Juan Carlos De Los Reyes 1 and Sergio González 1

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1 ESAIM: M2AN 43 (29) 8 7 DOI:.5/m2an/2839 ESAIM: Mathematical Modelling and Numerical Analysis PATH FOLLOWING METHODS FOR STEADY LAMINAR BINGHAM FLOW IN CYLINDRICAL PIPES Juan Carlos De Los Reyes and Sergio González Abstract. This paper is devoted to the numerical solution of stationary laminar Bingham fluids by pathfollowing methods. By using duality theory, a system that characterizes the solution of the original problem is derived. Since this system is illposed, a family of regularized problems is obtained and the convergence of the regularized solutions to the original one is proved. For the update of the regularization parameter, a pathfollowing method is investigated. Based on the differentiability properties of the path, a model of the value functional and a correspondent algorithm are constructed. For the solution of the systems obtained in each pathfollowing iteration a semismooth Newton method is proposed. Numerical experiments are performed in order to investigate the behavior and efficiency of the method, and a comparison with a penaltynewtonuzawaconjugate gradient method, proposed in [Dean et al., J. NonNewtonian Fluid Mech. 42 (27) 36 62], is carried out. Mathematics Subject Classification. 47J2, 76A, 65K, 9C33, 9C46, 9C53. Received June 5, 27. Revised June 2nd, 28. Published online October 6, 28.. Introduction Bingham models are used to analyze flows of materials for which the imposed stress must exceed a critical yield stress to initiate motion, i.e., they behave as rigid bodies when the stress is low but flow as viscous fluids at high stress. Examples of Bingham fluids include tooth paste, water suspensions of clay or sewage sludge. For the mathematical analysis of Bingham fluid flow we refer to [7,9,22]. In [22] the authors consider a variational formulation of the model and study qualitative properties of it. Existence and uniqueness of the solution and the structure of the flow are investigated. In [7] the authors further analyze the resulting inequality of the second kind and prove, among other results, the Lipschitz stability of the solution with respect to the plasticity threshold. Further in [4] and[9 ] the authors investigate the regularity of the solution for the cross section and cavity model, respectively. Bingham fluid flow in cylindrical pipes has been numerically treated by different methodologies. In [3], Chapter V, the authors propose a global εtype regularization of the model and prove the convergence of the regularized solutions towards the original one. Direct regularization of the primal problem by twice differentiable Keywords and phrases. Bingham fluids, variational inequalities of second kind, pathfollowing methods, semismooth Newton methods. Research partially supported by DAAD, EPN Quito and TU Berlin joint project: Ph.D. Programme in Applied Mathematics. Research Group on Optimization, Departmento de Matemática, EPN Quito, Ecuador. Article published by EDP Sciences c EDP Sciences, SMAI 28
2 82 J.C. DE LOS REYES AND S. GONZÁLEZ functions has also been considered in [23] in combination with Newton methods. Although this type of regularization allows the direct use of second order methods, important discrepancies of the regularized problem with respect to properties of the original model arise (cf. [6], p. 39). An alternative to the direct regularization of the primal problem consists in the socalled multiplier approach. In [3], the authors analyze the existence of multipliers by using duality theory and propose an Uzawatype algorithm for its numerical solution. Also by using duality theory, augmented Lagrangian methods are proposed in [8,9] and the unconditional convergence of the method is proven (see [9], Thm. 4.2). In the recent paper [6], the authors make a review of existing numerical approaches and propose a penaltynewtonuzawa conjugate gradient method for the solution of the problem. This approach is compared numerically with our method in Section 5. In this paper, we consider a Tikhonov regularization of the dual problem, which by duality theory implies a local regularization of the original one. The proposed local regularization allows the application of semismooth Newton methods and leads directly to a decoupled system of equations to be solved in each semismooth Newton iteration. This constitutes an important difference with respect to other primaldual second order approaches (see e.g. [6]), where an additional method has to be used in order to obtain a decoupled system, at the consequent computational cost. For the update of the regularization parameter a path following method is proposed and analyzed. The differentiability of the path and of the path value functional are studied. A model function that preserves the main properties of the value functional is proposed and a correspondent algorithm developed. After discretization in space, each regularized problem is solved by using a semismooth Newton method. These type of methods have been successfully applied to infinite dimensional complementarity problems like the Signorini or contact problem (see [4,2,24,25]), image restoration (see [6]), optimal control problems (see [5,7]), and, in general, to infinite dimensional optimization problems (see [6,7,9,26]). Pathfollowing strategies together with semismooth Newton methods have been investigated in [4,5,25] for variational inequalities of the first kind and constrained optimal control problems. These cases involve unilateral pointwise constraints on the state variable, which are regularized by a MoreauYosida technique. Differently from [4,5], our problem involves a variational inequality of the second kind. As a result, and in contrast to unilateral pointwise constrained problems, pointwise constraints on the Euclidean norm of the velocity gradient have to be considered. This fact adds new difficulties to the path analysis. In particular, extra regularity estimates for the regularized solutions have to be obtained in order to get differentiability of the path. Let us mention that, although the method developed in this article is concerned with Bingham fluid flow, the results can be extended to other variational inequalities of the second kind as well. The paper is organized as follows. In Section 2 the original problem is stated and, using Fenchel s duality theory, a necessary condition is derived. Since the characterizing system for the original problem is illposed, a family of regularized problems is introduced and the convergence of the regularized solutions to the original one is proved. In Section 3, the path value functional is introduced and the differentiability of the path and the value functional is investigated. A model function which preserves the qualitative properties of the pathvalue functional is constructed and an iterative algorithm is proposed. In Section 4 a semismooth Newton method to solve the complementarity system for each regularized problem is stated. In Section 5, numerical experiments which show the main features of the proposed algorithm are presented. 2. Problem statement and regularization Let be a bounded domain in R 2, with Lipschitz boundary Γ, and let f L 2 (). We are concerned with the following variational inequality of the second kind: find y H () such that a(y, v y)+gj(v) gj(y) (f,v y) 2, for all v H (), (2.) where a(y, v) :=μ y(x), v(x) dx, j(v) := v(x) dx and (, ) 2 stands for the scalar product in L2 (). The scalar product in R N and the Euclidean norm are denoted by, and, respectively. (, ) X stands
3 PATH FOLLOWING METHODS FOR STEADY LAMINAR BINGHAM FLOW IN CYLINDRICAL PIPES 83 for the scalar product in a Hilbert space X, and X for its associated norm. The duality pairing between a Banach space Y and its dual Y is represented by, Y,Y. Besides that, we will use the bold notation L 2 () := L 2 () L 2 (). Inequality (2.) models the stationary flow of a Bingham fluid in a pipe of cross section (see [7,3,22]). The variable y(x) stands for the velocity at x, f(x) for the linear decay of pressures, μ for the viscosity and g for the plasticity threshold of the fluid (yield stress). Problem (2.) corresponds to the necessary condition of the following unconstrained minimization problem, min J(y) := y H () 2 a(y, y)+gj(y) (f,y) 2. (P) Remark 2.. It can be shown (cf. [2], Thm. 6.) that there exists a unique solution y H () to problem (P). Moreover, if has a sufficiently regular boundary, it follows that y H 2 () H (), see [4], Theorem The Fenchel dual In this section, we obtain the dual problem of (P) by using Fenchel s duality in infinitedimensional spaces (see [8]). Let us start by defining the functionals F : H () R by F (y) := 2 a(y, y) (f,y) 2 and G : L 2 () R by G (q) :=g q(x) dx. It can be easily verified that these two functionals are convex, continuous and proper. We also define the operator Λ L(H (), L 2 ()) by Λv := v. Thanks to these definitions, we may rewrite problem (P) as inf {F (y)+g (Λy)}. (2.2) y H () Following [8], pp. 6 6, the associated dual problem of (2.2) isgivenby sup { F ( Λ q) G (q)}, (2.3) q L 2 () where Λ L(L 2 (),H ()) is the adjoint operator of Λ, and F : H () R and G : L 2 () R denote the convex conjugate functionals of F and G, respectively. We recall that given a Hilbert space H and a convex function ϕ : H R {, + }, the convex conjugate functional ϕ : H R {, + } is defined by Thus, we have that F ( Λ q) = sup ϕ (v )=sup{ v,v ϕ(v)}, for v V. v V v H () G (q) = sup p L 2 () { Λ q, v H (),H () } 2 a(v, v)+(f,v) 2, (2.4) {(q, p) L2() g } p(x) dx. (2.5) Note that in (2.3) we have already identified L 2 () with its dual. Now, let us calculate F ( Λ q). Let q L 2 () be given. From (2.4), we obtain that { F ( Λ q)= sup (q, Λv) L 2 v H () () } 2 a(v, v)+(f,v) 2,
4 84 J.C. DE LOS REYES AND S. GONZÁLEZ which implies, since { (q, Λv) L 2 () 2 a(v, v) +(f,v) 2 } is a concave quadratic functional in H (), that the supremum is attained at v(q) H () satisfying Using (2.6) withz = v(q), we obtain that a(v(q),z)+(q, Λz) L 2 () (f,z) 2 =, for all z H (). (2.6) F ( Λ q)= (q, v(q)) L 2 () 2 a(v(q),v(q)) + (f,v(q)) 2 = a(v(q),v(q)). (2.7) 2 Lemma 2.. The expression (q, p) L 2 () g p(x) dx, for all p L 2 () (2.8) is equivalent to q(x) g a.e. in. (2.9) Proof. Let us start by showing that (2.8) implies (2.9). Assume that (2.9) doesnothold, i.e., assume that S := {x :g q(x) < a.e.} has positive measure. Choosing p L 2 () such that p(x) := { q(x) in S in \ S leads to g p(x) dx q(x), p(x) dx = (g q(x) ) q(x) dx < S which is a contradiction to (2.8). Conversely, due to the fact that q(x) g a.e. in and thanks to the CauchySchwarz inequality, we obtain, for an arbitrary p L 2 (), that g p(x) dx q(x),p(x) dx (g q(x) ) p(x) dx. Lemma 2. immediately implies that { G if q(x) g, a.e. in (q) = (2.) + otherwise. Thus, using (2.7) and(2.) in(2.3) we obtain the dual problem sup J (q) := 2 a(v(q),v(q)) q(x) g where v(q) satisfies a(v(q),z) (f,z) 2 +(q, Λz) L 2 () =, for all z H (). (P ) Due to the fact that both F and G are convex and continuous, [8], Theorem 4., p. 59, and [8], Remark 4.2, p. 6, imply that no duality gap occurs, i.e., inf J(y) = sup J (q), (2.) y H () q(x) g a(v,z)+(q, z) L 2 =(f,z) () 2 and that the dual problem (P ) has at least one solution q L 2 ().
5 PATH FOLLOWING METHODS FOR STEADY LAMINAR BINGHAM FLOW IN CYLINDRICAL PIPES 85 Next, we will characterize the solutions of the primal and dual problems. From Fenchel s duality theory (see [8], Eqs. (4.22) (4.25), p. 6) the solutions y and q satisfy the following extremality conditions: Λ q F (y), (2.2) q G ( y). (2.3) Let us analyze (2.2). Since F is Gateaux differentiable in y, [8], Proposition 5.3, p. 23, implies that F (y) = {F (y)}. Thus,wehavethat(2.2) can be equivalently expressed as the following equation a(y, v) (f,v) 2 +(q, v) L 2 () =, for all v H (). (2.4) On the other hand, from (2.3) and the definition of the subdifferential it follows that ( ) g y(x) dx p(x) dx (q, y p) L 2 (), for all p L2 (). Then, for p =, we obtain that g y(x) dx (q, y) L 2 (), which implies, since q(x) g a.e. in and by Lemma (2.), that g y(x) dx =(q, y) L 2 (). This last expression is equivalent to { y(x) = or y(x), and q(x) =g y(x) y(x) (2.5) Lemma 2.2. Equations (2.9) and (2.5) can be equivalently expressed as the following equation max (σg, σq(x)+ y(x) ) q(x) =g (σq(x)+ y(x)), a.e. in, for all σ>. (2.6) Proof. We start by showing that (2.6) implies (2.9) and(2.5). From (2.6) it follows that σq(x)+ y(x) q(x) =g, a.e. in, max(σg, σq(x)+ y(x) ) which immediately implies (2.9). Let us split into the two following disjoint sets: {x :σg σq(x)+ y(x) } and {x :σg < σq(x)+ y(x) }. (2.7) On the set {x : σg σq(x) + y(x) }, wehavethatg(σq(x) + y(x)) σgq(x) =, and thus y(x) =. To see that y(x) on the set{x : σg < σq(x) + y(x) }, we assume the opposite and immediately obtain that g < q(x), which contradicts the fact that q(x) g a.e. in. Moreover, from (2.6), we have that g (σq(x) + y(x)) = σq(x) + y(x) q(x), (2.8) and it follows that g y(x) =( σq(x) + y(x) σg) q(x). (2.9)
6 86 J.C. DE LOS REYES AND S. GONZÁLEZ Considering the norms in (2.8) and(2.9), we find that ( σq(x) + y(x) σg) = y(x) and thus we are in the second case of (2.5). Reciprocally, assume that (2.9) holds and consider the two cases in (2.5). If y(x) =, we obtain that g (σq(x) + y(x)) = max (σg, σq(x) + y(x) ) q(x) = σgq(x). Similarly, if y(x) andq(x) =g y(x) y(x),wehavethat which implies that g (σq(x)+ y(x)) = g y(x) (σg + y(x) ), y(x) max (σg, σq(x)+ y(x) ) q(x) = max(σg, σg + y(x) ) g y(x) y(x) = g y(x) (σg + y(x) ). y(x) Thus, the equivalence follows. Summarizing, we may rewrite (2.2) and(2.3) as the following system { a(y, v)+(q, v)l 2 () =(f,v) 2, for all v H (), max (σg, σq(x) + y(x) ) q g(σq + y) =, a.e. in and for σ >. (S) We define the active and inactive sets for (S) bya := {x : σq(x)+ y(x) σg} and I := \A, respectively. Remark 2.2. The solution to (S) isnotunique(see[2], Rem. 6.3, and [3], Chap. 5) Regularization In order to avoid problems related to the nonuniqueness of the solution to system (S), we propose a Tikhonovtype regularization of (P ). With this regularization procedure, we do not only achieve uniqueness of the solution but also get a local regularization for the nondifferentiable term in (P). This technique has also been used for TVbased infconvolutiontype image restoration [6]. For a parameter > we consider the following regularized dual problem sup J (q) := 2 a(v(q),v(q)) 2 q 2 L 2 q(x) g where v(q) satisfies (P ) a(v(q),z)+(q, z) L 2 () (f,z) 2 =, for all z H (). Therefore, the regularized problem is obtained from (P ) by subtracting the term 2 q 2 L 2 from the objective functional. Further, it is possible to show that this penalization corresponds to a regularization of the primal problem. Consider the continuously differentiable function ψ : R 2 R, defined by ψ (z) := { g z g2 2 if z g 2 z 2 if z < g (2.2)
7 PATH FOLLOWING METHODS FOR STEADY LAMINAR BINGHAM FLOW IN CYLINDRICAL PIPES 87 By using this function, which is a local regularization of the Euclidean norm, we obtain the following regularized version of (P) min J (y) := a(y, y)+ ψ ( y)dx (f,y) y H () 2 2. (P ) Furthermore, we are able to state the following theorem. Theorem 2.3. Problem (P ) is the dual problem of (P ) and we have J (q )=J (y ), (2.2) where q and y denote the solutions to (P ) and (P ), respectively. Proof. In order to calculate the dual problem to (P ) we use the same argumentation used in Section 2. for the original problem (P). We only have to replace the functional G in (2.2) by G (q) = ψ (q)dx, (2.22) with ψ as in (2.2). Thus, for q L 2 (), we have G (q) = sup p L 2 () + {x : p(x) g, a.e.} {x : p(x) <g, a.e.} ] [ q(x),p(x) g p(x) + g2 dx (2.23) 2 [ q(x),p(x) 2 p(x) 2] dx. From (2.23) it is possible to conclude, by proceeding as in the proof of Lemma 2., thatg (q) = unless q(x) g. Suppose now that q(x) g. We define the functional Ψ : L 2 () R by Ψ(p) := {x : p(x) g, a.e} + {x : p(x) <g, a.e} ] [ q(x),p(x) g p(x) + g2 dx 2 [ q(x),p(x) 2 p(x) 2] dx. By introducing, for any p L 2 (), the function p L 2 () by p (x) := it is easy to verify that Ψ(p ) Ψ( p ), which yields { p (x) a.e. in {x : p(x) <g,a.e.} a.e. in {x : p(x) g, a.e.} sup p L 2 () Ψ(p) = sup p L 2 () Ψ(p). (2.24) p(x) g a.e. in Therefore, in order to calculate the supremum in (2.23), we only have to consider the last term in (2.24). Since this expression is a concave quadratical functional, the maximizer is easily calculated as p = p,which
8 88 J.C. DE LOS REYES AND S. GONZÁLEZ implies that { G (q) = 2 q 2 L 2 if q(x) g otherwise. (2.25) Note that this regularization procedure turns the primal problem into the unconstrained minimization of a continuously differentiable functional, while the corresponding dual problem is still the constrained minimization of a quadratic functional. Remark 2.3. Due to the regularization procedure, the objective functional of (P )resultsinal2 ()uniformly concave functional. Thus, (P ) admits a unique solution q L 2 () for each fixed >. Additionally, since a(, ) is a coercive and bicontinuous form and due to the fact that J is strictly convex and differentiable, [2], Theorem.6, implies that (P ) has also a unique solution. Theorem 2.4. Let y be the solution to (P ). Then y H 2 () H () and there exists a constant K>, independent of, such that y H 2 K( f L 2 + C). (2.26) for some C>. Proof. Note that y can be characterized as the solution of the following equation f μδy + ϕ(y ) in y = on Γ (2.27) where ϕ denotes the subdifferential of the convex and lower semicontinuous functional ϕ : L 2 () R { } defined by { ϕ(u) = ψ ( u)dx if ψ ( u) L () + elsewhere. Thus, [4], Lemma, implies the result. Remark 2.4. Theorem 2.4 implies that y H () and, since n =2, y L q (), for all q [, ). Moreover, from (2.26) we conclude that y is uniformly bounded in L q () for all q [, ). Next, we characterizethesolutionsto(p )and(p )(y and q, respectively). From Fenchel s duality theory, these solutions satisfy the following system: Λ q F (y ) (2.28) q G ( y ). (2.29) Note that both F and G are differentiable in y and y, respectively. Thus, F (y )and G ( y )consist only of the respective Gateaux derivatives. Since (2.28) is similar to equation (2.2), it is equivalent to a(y,v)+(q, v) L 2 () (f,v) 2 =, for all v H (). (2.3) On the other hand, due to the differentiability of G,equation(2.29) can be written as (q,p) L 2 () = y,p g dx + y,p dx, for all p L 2 () y A \A or equivalently as { q (x) = y (x) a.e. in \A, q (x) =g y(x) (2.3) y a.e. in A,
9 PATH FOLLOWING METHODS FOR STEADY LAMINAR BINGHAM FLOW IN CYLINDRICAL PIPES 89 where, A = {x : y (x) g, a.e.}. Consequently, the solutions (y,q ) of the regularized problems (P )and(p ) satisfy the system { a(y,v)+(q, v) L 2 () (f,v) 2 =, for all v H () max(g, y (x) )q (x) g y (x) =, a.e. in, for all >. (S ) Clearly q (x) = g on A and q (x) <gon I := \A.WecallsetsA and I the active and inactive sets for (S ), respectively. In the following theorem the convergence of the regularized solutions towards the original one is verified. Theorem 2.5. The solutions y of the regularized primal problems converge to the solution y of the original problem strongly in H () as. Moreover, the solutions q of the regularized dual problems converge to asolutionq of the original dual problem weakly in L 2 (). Proof. Let us start by recalling that (y, q) and(y, q ) satisfy equations (2.4) and(2.3) respectively. Thus, by subtracting (2.3) from(2.4), we obtain that μ (y y ), v dx = q q, v dx, for all v H (). (2.32) Further, choosing v := y y in (2.32), we have that μ (y y ) 2 dx = q q, (y y ) dx. (2.33) Next, we establish pointwise bounds for (q q)(x), (y y )(x), in the following disjoint sets A A, A I, A I and I I. On A A : Here, we use the facts that q(x) = q (x) = g, q(x) =g y(x) y(x) and q y (x) (x) =g y (x).thus,we have the following pointwise estimate (q q)(x), (y y )(x) q (x) y(x) g y (x) y (x), y (x) g y(x) y(x), y(x) (2.34) + q(x) y (x) g y(x) g y (x) g y(x) + g y (x) =. On A I : Here, we know that y (x) = q (x), q (x) <g, q(x) = g and q(x) =g y(x) y(x). Hence, we get (q q)(x), (y y )(x) g y(x) q (x) 2 g y(x) + g y (x) = q (x) 2 + g q (x) < (g 2 q (x) 2) < g 2. (2.35) On A I: Inthissetitholdsthat y(x) =andq (x) =g y(x) y (x). Then, we have that (q q)(x), (y y )(x) = g y (x) + q(x) y (x). (2.36) On I I: Here, we have that y(x) =, y (x) = q (x), q(x) g, and q (x) <g. (q q)(x), (y y )(x) = q (x) 2 + q(x) y (x) (g 2 q (x) 2) < g 2. (2.37)
10 9 J.C. DE LOS REYES AND S. GONZÁLEZ Since A A, A I, A I and I I provide a disjoint partitioning of, (2.33) and estimates (2.34), (2.35), (2.36) and(2.37) implythat μ (y y ) 2 dx< g 2 dx. (2.38) Thus, we conclude that y y strongly in H () as. On the other hand, since y y strongly in H (), (2.32) implies that q q, weakly in grad(h ()) L 2 (), (2.39) where grad(h ()) := {q L 2 () : v H () such that q = v}. 3. Pathfollowing method In this section, we investigate the application of continuation strategies to properly control the increase of. Our main objective is to develop an automatic updating strategy for the regularization parameter which guarantees an efficient and fast approximation of the solution to problem (P). For that purpose, we investigate the properties of the path (y,q ) H () L2 (), with (, ), and construct an appropriate model of the value functional, which will be used in an updating algorithm. 3.. The primaldual path In this part we introduce the primaldual path and discuss some of its properties. Specifically, Lipschitz continuity and differentiability of the path are obtained. Definition 3.. The family of solutions C = {(y,q ): [M, )} to (S ), with M a positive constant, considered as subset of H () L2 (), is called the primaldual path associated to (P )(P ). Lemma 3.. The path C is bounded in H () L2 (), i.e., thereexistc>, independent of, such that y H + q L 2 C. Proof. First, from the fact that q (x) g, for every >, we conclude that q is uniformly bounded in L 2 (). Furthermore, Theorem 2.4 implies that y H is uniformly bounded in H (). Therefore, C is bounded in H () L2 (). Theorem 3.2. Let [M, ). The function y is globally Lipschitz continuous in W,p (), for2 p< 2+min(s 2,ɛ), wheres>2 and ɛ depends on μ, and. Proof. Let, [M, ). We introduce the following notations δ y := y y, θ (x) :=max(g, y (x) ) and δ θ := θ θ. It is easy to verify that the following expression holds δ θ (x) y (x) y (x), a.e. in, which implies that δ θ (x) y (x) + δ y (x), a.e. in, (3.) and, similarly, δ θ (x) y (x) + δ y (x), a.e. in. (3.2) Next, we separate the proof in two parts. First, we prove the Lipschitz continuity of y in H (), and then, by introducing an auxiliar problem, we obtain the Lipschitz continuity in W,p (), for some p>2.
11 PATH FOLLOWING METHODS FOR STEADY LAMINAR BINGHAM FLOW IN CYLINDRICAL PIPES 9 In H (): From (S ), we know that ( y a(δ y,δ y ) = g y ), δ y θ θ ( ) y = g(), δ y θ L 2 () ( y + g L 2 () θ y ), δ y, θ L 2 () (3.3) which, since y (x) θ M, a.e. in, implies the existence of a constant K>such that a(δ y,δ y ) K δ y H ( y + g θ y ), δ y. (3.4) θ L 2 () Next, let us analyze the second term on the right hand side of (3.4). ( y g θ y ), δ y θ L 2 () ( ) θ δ y + δ θ y = g, δ y θ θ L 2 () = g δ y (x) 2 ( δθ y dx + g, δ y θ θ θ ). L 2 () (3.5) Since y (x) θ (x) a.e. in, CauchySchwarz inequality implies that ( ) δθ y g, δ y θ θ L 2 () g δθ y (x) θ (x)θ (x), δ y(x) dx g g δ θ (x) y (x) δ y (x) θ (x)θ (x) δ θ (x) δ y (x) θ (x) dx. dx Again, since y (x) θ (x) a.e.in,(3.) implies that ( ) δθ y g, δ y θ θ L 2 () g y δ y dx + g δ y 2 θ dx θ δ y 2 g δ y dx + g dx θ δ y 2 g M meas()/2 δ y H + g dx. θ (3.6) Finally, using (3.6) and(3.5) in(3.4), we have that a(δ y,δ y ) (K + g ) meas()/2 δ y M H, which, due to the coercivity of a(, ), implies the existence of a constant L> such that y y H L.
12 92 J.C. DE LOS REYES AND S. GONZÁLEZ In W,p (): First, note that (3.2) implies the existence of ζ(x) [, ] such that δ θ (x) =ζ(x)[ y (x) + δy(x) ], a.e. in. (3.7) From (S ), we have, for all v H (), that ( ) ( ) ( ) δy δθ y y a(δ y,v)+g, v g, v = g(), v, (3.8) θ L 2 () θ θ L 2 () θ L 2 () which, together with (3.7), implies that ( ) ( δy ζ(x) δy a(δ y,v)+g, v g θ L 2 () θ θ ( y g() θ ) y, v L 2 () = ), v + g L 2 () ( ζ(x) y θ θ ) y, v, (3.9) L 2 () for all v H (). Defining f := g( ) y θ +g ζ(x) y θ θ y,equation(3.9) motivates the introduction of the following auxiliar problem: find w H () such that a(w, v)+(β(w), v) L 2 () = ( f, v )L, for all v 2 () H (), (3.) [ w where β(w) :=g θ ζ(x) δy, w θ θ δ y y ]. Clearly, δ y is also solution of (3.). Note that f(x) y (x) g θ (x) + g ζ(x) y (x) θ (x)θ (x) y (x), a.e. in, which, since y(x) θ M and y (x) θ, a.e. in, implies that f(x) 2 g a.e. in. M Therefore, f L s 2 g M meas()/s, for s. (3.) Next, let us define the matrix A(x) R 2 2 by A(x) := y (x) δ y (x) x x y (x) δ y (x) x 2 x y (x) x y (x) x 2 δ y (x) x 2 δ y (x) x 2, a.e. in. Then, we can rewrite β(w) as [ ] I ζ(x) β(w) =g θ θ θ δ y A(x) w, (3.2) where I stands for the 2 2identity matrix. Moreover, we can rewrite the auxiliar problem (3.) as α(x) w, v dx = f, v dx, (3.3)
13 PATH FOLLOWING METHODS FOR STEADY LAMINAR BINGHAM FLOW IN CYLINDRICAL PIPES 93 ζ(x) where α(x) :=(μ + g θ (x)) I + g θ A(x), a.e. in. Multiplying α(x) byξ (x)θ (x) δ y(x) R2 2 and taking the scalar product with ξ, we obtain that α(x)ξ,ξ = μ ξ 2 + g ζ(x) θ (x) ξ 2 g θ (x)θ (x) δ y (x) δ y(x),ξ y (x),ξ, a.e. in. Furthermore, since ζ(x) and y (x) θ (x) a.e. in, the CauchySchwarz inequality implies that g ζ(x) θ (x)θ (x) δ y (x) δ ξ 2 y(x),ξ y (x),ξ g, a.e. in, θ (x) which, due to the fact that g θ (x), a.e. in, implies that μ ξ 2 α(x)ξ,ξ (μ +2) ξ 2, a.e. in. (3.4) Thus, [3], Theorem 2., p. 64, implies the existence of a constant c p such that w W,p c p f L s, for s>2and2 p<2+min(s 2,ɛ), (3.5) where w is the unique solution of (3.) andɛ depends on μ, and. Therefore, since δ y is solution of (3.), estimates (3.) and(3.5) imply the existence of L > such that y y W,p L, for 2 p<2+min(s 2,ɛ). Remark 3.2. Since y is Lipschitz continuous in W,p (), for some p>2, thereexists a weakaccumulation point ẏ W,p () of (y y )as, which is a strong accumulation point in H (). For the subsequent analysis and the remaining sections of the paper, we will use the following assumption. Assumption 3.3. Let [M, ). Thereexistε,ε 2 > and r> such that for all ( r, + r). meas({x A I : y (x) y (x) <ε })=, meas({x A I : y (x) y (x) <ε 2 })=, Lemma 3.4. Let [M, ) be fixed, and let ( r, + r). It holds that lim meas(a I ) = lim meas(a I )=. Proof. Let us introduce the set A ε := {x A I : y y ε }. From assumption (3.3), we get that meas(a I ) meas(a ε ). (3.6) Due to Chebyshev s inequality we get that ε meas(a ε ) y (x) y (x) dx A I y (x) dx + y (x) y (x) dx,
14 94 J.C. DE LOS REYES AND S. GONZÁLEZ which, by Lemma 3. and Theorem 3.2, implies that for some K >. Therefore, ε meas(a ε ) (meas()) /2 y H + (meas()) /2 y y H K, lim meas ({x : y (x) y (x) ε })=, and the result follows from (3.6). The other case is treated similarly. As a consequence of Lemma 3.4 we also obtain that lim meas(a A )=meas(a ) and lim meas(i I )=meas(i ), (3.7) which, since A =(A A ) (A \A )andi =(I I ) (I \I ), implies that lim (A \A ) = lim (I \I )=. (3.8) Proposition 3.5. Let >Mand ẏ + be a weak accumulation point of (y y ) in W,p (), forsome p>2, as. Thenẏ + satisfies (( ) ) ẏ + a(ẏ +,v)+g y y, ẏ + y 3 y χ A, v L 2 () + (( y + ẏ + ) χi, v ) =. (3.9) L 2 () Proof. See the Appendix. Proceeding as in Proposition 3.5, we also obtain that (( ) ) ẏ a(ẏ,v)+g y y, ẏ y 3 y χ A, v L 2 () + (( y + ẏ ) χi, v ) =, (3.2) L 2 () where ẏ stand for any weak accumulation point of (y y )inw,p (), for some p>2, as. Therefore, we obtain the following result. Theorem 3.6. The function y H () is differentiable at all [M,+ ), andẏ satisfies (( ) ) ẏ a(ẏ,v)+g y y, ẏ y 3 y χ A, v L 2 () + ( ( y + ẏ ) χ I, v ) =. (3.2) L 2 () Proof. Let z denote the difference between two accumulation points of () (y y )as. From (3.9) and(3.2) we obtain that (( ) ) z a(z,v) + g y y, z y 3 y χ A + zχ I, v =. L 2 ()
15 PATH FOLLOWING METHODS FOR STEADY LAMINAR BINGHAM FLOW IN CYLINDRICAL PIPES 95 Choosing v = z in the last expression, we obtain that ( ) μ z 2 H + z 2 L 2 (I + g z ) y y, z y 3 y, z L 2 (A ) =. (3.22) Since ( ) z y y, z y 3 y, z, L 2 (A ) we get, from (3.22), that z =. Consequently, accumulation points are unique and by (3.9) and(3.2) they satisfy (3.2) Path value functional In this section we study the value functional associated to (P ). We prove that the functional is twice differentiable with non positive second derivative, which implies concavity of the functional. Definition 3.3. The functional V () :=J (y ) defined on [M, ), M>, is called the path value functional. Let us start by analyzing the differentiability properties of V. Proposition 3.7. Let [M, ). The value functional V is differentiable at, with V () = 2 2 g 2 dx + y 2 dx. (3.23) 2 A I Proof. Let r> be sufficiently small and let ( r, + r). From (2.3) and by choosing v = y y,we find that 2 a(y + y,y y )+ 2 (q + q, (y y )) L 2 () (f,y y ) 2 =. (3.24) On the other hand, note that 2 a(y + y,y y )= 2 a(y,y ) 2 a(y,y ). Consequently, from (3.24), we obtain that V () V () = 2 a(y,y ) 2 a(y,y ) (f,y y ) 2 + [ψ ( y ) ψ ( y )] dx = [ψ ( y ) ψ ( y )] dx 2 (q + q, (y y )) L 2 (), where ψ is defined by (2.2). Then, from (S ), we conclude that V () V () = z dx, (3.25) where z is defined by z(x) := [ ψ ( y (x)) ψ ( y (x)) g y (x) 2 θ (x) + y ] (x), (y y )(x), θ (x)
16 96 J.C. DE LOS REYES AND S. GONZÁLEZ V () V () a.e. in. Next, we will analyze the limit lim. Using the disjoint partitioning of given by := A A, 2 := A I, 3 := A I,and 4 := I I,wegetthat V () V () = 4 j= j z j dx, where z j represents the value of z when restricted to each set j, j =,...,4. Now, we analyze each integral j z j dx separately. On : Here, we analyze the limit lim that θ (x) = y (x), θ (x) = y (x), ψ( y (x)) = g y (x) g2 2 Thus, we obtain the following pointwise a.e. estimate [ ] z (x) = g [ y (x) y (x) ]+ g2 2 g y (x) 2 y (x) + y (x) y (x), (y (x) y (x)) ( ) = g 2 [ y (x) y (x) ] y (x), y (x) y (x) y (x) z dx. We start by recalling that a.e. in,wehave and ψ( y (x)) = g y (x) g g2 2 [ ] and, therefore, z dx = g [ ][ y y y ], y dx + g2 dx. (3.26) 2 y y 2 Next, we estimate the two integrals in (3.26) separately. First, note that, since we are working in,wehave that y (x) g a.e. Therefore, we obtain the following pointwise estimate in : y (x), y (x) y (x) y (x) y (x) y (x) + y (x) y (x) y (x), y (x) y (x) y (x) y (x) y (x) y (x) + y (x) y (x), y (x) y (x) y (x) (3.27) 2 g y (x) y (x). Therefore, from CauchySchwarz inequality, Theorem 3.2 and (3.27), we have the following estimate g 2 [ y y ][ y ], y dx y y g y y 2 y, y y y dx y y 2 dx y y 2 H L 2. (3.28)
17 PATH FOLLOWING METHODS FOR STEADY LAMINAR BINGHAM FLOW IN CYLINDRICAL PIPES 97 Next, we analyze the second expression in the right hand side of (3.26). Since A =(A A ) (A \A ), we have that g 2 g2 g2 dx = dx dx. (3.29) 2 2 A 2 A \A Further, since, [M, ), we obtain g2 2 which, due to Lemma 3.4, implies that g 2 2 A \A dx A \A g2 2M 2 meas(a \A ), dx, as. (3.3) Thus, from (3.29), (3.3) and the Lebesgue s bounded convergence theorem, we conclude that g 2 2 dx g2 2 Therefore, from (3.26), (3.28) and(3.3), we conclude that dx, as. (3.3) A 2 lim z dx = 2 2 g 2 dx. (3.32) A On 4 : We study the limit of z 4 dx as. Let us recall that a.e. on 4, θ (x) =θ (x) =g, 4 ψ( y (x)) = 2 y (x) 2 and ψ( y (x)) = 2 y (x) 2. Thus, we obtain that z 4 (x) = [ 2 y (x) 2 2 y (x) 2 ] 2 y (x)+ y (x), y (x) y (x) = y (x), y (x), a.e., 2 which implies, since I =(I I ) (I \I ), that [ z 4 dx = ] y, y dx + y, y dx. (3.33) 4 2 I I \I Let us study the two integrals on the right hand side of (3.33) separately. Theorem 3.2 implies that y y strongly in H () as, and, therefore, y, y dx I y 2 dx. I (3.34) On the other hand, due to CauchySchwarz and Hölder inequalities, and since y (x) < g M and y (x) < g M a.e. in 4, we obtain that y, y dx I \I g2 M meas(i \I ),
18 98 J.C. DE LOS REYES AND S. GONZÁLEZ which, due to Lemma 3.4, implythat I \I y, y dx, as. (3.35) Finally, we obtain, from (3.33), (3.34) and(3.35), that lim z 4 dx = y 2 dx. (3.36) 4 2 I On 2 and 3 : We study the behavior of 2 z 2 dx and 3 z 3 dx, as. Let us start with 2 z 2 dx. First, note that a.e. in 2,wehavethatθ (x) = y (x), θ (x) =g, ψ( y (x)) = 2 y (x) 2 and ψ( y (x)) = g y (x) g2 2. Thus, we obtain that z 2 (x) =g y (x) g2 2 2 y (x) 2 g y (x) 2 y (x) + 2 y (x), y (x) y (x), a.e. in 2.Moreover,sincein 2 we have that y (x) <g, a.e., and due to the CauchySchwarz inequality, we have the following pointwise estimate: z 2 (x) g 2 y (x) g + g 2 y (x) y (x), y (x) + g2 2 g 2 y (x) g + y (x) 2 y (x) g + g2 2 (3.37) < g y (x) g + g2 2 a.e. in 2. Then, we divide the analysis in two cases: (i) y (x) g : Since in 2 we have that y (x) < g obtain that a.e., and due to Theorem 3.2 and (3.37), we { y g } z 2 dx g 2 y y dx + g2 2 meas( 2) g meas( 2 ) /2 y y H + g2 2 meas( 2) glmeas( 2 ) /2 + g2 2M meas( 2 2 ). (ii) y (x) g < : Since in 2 we have that y (x) g { y < g } z 2 dx g g 2 3 g2 2M meas( 2 2 ). a.e., we have that g dx + g2 2 meas( 2) (3.38) (3.39) Consequently, (3.37), (3.38), (3.39) and Lemma 3.4 imply that lim z 2 dx =. 2 (3.4) Analogously, we conclude that lim z 3 dx =. 3 (3.4) Then, the result follows from (3.32), (3.36), (3.4) and(3.4).
19 PATH FOLLOWING METHODS FOR STEADY LAMINAR BINGHAM FLOW IN CYLINDRICAL PIPES 99 Proposition 3.8. Let [M, ). The function V () is twice differentiable at, with its second derivative given by V () = 3 g 2 dx + y, ẏ dx, (3.42) A I where ẏ is defined in Proposition 3.6. Moreover, V (), for all [M, ). Proof. Letusfirstprove(3.42). Since q (x) = g, a.e. ina,and q (x) = y (x) a.e. in I,wecanwrite V () = 2 2 q 2 dx. (3.43) From (3.43) we conclude that V () V () = 2 2 q 2 dx 2 2 q 2 dx. V We are concerned with the limit lim () V (). We introduce the notation I j = 2 2 j q 2 dx 2 2 j q 2 dx, j =,...,4, where the sets j, j =,...,4 are defined as in Proposition 3.7, andweanalyze the integrals I j separately. On (lim I ): Let us start by recalling that a.e. on,wehavethat q (x) = q (x) = g. Thus, since A =(A A ) (A \A ), we get () I = () g 2 ( 2 2 ) dx = g 2 ( + ) A 2 2 dx g 2 ( + ) 2 A \A 2 2 dx. 2 Since, [M, ), we get that g 2 ( + ) A \A 2 2 dx g2 ( + ) 2 2M 4 meas(a \A ), which, due to Lemma 3.4, implies that A \A g 2 ( + ) dx as. (3.44) Thus, (3.44), Lemma 3.4 and the Lebesgue s bounded convergence theorem imply that lim ( ) I = 3 g 2 dx. (3.45) A On 4 (lim I 4): First, note that q (x) = y (x) and q (x) = y (x) a.e. on 4. Thus, since I =(I I ) (I \I ), we obtain that () I 4 = 2() I [ y 2 y 2] [ dx y 2 y 2] dx. (3.46) 2() I \I Next, let us analyze the two integrals of the right hand side of (3.46) separately.
20 J.C. DE LOS REYES AND S. GONZÁLEZ [ (i) lim 2( ) I y 2 y 2] dx: Notethat [ y 2 y 2] ( ) ( ) y y y y dx = y, + y, dx, I I }{{} :=J which, implies that J 2 I y, ẏ dx ( ) y y I y, y, ẏ dx + ( ) y y I y, ẏ dx. Next, we separately analyze the two terms in the right hand side of the inequality above. Since ẏ is an accumulation point of y y in H () as (see Rem. 3.2 and Thm. 3.6), we have that ( ) y y y, ẏ dx as. (3.47) I On the other hand, we have that I ( ) y y y, y, ẏ dx ( ) y y y, ẏ dx + I which, since ẏ is an accumulation point of y y I I ( ) y y y y, dx, in H () as, implies that ( ) y y y, y, ẏ dx as. (3.48) Furthermore, Theorem 3.2 yields that ( ) y y y y, dx = y y 2 H L 2. (3.49) I Consequently, (3.47), (3.48) and(3.49) imply lim [ y 2 y 2] dx = y, ẏ dx. (3.5) 2() I I [ (ii) lim 2( ) I \I y 2 y 2] dx: First note that I \I y 2 y 2 dx I \I y y y dx + I \I y y y dx.
21 PATH FOLLOWING METHODS FOR STEADY LAMINAR BINGHAM FLOW IN CYLINDRICAL PIPES Therefore, from Theorem 3.2, Remark 2.4 and Hölder inequality, we have that y I \I 2 y 2 dx meas(i \I ) ( /4 y L 4 + y L 4) y y H L 2K meas(i \I ) /4, and then, from Lemma 3.4, we conclude that lim Finally, (3.46), (3.5) and(3.5) implythat [ y 2 y 2] dx =. (3.5) 2() I \I lim I 4 = y, ẏ dx. (3.52) I On 2 and 3 : We analyze the limits lim I 2 and lim I 3. We start by analyzing lim I 2. Thus, we recall that in 2,wehavethat y (x) g. Then, from Theorem 3.2, Remark2.4 and Hölder inequality, we conclude that I g 2 2 y 2 dx 2 2 y 2 y 2 dx LKmeas(2 ) /4, which, due to Lemma 3.4, implies that lim I 2 =. (3.53) Analogously, we conclude that lim I 3 =. (3.54) Finally, (3.45), (3.52), (3.53), and (3.54) imply(3.43). Now, we prove that V (). Using v =ẏ χ I in (3.2), we obtain that a(ẏ, ẏ χ I )+ ( ) y, ẏ χ + ( ) I ẏ L 2 () χ I, ẏ χ =, I L 2 () which implies that (μ + ) ẏ 2 dx = I y, ẏ dx. I Thus, we can easily conclude that y, ẏ dx, I which yields that V () = 3 g 2 dx + y, ẏ dx. A I
22 2 J.C. DE LOS REYES AND S. GONZÁLEZ 3.3. Model functions and pathfollowing algorithm In this section, following [4], we propose model functions which approximate the value functional V () and share some of its qualitative properties. These model functions will then be used for the development of the path following algorithm. From Theorem 3.2 and Propositions 3.7 and 3.8, it follows that V (), [M, ) is a monotonically increasing and concave function. We then propose the model functions m() =C C 2 μ + G, (3.55) with C R, C 2 andg, which share the main qualitative properties of V (), i.e., ṁ() and m(). To motivate the introduction of these model functions, let us take the test function v = y χ I in (3.2). We get that a(ẏ,y χ I )+ ( ) y χ I, y χ I L 2 () + ( ẏ χ I, y χ I =. (3.56) )L 2 () From the definition of a(, ), we obtain that (μ + ) ( ) ẏ, y χ I L 2 () + y 2 dx =. I Consequently, by using Propositions 3.7 and 3.8, weobtain (μ + ) V ()+ 3 g 2 dx +2 V () 2 g 2 dx =, A A which implies that (μ + ) V ()+2 V ()+μ 3 g 2 dx =. (3.57) A Note that A g 2 is a function of which is uniformly bounded from above by g 2 meas(). Replacing V by m and the dependent term A g 2 by 2G, we obtain the differential equation (μ + ) m()+2ṁ()+2 3 μg =, (3.58) whose solutions are the family of functions given by (3.55). In order to determine C, C 2 and G, we fix a reference value r >, r, for which the value V ( r )is known. Then, we use the following conditions m() =V (), m( r )=V ( r ), ṁ() = V (). Solving the resulting system of nonlinear equations C C 2 μ + G = V (), C C 2 G = V ( r ), μ + r r C 2 (μ + ) 2 + G 2 = V (),
23 PATH FOLLOWING METHODS FOR STEADY LAMINAR BINGHAM FLOW IN CYLINDRICAL PIPES 3 we obtain that G = r(v ( r ) V ()) r r 2 ϑ(μ + ), (3.59) μ( r ) [ ] where ϑ := r V () r(v ( r) V ()) ( r ). Consequently, the parameters C and C 2 are given by C 2 =(μ + ) 2 ( V () G 2 ), C = V ()+ C 2 μ + + G Once we have determined the values of the coefficients of the model, we are able to propose the updating strategy for. Let {τ k } satisfy τ k (, ) for all k N and τ k ask, and assume that V ( k )is available. Following [4], the idea is to have a superlinear rate of convergence for our algorithm, i.e., given k, theupdatevalue k+ should ideally satisfy V V (k+ ) τk V V (k ), (3.6) where V := lim V (). Since V and V ( k+ ) are unknowns, we approximate these values by lim k m( k ) and m( k+ ), respectively. Hereafter, we use the notation C,k, C 2,k and G k for the coefficients on the model function (3.55) related with each k. Further, note that lim k m( k )=C,k.Thus,(3.6) is replaced by C,k m( k+ ) τ k C,k m( k ). (3.6) Calling β k := τ k C,k m( k ) and solving the equation C,k m( k+ )=β k, we obtain that k+ = D k 2 + D 2 k 4 + μg k β k, (3.62) where D k = (C 2,k+G k ) β k μ. Next, we write a pathfollowing algorithm which uses the update strategy for, given by(3.62). Algorithm PF.. Select r and compute V ( r ). Choose > max(, r ) and set k =. 2. Solve { a(yk,v)+(q k,v) L 2 () (f,v) 2 =, for all v H () max(g, k y k (x) )q k (x) =g k y k (x), a.e. in. 3. Compute V ( k ), V (k ) and update k by using k+ = D k 2 + D 2 k 4 + μg k β k (3.63) 4. Stop, or set k := k + andgotostep2. 4. Semismooth Newton method In this section we state an algorithm for the efficient solution of (3.63). Since no smoothing operation takes place in the complementary function in (3.63), it is not possible to get Newton differentiability in infinite
24 4 J.C. DE LOS REYES AND S. GONZÁLEZ dimensions (see [27], Sect. 3.3). Therefore, we consider a discretized version of system (3.63), and propose a semismooth Newton method to solve this problem. Specifically, we state a primaldual scheme to solve system (3.63) and prove local superlinear convergence of the method. By involving the primal and the dual variables in the same algorithm, we compute the solutions to the discrete versions of (P )and(p ) simultaneously. The algorithm proposed is a particular case of the Newton type algorithms developed in [6]. Let us introduce the definition of Newton differentiability Definition 4.. Let X and Z be two Banach spaces. The function F : X Z is called Newton differentiable if there exists a family of generalized derivatives G : X L(X, Z) such that lim h h X F (x + h) F (x) G(x + h)h Z =. Throughout this section we denote discretized quantities by superscript h. For a vector v R n we denote by D(v) :=diag(v) then ndiagonal matrix with diagonal entries v i. Besides that, we denote by the Hadamard product of vectors, i.e., v w:= (v w,...,v n w n ). We use a finite element approximation of system (3.63) and consider the spaces V h := { η C() : η T Π, T T h}, W h := { q h := (q h,q h 2 ) L 2 () : q h T,q h 2 T Π, T T h}, to approximate the velocity y h and the multiplier q h, respectively. Here, Π k denotes the set of polynomials of degree less or equal than k and T h denotes a regular triangulation of. Thus, the discrete analogous of (3.63) is given by { A h μ y + B h q f h = max ( ge h,ξ( h y) ) q g h (4.) y =, for >, where A h μ Rn n is the stiffness matrix, e h R 2m is the vector of all ones and B h R n 2m is obtained in the usual way, from the bilinear form (, ) L 2 () and the basis functions of V h and W h. Here, y h R n and q h R 2m are the solution coefficients of the approximated regularized primal and dual solutions y h V h and q h W h, respectively. Further, we construct the right hand side f h using the basis functions ϕ i V h,i =,...,n, (see [], Sect. 6). The discrete version of the gradient is given by ( ) h h := 2 h R 2m n, (4.2) where h := ϕi(x) Tk x and 2 h := ϕi(x) Tk x 2,fori =,...,n; k =,...,m.notethat ϕi(x) Tk x and ϕi(x) Tk x 2 in each triangle T k, respectively. Consequently, we obtain that h y = y h (x). Hereafter, the matrix A h μ is assumed to be symmetric and positive definite. The function ξ : R2m R 2m is defined by are the constant values of ϕi(x) x and ϕi(x) x 2 (ξ(p)) i =(ξ(p)) i+m := (p i,p i+m ) for p R 2m, i =,...,m. System (4.) can also be written as the following operator equation: [ ] F (y h,q h A h μy h + B h q h f h ):= max ( ge h,ξ( h y h ) ) q h g h y h =. (4.3) It is well known (see e.g. [7,27]) that the maxoperator and the norm function ξ involved in (4.3) are semismooth. Furthermore, this is also true for the composition of semismooth functions that arises in (4.3). A particular
25 PATH FOLLOWING METHODS FOR STEADY LAMINAR BINGHAM FLOW IN CYLINDRICAL PIPES 5 element of the generalized Jacobian of max(, ) :R N R N is the diagonal matrix G max R N N defined by (G max (v)) ii := { if vi, if v i <, for i N. (4.4) Consequently, given approximations yk h and qh k the Newton step to (4.3) at(yh k,qh k )isgivenby: [ ][ ] [ A h μ B h ( χ Ak+ D(qk h)p ( δy A h ) ) μ yk h h h yk h gi2l h D(m h k ) = Bh qk h + f ] h δ q D(m h k )qh k + g h yk h, (4.5) where m h k := max ( ge h,ξ( h yk h)) R 2m,andχ A = D(t h k ) R2m 2m with ( t h k ) i := { if ξ( h y h k ) i g else. (4.6) Further, P h R 2m 2m denotes the generalized Jacobian of ξ, i.e., forp R 2m,wehavethat ξ i ξ i P h p j p j+m (p) :=, where the block diagonal matrices are defined by ξ i+m p j ξ i+m p j+m ξ i = ξ { p i i+m := δ (p i,p i+m) if (p i,p i+m ) ij p j p j ε if (p i,p i+m ) = for i, j =,...,m, ξ i = ξ { p i+m i+m := δ (p i,p i+m) if (p i,p i+m ) ij p j+m p j+m ε 2 if (p i,p i+m ) = for i, j =,...,m, with ε and ε 2 real numbers such that (ε,ε 2 ). From the invertibility of D(m h k ) we obtain that δ q = q h k + D(m h k) ( g h y h k + C h k h δ y ), (4.7) where C h k := gi 2l χ Ak+ D(q h k )P h ( h y h k ). Thus, the remaining equation for δy can be written as where the matrix Ξ,k and the right hand side η,k are given by Ξ,k δ y = η,k, (4.8) Ξ,k := A h μ + B h D(m h k) Ck h h, η,k := A h μyk h + M h f h gb h D(m h k) h yk. h It can be verified (cf. [6], p. 8) that the matrix Ξ,k is symmetric at the solution. Thanks to [6], Lemma 3.3, we know that the condition ξ(qk h) i g, fori =,...,m, must hold to guarantee the positive definiteness of the matrix Ch k. Moreover, we can assert that if { the last } condition is fulfilled, the matrix Ξ,k is positive definite, λ min (Ξ,k ) λ min (A h μ) > and the sequence is uniformly bounded. Ξ,k k N
26 6 J.C. DE LOS REYES AND S. GONZÁLEZ Due to these results, we know that if ξ(qk h) i g holds for all i =,...,m, the solution of (4.8) existsfor all k and it is a descent direction for the objective functional in (P ). However, this condition is unlikely to be fulfilled by all i m and k N. To overcome this difficulty, Hintermüller and Stadler [6] constructed a globalized semismooth Newton algorithm by modifying the term involving D(qk h)p h ( y yk h ) for indices i in which ξ(qk h) i >g. This is done by replacing qk h by g max ( g, ξ(q h k ) i) ((q h k ) i, (q h k ) i+m), when assembling the system matrix Ξ,k. Thus, we guarantee that ξ(qk h) i g for i =,...,m. Further, we obtain a modified system matrix, denoted by Ξ +,k, which replaces Ξ,k in (4.8). This new matrix is positive } definite for all and the sequence {(Ξ +,k ) is uniformly bounded. k N Algorithm SSN.. Initialize (y h,qh ) Rm R 2m and set k =. 2. Estimate the active sets, i.e., determine χ Ak+ R 2m 2m. 3. Compute Ξ +,k if the dual variable is not feasible for all i =,...,m;otherwisesetξ+,k =Ξ,k. Solve Ξ +,k δ y = η,k. 4. Compute δ q from (4.7). 5. Update yk+ h := yh k + δ y and qk+ h := qh k + δ q. 6. Stop, or set k := k + and go to step 2. Following [6], Lemma 3.5, we know that qk h qh and yk h yh implies that Ξ +,k converges to Ξ,k as k. Thus, thanks to this result we can state the following theorem. Theorem 4.. The iterates (y h k,qh k ) of Algorithm SSN converge superlinearly to (yh,q h ) provided that (y h,q h ) is sufficiently close to (y h,qh ). Proof. We refer the readers to [6], Theorem 3.6, for the complete proof. The projection procedure, which yields the matrix Ξ +,k, assures that in each iteration of Algorithm SSN, δ y =(Ξ +,k ) η,k constitutes a descent direction for the objective functional in (P ). Additionally, steps 3. and 4. of the algorithm involve a decoupled system of equations for δ y and δ q, which is obtained directly, due to the regularization proposed and the structure of the method. Moreover, the computation of δ q through (4.7) turns out to be computationally efficient, since only the inverse of a diagonal matrix is needed. 5. Numerical results In this section we present numerical experiments which illustrate the main properties of the pathfollowing and the semismooth Newton methods applied to the numerical solution of laminar Bingham fluids. The experiments have been carried out for a constant function f, representing the linear decay of pressure in the pipe. The parameter is updated using the pathfollowing strategy defined in Section 3.3. Unless we specify the contrary, we stop the Algorithm PF as soon as r h k := (r,h k,r2,h k,r3,h k ) is of the order 7,where r,h k = y h k +(A h μ ) (B h q h k f h ) H,h / f h L 2,h r 2,h k = max(ge h,ξ(q h k + h y h k )) q h k g(q h k + h y h k ) L 2,h r 3,h k = max(, q h k g) L 2,h
27 PATH FOLLOWING METHODS FOR STEADY LAMINAR BINGHAM FLOW IN CYLINDRICAL PIPES Figure. Example : flow of a Bingham fluid defined by g =,μ =andf =(left)and velocity profile along the diagonal y(x,x ) (right). Figure 2. Example : final inactive set I. with A h μ, B h, h and ξ defined as in (4.). Here H,h, L 2,h and L 2,h denote the discrete versions of H, L 2 and L 2 respectively. r,h k and r 2,h k describe the improvement of the algorithm towards the solution of the discrete version of the optimality system (S), while r 3,h k measures the feasibility of q h k. We use the mass matrix to calculate the integrals related to the space V h and a composite trapezoidal formula for the integrals associated to the space W h. Additionally we use the sequence τ k =. k Example In our first example, we focus on the behavior of Algorithm PF. We consider := ], [ ], [ and compute the flow of a Bingham fluid defined by μ = g =andf =. We work with a uniform triangulation, with h =.46 ( /28), where h is the radius of the inscribed circumferences of the triangles in the mesh. In this example, we use the initial values r =and =. The inner Algorithm SSN for = is initialized with the solution of the Poisson problem A h μy h = f h together with q h = and is finished if the residual δ y is lower than ɛ,whereɛ denotes the machine accuracy. The resulting velocity function is displayed in Figure and the final inactive set in Figure 2. The value of the regularization parameter k reaches a factor of 3 in three iterations and we obtain a maximum velocity of.29. The graphics illustrate the expected mechanical properties of the material, i.e., since the shear stress transmitted by a fluid layer decreases toward the center of the pipe, the Bingham fluid moves like a solid in that sector. Besides that, Figure shows that there are no stagnant zones in the flow (see [22]).
28 8 J.C. DE LOS REYES AND S. GONZÁLEZ Table. updates and convergence behavior of Algorithm PF for a Bingham fluid defined by g =,μ =andf =. #it. k rk h y h k+ y h H k ν,h k h.8 e e e e e e e e e 4 Table 2. Number of iterations of Algorithm SSN in each pathfollowing iteration. k.8 e+3.82 e+7.9 e+3 #it. 7 6 =4 Table 3. Number of iterations of Algorithm SSN without any automatic updating strategy. k.8 e+3.82 e+7.9 e+3 #it.ssn 3 33 fails to converge In Table we report the values of the regularization parameter k and the residuals r h k and ν h k = y h k+ y h k H,h y h k y h k H,h From the behavior of rk h, it is possible to observe a superlinear convergence rate of the Algorithm PF according to the strategy proposed in (3.6). Furthermore, the behavior of νk h implies a superlinear convergence rate of y k towards the solution, as k increases. These data are depicted in Figure 3, where the two magnitudes are plotted in a logarithmic scale. In Table 2, we show the number of inner iterations that Algorithm SSN needs to achieve convergence in each iteration of Algorithm PF and the total number of SSN iterations needed. It can be observed that the path following strategy allows to reach large values of k and, consequently, to obtain a better approximation of the solution of the problem. In contrast to these results, in Table 3 we show the number of iterations that Algorithm SSN needs to achieve convergence without any updating strategy. In this case, the algorithm does not only need more iterations for each value of k, but also fails to converge for large values of it. Finally, in Figure 4 we plot and compare the path value functional V () (solid line) and the model functions m( k ) calculated from the values C,k, C 2,k and G k given in each iteration of the algorithm. It can be observed that as k increases, m( k ) becomes a better model for V (). However, even for small values of k, the model functions stay close to the value functional Example 2 In this example, we compare the numerical behavior of Algorithm PF versus a penaltynewtonuzawaconjugate gradient method proposed by Dean et al. in[6]. We consider the flow of a Bingham fluid in the cross section of a cylindrical pipe, given by the disk defined by := {x =(x,x 2 ) R 2 : x 2 + x2 2 <R2 },wherer>. It is well known (see [2], Ex. 2, p. 8) that in
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