), the result is shown as in Lemma 3.2, with the inclusion of the constant 1/C. Lemma 4.3. For the choices of T h, T, h, and U h above, there exists a positive constant β Ch such that inf sup [ C(τ h ), (u h, λ h )] β Ch. (4.18) (u h,λ h ) U h R τ h T τ h,h T (u h, λ h ) U R Proof. The proof is found in [4]. From [12, 28] the standard approximation properties hold: for all (ς, τ, v) (W m,r ()) 2 2 ( W m,r () ) 2 2 (W m,r ()) 2 with τ ( W m,r () ) 2, there exists (ςh, τ h, v h ) T 0 h T, h U h satisfying ς ς h T Ch m ς m,r,, ς (W m,r ()) 2 2, (4.19) τ τ h T Ch m τ m,r,, τ ( W m,r () ) 2 2, (4.20) (τ τ h ) T Ch m τ m,r,, ( τ) ( W m,r () ) 2, (4.21) v v h U Ch m v m,r,, v (W m,r ()) 2. (4.22) Theorem 4.3. Let f ( L r () ) 2 and uγ ( W 1 1/r, r (Γ) ) 2. Let (φ, ψ, u, λ) T 0 T U R solve (3.1)-(3.3) and let (φ h, ψ h, u h, λ h ) T 0 h T, h U h R solve (4.2)-(4.4). Assume 1 m k

14 and (φ, ψ, u) (W m,r ()) 2 2 ( W m,r () ) 2 2 (W m,r ()) 2 with ψ ( W m,r () ) 2. Then there exists a positive constant C such that φ φ h 2 T {h C mr E(φ, φ h ) r φ r m,r, )} + h ( φ 2m m,r, + u m,r, + ψ m,r, + ψ m,r,, (4.23) { ψ ψ h T C h m ( ) ψ m,r, + ψ m,r, ( + E(φ, φ h ) g(φ) g(φ h ) ) 1/r } φ φh d, (4.24) u u h U + λ λ h C φ φ h T. (4.25) Proof. The result follows directly from Theorem 4.2 and properties (4.19) (4.22). Remark 4.5. The extension of Remark 4.1 to these approximation spaces is given by: If 1/( φ + φ h ) C for some constant C > 0 and φ φ h φ φ h T, the estimates (4.23) (4.25) may be written as φ φ h T + ψ ψ h T + u u h U + λ λ h } C h { φ m m,r, + u m,r, + ψ m,r, + ψ m,r,. (4.26) 4.3. Postprocessing Computation of the Pressure The constitutive relation (2.1) indicates that ψ = g(φ) pi. For trace-free φ, this implies that tr(ψ) = tr(pi) = n p. Thus the true pressure p L r () is given by p = 1 ntr(ψ) when φ is trace-free. The following result follows directly from Theorem 4.3. Corollary 4.1. Let p P = L r () be given by p = 1 ntr(ψ) where ψ is part of the solution to (3.1)-(3.3). Let P h = {p P p K P k+1 (K), K T h }, and let p h be given by p h = 1 2 tr(ψ h) where ψ h is part of the solution to (4.2)-(4.4). Then p h P h and there is a constant C > 0 such that Proof. Note that p p h P = 1 2 tr(ψ) 1 2 tr(ψ h) p p h P C ψ ψ h T. (4.27) 0,r = 1 2 tr(ψ ψ h ) 0,r C ψ ψ h 0,r. 14

15 5. Solution of Linear Systems Arising in the Dual-Mixed Formulation In this section, the solution of linear systems that arise in the computation of solutions to the problem (4.2) (4.4). For ease of notation, the subscript h on discrete approximation spaces and computed approximations will be omitted. For general constitutive laws of the type (2.1), the system of equations that arises from the finite element discretization is nonlinear, and requires an iterative solution of a linearized form of the problem. The linearization given here is fixed-point in nature. As the nonlinearity occurs only in the operator Ã(φ), the linearization is accomplished by lagging the nonlinear part of the viscosity: for g(φ) = ν( φ )φ, the fixed-point iteration consists of computing the ith solution (φ i, ψ i, u i, λ i ) of (3.1) (3.3) with g(φ i ) = ν( φ i 1 )φ i. An initial iterate (φ 0, ψ 0, u 0, λ 0 ) for the fixed-point scheme is found by solving the Stokes problem (r = 2). The fixed-point iteration is continued until (φ i, ψ i, u i, λ i ) (φ i 1, ψ i 1, u i 1, λ i 1 ) is small enough. At each iteration, a linear system of equations must be solved. The structure of the linear system will depend on the choice of discretization method, in particular the method of numbering or grouping the unknown quantities. When all unknowns are grouped together on each simplex, the finite element method yields a sparse system of linear equations that may have a narrow bandwidth, depending on the type of problem and the structure of the computational mesh. For many classical problems, specific solvers can be constructed to exploit the banded structure and compute solutions in little time. However, systems arising from (4.2) (4.4) include the scalar Lagrange multiplier λ, which will pollute the banded structure. Figure 5.1 gives the sparsity pattern of a sample coefficient matrix on a square mesh for lowest-order (k = 0) elements with this pseudobanded structure. The rightmost column and bottom row contain the nonzero entries due to the Lagrange multiplier. Figure 5.1: Sparsity plot of sample pseudobanded coefficient matrix. When unknowns are treated inidually across all of T h, the problem (4.2) (4.4) lends a 15

16 particular block structure, given by (see Figure 5.2) A B T 0 φ 0 Ax = B 0 C T ψ = u Γ, τ Γ = b, (5.1) 0 C 0 (u, λ) f, v where u Γ, τ Γ = Γ u Γ (τ n) dγ and f, v = f v d. Figure 5.2: Sparsity plot of sample block coefficient matrix. Here, A is l l symmetric positive definite and both B (m l) and C (n m) have full rank, i.e., rank (B) = m and rank (C) = n. The matrix A is symmetric indefinite, and it is shown in [31] using Sylvester s Law of Inertia that A has l +n positive eigenvalues and m negative eigenvalues. The Uzawa method [32, 33] is a common approach for solving indefinite linear systems that arise in saddle-point problems. Consider the general indefinite linear system [ ] [ [ ] M Q T x w = (5.2) Q 0 y] z where M is symmetric positive definite. Block elimination applied to (5.2) implies that y is a solution of the linear system QM 1 Q T y = QM 1 w z. An Uzawa conjugate gradient method [34] for solving (5.2) consists of the following: for some initial y 0, compute x 0 = M 1 (w Q T y 0 ), g 0 = z Qx 0, h 0 = g 0, then for n 0 compute 1. ξ n = M 1 Q T h n 2. ρ n = g n 2 /(Qξ n, h n ) 3. y n+1 = y n ρ n h n 4. g n+1 = g n ρ n Qξ n 16

17 5. γ n = g n+1 2 / g n 2 6. h n+1 = g n+1 + γ n h n until y n converges. Steps 1 3 of the above algorithm represent the descent step for y n, and steps 4 6 compute the new descent direction. Upon convergence, the solution component x can be computed from the first equation of (5.2), i.e. x = M 1 (w Q T y). The solution of a linear system whose matrix is M is also required at each step of the above algorithm. As (5.1) represents a twofold saddle point problem, it might seem intuitive to construct a solution approach that consists of a nested Uzawa-like [ ] algorithm. However, (5.1) is unsuitable A B T for such an approach, as the upper-left block is singular, and the Uzawa method requires B 0 this block to be symmetric positive definite Solution of Pseudobanded System via Bordering Algorithm In studies of solving linear systems arising in numerical continuation problems, Keller [35, 36] describes a bordering algorithm that can be used to solve N + 1 N + 1 systems of the form [ [ ] [ ] M b x w =. (5.3) c d] y z This approach will compute the solution to nonsingular systems (5.3) even in cases where the N N block M is singular. The bordering algorithm consists of the following steps: 1. Determine ξ such that Mξ = b. 2. Determine γ such that Mγ = w. 3. Then y = (c T γ + z)/(d c T ξ) and x = γ yξ. For pseudobanded systems arising from the Stokes or nonlinear Stokes problem, utilization of this approach allows for the use of tailored solvers for the banded matrix M in the first two steps of the above algorithm Solution of Block System via Penalty or Fixed-Point Iteration Existing methods for solving linear systems of the particular structure (5.1) are studied in [31] and [37]. In [31], the authors present a preconditioned minimum residual method in which the convergence behavior is improved through a scaling of basis functions. In [37], the authors present a solution method for problems of (5.1) type in which the indefinite system is transformed to a positive definite system and then the conjugate gradient method is applied using a specific inner product defined in the transformation. Both of these methods rely on the construction of a preconditioner for the A block of (5.1) that cannot be the explicit inverse of A. However, due to the choice of discontinuous finite element spaces, in the lowest-order case A is strictly diagonal and SPD on any computational mesh. Thus a method that exploits the immediate availability of A 1 is desired. Interchanging the last two rows and columns of (5.1) yields the system A 0 B T φ C (u, λ) = f, v. (5.4) B C T 0 ψ u Γ, τ Γ 17

18 Figure 5.3: Rearranged block coefficient matrix. The sparsity of this coefficient matrix in the lowest-order case (square mesh) is given in Figure 5.3. If the upper-left 2 2 block can be made symmetric positive definite while preserving its diagonal structure, then the Uzawa method could be applied to (5.4). This would be particularly advantageous as the solutions of Mx = b can be computed immediately given that M is diagonal. This could be accomplished with the addition of an inner product-like term on U R to the formulation (this term may not be a true inner product as U is not an inner product space for 1 < r < 2). Two approaches for doing this are through a penalty method or a method that adds extra terms to the existing fixed-point iteration (for 1 < r < 2) Penalty Method Penalty methods are commonly used in the numerical solution of finite element approximations of the incompressible Stokes and Navier-Stokes equations [12, 38] as well as for nonlinear power-law Stokes problems [39, 40] as a stabilization approach. In the Stokes and Navier-Stokes case, the penalty is usually implemented by adding an inner product on the pressure space with a coefficient that depends on a parameter ε > 0. However, traditional penalty methods for the Stokes problem can adversely affect the condition number of linear systems, making them less suitable for iterative solution methods [12]. In the current context, the penalty term is introduced to create additional nonzero diagonal entries in the coefficient matrix. A penalty method for the discrete formulation (4.2) (4.4) is accomplished by adding an inner product-like term on U h R with a coefficient of ε. This method would transform the formulation (4.2) (4.4) by changing (4.4) to ελ h η h + ε u h r 2 u h v h d v h ψ h d + η h tr(ψ h ) d = f v h d, (v h, η h ) U h R. (5.5) When 1 < r < 2, the nonlinearity that is introduced in (5.5) can be handled in the same manner as the nonlinearity in the constitutive law. For values ε > 0, this penalized method would then 18

19 result in a coefficient matrix that has a symmetric positive definite upper left block and the Uzawa method previously described can then be applied. As U h is constructed of piecewise discontinuous polynomials, in the lowest-order case this upper left block will be strictly diagonal, thus the Uzawa algorithm can be used with minimal cost of computing M 1. Figure 5.4 gives a sparsity plot of the penalized coefficient matrix in the lowest-order case. Figure 5.4: Rearranged block coefficient matrix with penalty or lagged term Fixed-Point Method As described above, when 1 < r < 2 the system of equations resulting from (4.2) (4.4) is nonlinear, and an iterative solution method is required. The inner product-like terms would be added to both sides of (4.4) with the right-hand side terms completely lagged. Thus, the computation of the ith iterate would be modified so that (4.4) consisted of λ i h η h + u i 1 h r 2 u i h v h d = f v h d + λ i 1 h η h + v h ψ i h d + η h tr(ψ i h ) d u i 1 h r 2 u i 1 h v h d, (v h, η h ) U h R. (5.6) However, this method will converge only linearly, even if a Newton iteration is used for the nonlinearity in the constitutive law. The sparsity pattern of the coefficient matrix in the lowestorder case for this approach is also represented in Figure 5.4. Further analysis of these solution approaches, including the effect of this penalty method on the conditioning of the linear system, will be studied in subsequent work. 6. Numerical Experiments In this section, data that illustrate the reduction in number of unknowns due the traceless gradient method are given, and computations that support the theoretical error estimates outlined 19

20 in Section 4 are also presented. Error results for the original solution method of [3, 4] are also given for comparison. Computations are performed using the FreeFEM++ finite element software package [41]. All computations below are performed in the lowest-order case (k = 0). For these computations, approximations are computed for a Ladyzhenskaya law fluid with ν 0 = 0 and ν 1 = 1.0 (the problem originates in [3, 4]). The computational domain is = [0, 2] [0, 2], with f and u Γ chosen so that the exact solution of (2.12)-(2.14) is given by [ ] u1 u = and p = x u 1 + x 2, 2 with u 1 = (4.0 x 1 x 2 ) α and u 2 = u 1 for α = 8/3. Computations are performed on uniform meshes of decreasing size h. The number of global degrees of freedom N do f for both the traceless gradient method and the original method are given in Table 6.1. Error results for u, φ(= u), ψ and p, are shown in the tables below. h N do f, original method N do f, traceless method reduction in N do f / / / / / / Table 6.1: Comparison of degrees of freedom for traceless gradient and original solution method. Table 6.2 gives the error results for both methods. The results in these tables indicate that for this problem, the traceless gradient method and the original method both compute approximations to u, φ, ψ with identical accuracies and rates, while the tracelss gradient method only requires the solution of linear systems that have around 80% of the unknowns of the original method. Additionally, for this problem the postprocessing computation of p h of the traceless gradient method is more accurate than the original approximation method, with smaller errors and a larger rate of convergence. The errors for the pressure approximations are displayed in Figure 6.1. References [1] C. D. Han, Multiphase Flow in Polymer Processing, Academic Press, New York, [2] O. A. Ladyzhenskaya, New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them, in: Boundary Value Problems of Mathematical Physics V, American Mathematical Society, Providence, RI, [3] G. N. Gatica, M. González, S. Meddahi, A low-order mixed finite element method for a class of quasi-newtonian Stokes flows. I. A priori error analysis, Comput. Methods Appl. Mech. Engrg. 193 (9-11) (2004) [4] V. J. Ervin, J. S. Howell, I. Stanculescu, A dual-mixed approximation method for a three-field model of a nonlinear generalized Stokes problem, Comput. Methods Appl. Mech. Engrg. 197 (33 40) (2008) [5] J. Baranger, K. Najib, D. Sandri, Numerical analysis of a three-fields model for a quasi-newtonian flow, Comput. Methods Appl. Mech. Engrg. 109 (3-4) (1993) [6] H. Manouzi, M. Farhloul, Mixed finite element analysis of a non-linear three-fields Stokes model, IMA J. Numer. Anal. 21 (1) (2001) [7] R. Bustinza, G. N. Gatica, M. González, A mixed finite element method for the generalized Stokes problem, Internat. J. Numer. Methods Fluids 49 (8) (2005)

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