Generalized Newtonian Fluid Flow through a Porous Medium

Transcription

1 Generalized Newtonian Fluid Flow through a Porous Medium V.J. Ervin Hyesuk Lee A.J. Salgado April 24, 204 Abstract We present a model for generalized Newtonian fluid flow through a porous medium. In the model the dependence of the fluid viscosity on the velocity is replaced by a dependence on a smoothed (locally averaged) velocity. With appropriate assumptions on the smoothed velocity, existence of a solution to the model is shown. Two examples of smoothing operators are presented in the appendix. A numerical approximation scheme is presented and an a priori error estimate derived. A numerical example is given illustrating the approximation scheme and the a priori error estimate. Key words. Darcy equation, Generalized Newtonian fluid AMS Mathematics subject classifications. 65N30, 75D03, 76A05, 76M0 Introduction Of interest in this article is the modeling and approximation of generalized Newtonian fluid flow through a porous medium. Darcy s modeling equations for a steady-state fluid flow through a porous medium,, are ν eff K u + p = 0, in, (.) u = 0, in. (.2) where u and p denote the velocity and pressure of the fluid, respectively. K(x) in (.) represents the permeability of the medium at x, which is assumed to be a symmetric, positive definite tensor. As our investigations are not concerned with K, we assume that K is of the form k(x)i where k(x) is a Lipschitz continuous, positive, bounded and bounded away from zero, scalar function. ν eff in (.) represents the effective viscosity of the fluid. Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA. (vjervin@clemson.edu) Department of Mathematical Sciences, Clemson University, Clemson, SC , U.S.A. (hklee@mail.clemson.edu) Partially supported by the NSF under grant no. DMS Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A. (asalgad@utk.edu)

2 In the case of a Newtonian fluid we have that ν eff is a positive constant. For a generalized Newtonian fluid ν eff is a function of u. Two such examples are Power Law Model: ν eff ( u ) = c ν u r 2, Cross Model: ν eff ( u ) = ν + ν 0 ν, (.3) + c ν u 2 r where c ν, ν 0, ν and r are fluid dependent constants. For shear thinning fluids < r < 2. (In modeling the viscosity of shear thinning fluids the Power Law model suffers the criticism that as u 0 ν eff.) For the case of a Newtonian fluid (.), (.2) are well studied. The two standard approaches in analyzing (.), (.2) are: (i) study (.), (.2) as a mixed formulation problem for u and p (either (u, p) H div () L 2 (), or (u, p) L 2 () H ()), or (ii) use (.2) to eliminate u in (.) to obtain a generalized Laplace s equation for p. For generalized Newtonian fluids, with ν eff = ν eff ( u ), assumptions are required on ν eff in order to establish existence and uniqueness of solutions. Typical assumptions are uniform continuity of ν eff ( u )u and strong monotonicity of ν eff ( u ) [7, 8, 0], i.e., there exists C > 0 such that ν eff ( u )u ν eff ( v )v C u v, u, v IR d, (.4) (ν eff ( u )u ν eff ( v )v) (u v) C (u v) (u v), u, v IR d. (.5) A more general setting where the fluid rheology is defined implicitly has been analyzed in [5, 6]. The case where the fluid viscosity depends on the shear rate and pressure has been studied in [3, 2]. For both of these cases additional structure beyond (.4) and (.5) is required in order to establish existence and uniqueness of a solution. A nonlinear Darcy fluid flow problem, with a permeability dependent upon the pressure was investigated by Azaïez, Ben Belgacem, Bernardi, and Chorfi [2], and Girault, Murat, and Salgado []. For a Lipschitz continuous permeability function, bounded above and bounded away from zero, existence of a solution (u, p) L 2 () H () was established. Important in handling the nonlinear permeability function, in establishing existence of a solution, was the property that p H (). In [2] the authors also investigated a spectral numerical approximation scheme for the nonlinear Darcy problem, assuming an axisymmetric domain. A convergence analysis for the finite element discretization of that problem was given in []. Our interest in this paper is in relaxing the assumptions (.4) and (.5). Specifically, our interest is assuming that ν eff ( ) is only Lipschitz continuous and both bounded above and bounded away from zero. However, relaxing the conditions (.4) and (.5) requires us to make an additional assumption regarding the argument of ν eff ( ). In order to obtain a modeling system of equations for which a solution can be shown to exist, we replace u in ν eff ( u ) by a smoothed velocity, u s. The approach of regularizing the model with the introduction of u s is, in part, motivated by the fact that the Darcy fluid flow equations can be derived by averaging, e.g. volume averaging [6], homogenization [], or mixture theory [4]. Presented in the Appendix are two smoothing operators for u. One is a local averaging operator, whereby u s (x) is obtained by averaging u in a neighborhood of x. The second smoothing operator, which is nonlocal, computes u s (x) using a differential filter applied to u. That is, u s is given by the solution to an elliptic differential equation whose right hand side is u. For establishing the existence 2

3 of a solution to (.)-(.2), the key property of the smoothing operators is that they transform a weakly convergent sequence in L 2 () into a sequence which converges strongly in L (). For the mathematical analysis of this problem it is convenient to have homogeneous boundary conditions. This is achieved by introducing a suitable change of variables. For example, assuming = Γ in Γ Γ out, in the case the specified boundary conditions are u ( n) = g in on Γ in, u n = 0 on Γ, p = p out on Γ out, we introduce functions b(x) and p b (x) defined on satisfying b = 0, in, b n = g in, on Γ in, b t i = 0, on Γ in, b = 0, on \Γ in, where t i, i =,..., (d ) denotes an orthogonal set of tangent vectors on Γ in. p b = 0, in, p b = p out, on Γ out, p b n = 0, on \Γ out. (In case the pressure is specified on the inflow boundary Γ in, then b = 0, and the definition of p b is appropriately modified.) With the change of variables: u = u 0 + b and p = p 0 + p b, and subsequent relabeling u 0 = u, p 0 = p and f = p b we obtain the following system of modeling equations: where β( u s + b ) = ν eff ( u s + b ) k. β( u s + b )u + β( u s + b )b + p = f, in, (.6) u = 0, in, (.7) u n = 0, on Γ in Γ, (.8) p = 0, on Γ out, (.9) In the next section we show that, under suitable assumptions on β( ) and u s, there exists a unique solution to (.6)-(.9). An approximation scheme is presented in Section 3, and an a priori error estimate derived. A numerical example illustrating the approximation scheme and the a priori error estimate is presented in Section 4. 2 Existence and Uniqueness In this section we investigate the existence and uniqueness of solutions to the nonlinear system equations (.6)-(.9). We assume that IR d, d = 2 or 3, is a convex polyhedral domain and for vectors in IR d denotes the Euclidean norm. Throughout, we use C to denote a generic nonnegative constant, independent of the mesh parameter h, whose actual value may change from line to line in the analysis. 3

4 We make the following assumptions on β( ) and u s. Assumptions on β( ) Aβ : β( ) : IR + IR +, Aβ2 : 0 < β min β(s) β max, s IR +, Aβ3: β is Lipschitz continuous, β(s ) β(s 2 ) C β s s 2. Assumptions on u s Au s : For u L 2 (), u s L () C s u L 2 (), Au s 2: For {u n } n= L2 (), with u n converging weakly to u L 2 (), then {u s n} n= u s in L (), Au s 3: The mapping u u s is linear. Weak formulation of (.6)-(.9) Let X = {v H div () : v n = 0, on Γ in Γ}. We use (f, g) := f g d, and f := (f, f) /2 converges to to denote the L 2 inner product and the L 2 norm over, respectively, for both scalar and vector valued functions. Additionally, we introduce the norm v X = ( ) /2 ( v v + v v) d. Remark: For v H div () it follows that v n H /2 ( ). For the interpretation of the condition v n = 0 on Γ in Γ see [9, 5]. We restate (.6)-(.9) as: Given b, f L 2 (), find (u, p) X L 2 (), such that for all v X and q L 2 () (β( u s + b )u, v) (p, v) + (β( u s + b )b, v) = (f, v), (2.) (q, u) = 0. (2.2) For the spaces X and L 2 () we have the following inf-sup condition inf sup (q, v) c 0 > 0. (2.3) q L 2 () v X q v X We begin by establishing boundedness of any solution to (2.)-(2.2). Lemma 2. Any solution (u, p) X L 2 () to (2.)-(2.2) satisfies u X + p C ( b + f ). (2.4) Proof: From (2.2) and that X L 2 () we have that any solution u to (2.)-(2.2) satisfies u = 0. (2.5) 4

5 With the choice v = u, q = p, subtracting (2.2) from (2.), and using assumption Aβ2 yields (β( u s + b )u, u) = (β( u s + b )b, u) + (f, u), β min u 2 β max b u + f u. (2.6) Combining (2.5) and (2.6) we obtain the stated bound for u. The estimate for p is obtained using the inf-sup condition (2.3). p (p, v) sup c 0 v X = sup v X c 0 v X ( β( u s + b )u + β( u s + b )b + f ) c 0 c 0 (β max ( u + b ) + f ), from which the stated bound follows. (β( u s + b )u, v) + (β( u s + b )b, v) (f, v) v X Define Z = { v X : (q, v) = 0, q L 2 () }. Because of the inf-sup condition (2.3), the weak formulation (2.)-(2.2) can be equivalently stated as: Given b, f L 2 ()), find u Z, such that for all v Z (β( u s + b )u, v) + (β( u s + b )b, v) = (f, v). (2.7) Remark: For v Z, v X = v, as v = 0. To establish the existence of a solution to (2.7) we use the Leray-Schauder fixed point theorem. To do this we show that a solution to (2.7) is a fixed point of a compact mapping Φ. Theorem 2. For β( ) and u s satisfying assumptions Aβ Aβ3 and Au s Au s 2, respectively, there exists a solution u to (2.7). Proof: Let Φ : Z Z be defined by Φ(u) = w, where w satisfies (β( u s + b )w, v) + (β( u s + b )b, v) = (f, v). (2.8) That Φ is well defined follows from Aβ2 and the Lax-Milgram theorem. To show that Φ is a compact operator, let {u n } n= denote a bounded sequence in Z. From {u n} n= we can extract a subsequence, which we again denote as {u n } n=, such that {u n} n= converges weakly to u Z. For w n = Φ(u n ), using (2.8) (β( u s + b )w, v) (β( u s n + b )w n, v) = (β( u s + b )b, v) + (β( u s n + b )b, v) (β( u s n + b ) (w w n ), v) = ((β( u s + b ) β( u s n + b )) w, v) ((β( u s + b ) β( u s n + b )) b, v). 5

6 With v = w w n, and using Aβ2 and Aβ3 β min w w n 2 C β ( u s + b u s n + b ) w w w n + C β ( u s + b u s n + b ) b w w n C β u s u s n w w w n + C β u s u s n b w w n C β d u s u s n L () w w w n + C β d u s u s n L () b w w n w w n X = w w n C β d u s u s β n L () ( w + b ), min from which, with Au s 2, we can conclude that Φ is a compact operator. For r = βmax β min ( b + f ), from Lemma 2. we have that Φ(u) r, u Z. Then, applying the Leray-Schauder fixed point theorem [7] we obtain that there exists a u Z such that u = Φ(u). Under small data conditions we have the following theorem guaranteeing uniqueness of solutions to (2.7). Theorem 2.2 { With the stated assumptions Aβ Aβ3 and Au s Au s 2, and the condition that } b max β min /β max, β min /(C β d Cs ), if a solution u to (2.7) exists satisfying { } β min β u < max, min b, (2.9) β max C β d Cs then there is no other solution to (2.7). Proof: Suppose that both u and w Z satisfy (2.7), i.e., together with (2.7) we have that (β( w s + b )w, v) + (β( w s + b )b, v) (f, v) = 0, v Z. (2.0) With v = u w, subtracting (2.0) from (2.7) and using the bounds for β( ) we obtain (β( w s + b )(u w), (u w)) + ((β( u s + b ) β( w s + b ))u, (u w)) + ((β( u s + b ) β( w s + b ))b, (u w)) = 0 (2.) β min u w 2 β max u u w + β max b u w (β min β max ( u + b )) u w 0 (2.2) w = u, provided u < β min β max b. (2.3) Alternatively, from (2.), using Aβ3 and Au s, β min u w 2 C β d u s w s L () u u w + C β d u s w s L () b u w C β d Cs ( u + b ) u w 2 (β min C β d Cs ( u + b )) u w 2 0 β min w = u, provided u < b. C β d Cs 6

7 3 Finite Element Approximation In this section we investigate the finite element approximation to (u, p) satisfying (.6)-(.9). Let T h be a triangulation of made of triangles (in IR 2 ) or tetrahedrons (in IR 3 ). computational domain is defined by = K Th K. We assume that there exist constants c, c 2 such that Thus, the c h h K c 2 ρ K, where h K is the diameter of triangle (tetrahedron) K, ρ K is the diameter of the greatest ball (sphere) included in K, and h = max K Th h K. For k IN, let P k (A) denote the space of polynomials on A of degree no greater than k, and RT k (T h ) the (Piola) affine transformation of the Raviart-Thomas elements of order k on the unit triangle. We define the finite element spaces X h, X s h and Q h as follows. Note that as X h X h := {RT k (T h ) X}, (3.) X s h := { v X C 0 () : v K P l (K), K T h }, (3.2) Q h := { q L 2 () : q K P k (K), K T h }. (3.3) Additionally, let Z h := {v X h : (q, v) = 0, q Q h }. (3.4) Q h, for v Z h we have that v = 0, thus v X = v. For X h and Q h defined in (3.) and (3.3), the following discrete inf-sup condition is satisfied inf q Q h (q, v) sup c 0 > 0. (3.5) v X h q Q v X With X h, Z h, Q h defined above, we have the following approximation properties [4, 3]. For u Z H k+ () and p H k+ () inf v Z h u v X = inf v Z h u v C inf v X h u v = C h k+ u H k+ (), (3.6) inf q Q h p q C h k+ p H k+ (). (3.7) The approximation scheme we investigate is: Given b, f L 2 (), determine (u h, p h ) X h Q h, satisfying (β( u s h + b )u h, v) (p h, v) + (β( u s h + b )b, v) = (f, v), v X h (3.8) (q, u h ) = 0, q Q h. (3.9) Regarding u s h, note that applying a smoother to a function v X h (typically) does not result in v s Xh s. Therefore, we let ũs h Hl+ () C 0 () denote the result of the smoother applied to u h, and define u s h (x) = I hũ s h (x), (3.0) 7

8 where I h : C 0 () Xh s denotes an interpolation operator. We assume that the smoothed velocity ũ s h dependent on ũ s h, C ũ s such that h is sufficiently regular such that there exists a constant The precise dependence of Cũs h ũ s h I hũ s h L () Cũs h h l+. (3.) on ũ s h will depend on the particular smoother used. The existence, uniqueness, and boundedness of the solutions (u n h, pn h ) to (3.8)-(3.9) are established in a completely analogous manner as for the continuous problem. Corollary 3. (See Lemma 2..) Any solution (u, p) X h Q h to (3.8)-(3.9) satisfies u h X + p h C ( b + f ). (3.2) Corollary 3.2 (See Theorem 2..) For β( ) and u s h satisfying assumptions Aβ Aβ3 and Aus Au s 2, respectively, there exists a solution (u h, p h ) to (3.8)-(3.9). Proof: The existence of u h is established as that for u in Theorem 2.. The existence of p h then follows from the discrete inf-sup condition (3.5). In the next lemma we present the a priori error estimate for the approximation given by (3.8)-(3.9). Lemma 3. For (u, p) H k+ () X H k+ () satisfying (2.)-(2.2), (u h, p h ) satisfying (3.8)- (3.9), and u satisfying the small data condition C β d Cs ( u + b ) < β min, (3.3) and assuming that Cũs h given in (3.) is bounded by a constant C u, we have that there exists C > 0 such that ( u u h X + p p h C h k+ u H k+ () + h k+ p H k+ () + C u h l+). (3.4) Remark: The condition (3.3) guarantees uniqueness of the solution to (3.8)-(3.9), see Theorem 2.2. Proof: We have that the solutions u h and u to (3.8)-(3.9) and (2.)-(2.2), respectively, satisfy the following equations for all v Z h : and (β( u s h + b )u h, v) + (β( u s h + b )b, v) = (f, v), (3.5) (β( u s h + b )u, v) + (β( us h + b )b, v) = (f, v) ((β( us + b ) β( u s h + b )) u, v) ((β( u s + b ) β( u s h + b )) b, v). (3.6) 8

9 With e = u u h, subtracting equations (3.5) and (3.6) we obtain (β( u s h + b )e, v) = ((β( us + b ) β( u s h + b )) u, v) ((β( u s + b ) β( u s h + b )) b, v), v Z h. (3.7) For U Z h, let e = (u U) + (U u h ) := Λ + E. Then, for v = E, (3.7) becomes (β( u s h + b )E, E) = (β( us h + b )Λ, E) ((β( us + b ) β( u s h + b )) u, E) ((β( u s + b ) β( u s h + b )) b, E). (3.8) Next we bound each of the terms in (3.8). (β( u s h + b )E, E) β min E 2. (3.9) (β( u s h + b )Λ, E) β max Λ E ɛ E 2 + 4ɛ β2 max Λ 2. (3.20) ((β( u s + b ) β( u s h + b )) u, E) (β( us + b ) β( u s h + b )) u E C β d u s u s h L () u E C β d ( u s ũ s h L () + ũ s h us h L ()) u E C β d ( Cs u u h + ũ s h I hũ s h L ()) u E C β d ( Cs ( Λ + E ) + ũ s h I hũ s h L ()) u E C β d Cs u E 2 + ɛ 2 E 2 + 2ɛ 2 C 2 β d u 2 ( C 2 s Λ 2 ) + ũ s h I hũ s h 2 L () (3.2). A similar bound to that given in (3.2) holds for the third term on the right hand side of (3.8). Combining the estimates (3.9)-(3.2) with (3.8) we have ) (β min ɛ C β d Cs ( u + b ) 2ɛ 2 ( 4 ɛ β 2 max + 2 ɛ 2 C 2 β d C2 s ( u 2 + b 2 ) E 2 ) Λ ɛ 2 C 2 β d ( u 2 + b 2 ) ) ũ s h I hũ s h 2 L (). (3.22) Hence, in view of the stated hypothesis (3.3), there exists C > 0 such that E C ( Λ + ũ s h I hũ s h L ()). Finally, from the triangle inequality and (3.6) we have that ( u u h X = u u h Λ + E C h k+ u H k+ () + C u h l+). (3.23) To obtain the error estimate for the pressure, let P Q h. Then, from (3.5) we have that there exists v X h such that c 0 P p h (P p h, v) v X = (P, v) (p h, v) v X. 9

10 Using (3.8) and (2.) we obtain c 0 v X P p h (P, v) + (f, v) (β( u s h + b )u h, v) (β( u s h + b )b, v) = (P, v) (p, v) + (β( u s + b )u, v) + (β( u s + b )b, v) (β( u s h + b )u h, v) (β( u s h + b )b, v) = (P p, v) + (β( u s h + b ) (u u h), v) + ((β( u s + b ) β( u s h + b )) u, v) + ((β( u s + b ) β( u s h + b )) b, v) c 0 P p h P p + β max u u h ( + C β d Cs u u h + ũ s h I hũ h L ()) ( u + b ). Using the triangle inequality, (2.4), (3.7), (3.) and (3.23) we obtain the stated estimate for p p h. Remark: The L () norm used for the term (ũ s h I hũ s h ) and the L2 () norm used for u and b in (3.22) may be interchanged, assuming that the functions u and b are sufficiently regular. 4 Numerical Computations In this section we present a numerical example to demonstrate the numerical approximation scheme (3.8)-(3.9), and investigate the a priori error estimate (3.4). Let = (, ) (0, ), β(s) = v + (v 0 v )/( + ks 2 r ), with parameters v =, v 0 = 5, k =, and r = /2. (β( ) represents the Cross model for the effective viscosity for a generalized Newtonian fluid.) The true solution u and p are taken to be [ ] sin(πx) cos(πy) u(x, y) =, p(x, y) = xy. (4.) cos(πx) sin(πy) For this choice of u, u 0, hence a right hand side function is added to (3.9). The boundary conditions used are u n along {} (0, ), (, ) {}, { } (0, ), with p = 0 weakly imposed along (, ) {0}. A computation mesh corresponding to mesh parameter h = /4 is presented in Figure 4.. Plots of β( u ), u and p are given in Figures 4.2, 4.3 and 4.4, respectively. Example. For u s h, the interpolate of ũs h (the smoothed function of u h), we compute a continuous, piecewise quadratic, velocity by taking a simple average of u h at the nodal points of u s h. Computations were performed using RT 0 discp 0, RT discp, and RT 2 discp 2 elements for the velocity and pressure. (By RT k we are referring to Raviart-Thomas elements of degree k, and discp k refers to the space of discontinuous scalar functions which are polynomials of degree less that or equal to k on each triangle in the triangulation.) The results, together with the experimental convergence rates are presented in Table 4.. The experimental convergence rates are consistent with those predicted by (3.4) for l = 2. (Regarding the O(h 4 ) experimental convergence rate for the pressure using RT 2 discp 2 elements, note that the true solution for the pressure lies in the discp 2 approximation space.) Example 2. In order to investigate the dependence of the approximation on the interpolant of the smoother, 0

11 Figure 4.: Computational mesh for h = /4. Figure 4.2: Plot of β( u ). Velocity Figure 4.3: Plot of the velocity flow field u. Figure 4.4: Plot of the pressure function p. in this case we take u s h to be a continuous, piecewise linear function, obtained by taking a simple average of ũ s h at the vertices of the triangles in the triangulations. The results obtained using RT discp, and RT 2 discp 2 approximating elements are presented in Table 4.2. In this case (l = ) we observe optimal convergence for RT discp (and RT 0 discp 0, results not included). However, the experimental convergence rates for the RT 2 discp 2 approximation is limited to 2 for the velocity and pressure, consistent with (3.4). A Example of a local smoothing function In this section we give an example of a local smoothing function which satisfies properties Au s and Au s 2 presented in Section 2. The smoothing function is a simple averaging operator. We use the term domain to refer to an open connected set in IR n. For simplicity we present the case for a scalar function u(x). smoother is simply applied to each of the coordinate functions. For a vector valued function the

12 h u u h L 2 () Cvg. rate (u u h ) L 2 () Cvg. rate p p h L 2 () Cvg. rate X h = RT 0 Q h = discp 0 / E E E-2.29 / E E E-2.0 /8.790E E E-2.08 /0.433E E E-2.05 /2.95E E E-2 Pred X h = RT Q h = discp / E E E / E E E /8.456E E E / E E E / E E E-04 Pred X h = RT 2 Q h = discp 2 /4 6.66E E E /6.905E E E / E E E / E E E /2 2.32E E E-05 Pred Table 4.: Example, u s h a quadratic interpolant of ũs h. h u u h L 2 () Cvg. rate (u u h ) L 2 () Cvg. rate p p h L 2 () Cvg. rate X h = RT Q h = discp / E E E-2.99 /6 3.50E E E /8.808E E E /0.70E E E /2 8.69E E E-3 Pred X h = RT 2 Q h = discp 2 / E E E-2.7 /6.727E E E / E E E /0 6.30E E E / E E E-3 Pred Table 4.2: Example 2, u s h a linear interpolant of ũs h. Let denote a bounded domain in IR n and L() the Lebesgue measurable sets in. Let δ > 0 denote the (fixed) volume measure over which we average a function to obtain its smoothed value. 2

13 For x the typical averaging volume which comes to mind is B(x, r δ ), where B(x, r δ ) denotes the ball centered at x of radius r δ having volume δ. As δ is fixed the difficulty in using B(x, r δ ) arises for points whose distance from is less that r δ. This requires us to consider averaging volumes other than balls. Namely, for each point x we associate a domain V (x) having a volume of δ. We require that the association of x with V (x) be continuous. This continuity is formally described in the next paragraph. Let ν denote the Lebesgue measure in IR n. For S, S 2 L(), introduce the metric d(s, S 2 ) defined by d(s, S 2 ) := ν(s S 2 ), where S S 2 := (S \S 2 ) (S 2 \S ). (A.) Now, let V : L() satisfy: (i) V (x) is a domain with ν(v (x)) = δ for all x, and (ii) d(v (x), V (y)) = ν(v (x) V (y)) C V x y for all x, y, where C V a fixed constant. For convenience we denote the domain V (x) as V x. Definition: Local Smoothing Operator For u L 2 (), define u s as u s (x) = δ We have the following properties for u s (x). V x u(z) d. (A.2) Lemma A. For u L 2 (), u s defined by (A.2) satisfies the following properties. (i) u s L () δ /2 u L 2 (). (ii) u s : IR is uniformly continuous. (iii) Suppose that {u n } n= L2 () and that u n converges weakly to u L 2 (). Then {u s n} n= converges to u s in L (). Proof: Let S L 2 () denote the characteristic function of the domain S. From (A.2), for x u s (x) = u(z) d = Vx u(z) d δ V x δ ( ) /2 ( ) /2 ( Vx ) 2 d u(z) 2 d δ which establishes (i). = δ /2 u L 2 (), For x, y, u s (x) u s (y) δ = δ ( Vx Vy u(z) d ( Vx Vy ) 2 d ) /2 ( = δ u L 2 () d(v (x), V (y)) /2 = C/2 V δ u L 2 () x y /2, ) /2 u(z) 2 d 3

14 which establishes the uniform continuity of u s. As u s is bounded on then u s can be continuously extended to. To establish (iii), as {u n } converges weakly, let sup n u n = M <. In addition, for ɛ > 0, σ = ( ) 2, ɛ/(6 M C /2 V ) let {zi } N i= denote a σ-net of, i.e., for all x there exists an i x {, 2,..., N} such that x z ix < σ. Now, u s n(x) u s (x) = = (u n (y) u(y)) d V x (u n (y) u(y)) d + (u n (y) u(y)) d V zix V x\v zix (u n (y) u(y)) d V zix + u n (y) u(y) d. V x V zix (A.3) Since {u n } converges weakly to u in L 2 (), for all w L 2 () there exists N w such that for n > N w (u n u) w d < ɛ 3. (A.4) Let N = max i=,2,...,n { N Vzi }. Then, for n > N (u n (y) u(y)) d V zix = For the second term on the right hand side of (A.3) we have (u n (y) u(y)) Vzi d < ɛ 3. V x V zix u n (y) u(y) d ( V x V zix u n (y) u(y) 2 d 2M ν(v x V zix ) /2 ) /2 ( 2M C /2 V x z ix /2 2M C /2 V σ /2 V x V zix d ) /2 = ɛ 3. (A.5) Thus, from (A.3)-(A.5) it follows that for all x, for n > N u s n(x) u s (x) < 2 3 ɛ, i.e., us n u s L () < 2 3 ɛ < ɛ. A. Regularity of u s (for u L ()) If, in place of u L 2 (), we have u L () then u s defined by (A.2) is a H () function. To establish this regularity result we begin by citing a characterization of the W,p (IR n ) function space. 4

15 Theorem A. ( [8], Theorem 2..6) Let < p <. Then u W,p (IR n ) if and only if u L p (IR n ) and ( u(x + h) u(x) p /p IR n h dx) = h u(x + h) u(x) L p (IR n ) (A.6) remains bounded for all h IR n. Theorem A.2 If u L () then, for u s defined by (A.2), u s H (). Proof: In order to apply Theorem A. we need to define an extension of u to IR n. Let { u(x), x ũ(x) = 0, x, and Ṽ : IRn L(IR n ) denote an extension of V satisfying properties (i) and (ii) (with replaced by IR n ), and additionally that there exists constants C > 0 and C 2 0 such that (iii) diameter(ṽ (z)) C for all z IR n, and (iv) sup z IR n inf y Ṽ (z) z y C 2. Let B denote the bounded set, B := {x IR n : inf y x y < + C + C 2 } support(ũ s ). Note that for x IR n \ B and h <, ũ s (x + h) = 0. Now, for h, ũ s (x + h) ũ s (x) IR n h For h <, ũ s (x + h) ũ s (x) IR n h 2 2 dx dx = = = ( ) 2 h 2 IR (ũs (x + h)) 2 dx + n IR (ũs (x)) 2 dx n 4 h 2 IR (ũs (x)) 2 dx 4 n h 2 ũs 2 L ( B ) ν( B) 4 ν( B ) ũ 2 L (IR n = 4 ν( ) B ) u 2 L (). (A.7) h 2 ũ s (x + h) ũ s (x) 2 dx B ( h 2 Ṽx+h (z) Ṽx (z) δ B h 2 h 2 ũ(z) B δ 2 ũ 2 L ( B ) B ( B ) dz Ṽx+h (z) Ṽx (z) d(ṽx+h, Ṽx) 2 dx δ 2 ũ 2 L ( B ) B h 2 δ 2 ũ 2 L ( B ) C2 V h 2 ν( B ) 2 dx 2 dz) dx = δ 2 C2 V ν( B ) ũ 2 L ( B ) = δ 2 C2 V ν( B ) u 2 L (). (A.8) From (A.7) and (A.8), together with Theorem A., we obtain that ũ s H (IR n ). As u s = ũ s, it then follows that u s H (). 5

16 B Example of a differential smoothing function As an alternative to the local averaging filter discussed in Section A, in this section we present a differential smoothing filter. Let X s = H 0 () = {v H () : v = 0 on } X. (B.9) Definition: Differential Smoothing Operator For u L 2 (), define u s X s as ( u s, v) = (u s, v), v X s. (B.0) The well posedness of u s follows from an application of the Lax-Milgram theorem. Next we show that this smoothing operation satisfies properties Au s and Au s 2 presented in Section 2. Lemma B.2 For u L 2 (), u s defined by (B.0) satisfies the following properties. (i) u s L () C u L 2 (). (ii) Suppose that {u n } n= L2 (), and that u n converges weakly to u L 2 (). The {u s n} converges to u s in L (). Proof: From (B.0) we have that u s X s, and as u L 2 (), from the shift theorem (together with a sufficiently smooth ), it follows that u s H 2 () X s, with u s H 2 () C u. (B.) Using the embedding of H 2 () in L () we establish (i). Let W : L 2 () H 2 () X s, W(u) := u s, denote the filter mapping. Then from (B.) W is a bounded (linear) transformation from L 2 () H 2 () X s. Let W : ( H 2 () X s) L 2 () denote the adjoint operator of W. (The existence of W follows immediately from the Riesz Representation Theorem.) Now, for η ( H 2 () X s) u s n u s, η H 2,(H 2 ) = W(u n) W(u), η H 2,(H 2 ) = W(u n u), η H 2,(H 2 ) = (u n u, W (η)) 0 as n, as u n converges weakly in L 2 () to u. Hence as H 2 () X s is compactly embedded in L () X s, then u s n converges to u s strongly in L (). References [] G. Allaire. Homogenization of the Navier-Stokes equations with a slip boundary condition. Comm. Pure Appl. Math., 44(6):605 64, 99. 6

17 [2] M. Azaïez, F. Ben Belgacem, C. Bernardi, and N. Chorfi. Spectral discretization of Darcy s equations with pressure dependent porosity. Appl. Math. Comput., 27(5): , 200. [3] S.C. Brenner and L.R. Scott. The mathematical theory of finite element methods, volume 5 of Texts in Applied Mathematics. Springer-Verlag, New York, second edition, [4] F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Springer-Verlag, New York, 99. [5] M. Bulíček, P. Gwiazda, J. Málek, and A. Świerczewska-Gwiazda. On steady flows of incompressible fluids with implicit power-law-like rheology. Adv. Calc. Var., 2(2):09 36, [6] L. Diening, C. Kreuzer, and E. Süli. Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. SIAM J. Numer. Anal., 5(2):984 05, 203. [7] V.J. Ervin, E.W. Jenkins, and S. Sun. Coupled generalized nonlinear Stokes flow with flow through a porous medium. SIAM J. Numer. Anal., 47(2): , [8] V.J. Ervin, E.W. Jenkins, and S. Sun. Coupling nonlinear Stokes and Darcy flow using mortar finite elements. Appl. Numer. Math., 6():98 222, 20. [9] J. Galvis and M. Sarkis. Non matching mortar discretization analysis for the coupling Stokes Darcy equations. Electron. Trans. Numer. Anal., 26: , [0] G.N. Gatica, R. Oyarzúa, and F.-J. Sayas. A twofold saddle point approach for the coupling of fluid flow with nonlinear porous media flow. IMA J. Numer. Anal., 32(3): , 202. [] V. Girault, F. Murat, and A. Salgado. Finite element discretization of Darcy s equations with pressure dependent porosity. M2AN Math. Model. Numer. Anal., 44(6):55 9, 200. [2] A. Hirn, M. Lanzendörfer, and J. Stebel. Finite element approximation of flow of fluids with shear-rate- and pressure-dependent viscosity. IMA J. Numer. Anal., 32(4): , 202. [3] J. Málek, J. Nečas, and K.R. Rajagopal. Global analysis of the flows of fluids with pressuredependent viscosities. Arch. Ration. Mech. Anal., 65(3): , [4] K. R. Rajagopal. On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Math. Models Methods Appl. Sci., 7(2):25 252, [5] L. Traverso, T.N. Phillips, and Y. Yang. Mixed finite element methods for groundwater flow in heterogeneous aquifers. Computers & Fluids, 88:60 80, 203. [6] S. Whitaker. Flow in porous media I. A theoretical derivation of Darcy s law. Transp. Porous Media, :3 25, 986. [7] E. Zeidler. Applied functional analysis, volume 08 of Applied Mathematical Sciences. Springer- Verlag, New York, 995. Applications to mathematical physics. [8] W.P. Ziemer. Weakly differentiable functions, volume 20 of Graduate Texts in Mathematics. Springer-Verlag, New York, 989. Sobolev spaces and functions of bounded variation. 7