Optimal error estimates of the penalty finite element method for micropolar fluids equations

XX Congreso de Ecuaciones Diferenciales y Aplicaciones X Congreso de Matem´ atica Aplicada Sevilla, 24-28 septiembre 2007 (pp. 1–8) Optimal Error Estimates of the Penalty Finite Element Method for Micropolar Fluids Equations E. Ortega-Torres1, M. Rojas-Medar2 1Dpto. de Matem´aticas, Universidad de Antofagasta, Casilla 170, Antofagasta-Chile. E-mail: [email protected]. 2Dpto. de Ciencias B´asicas, Universidad del Bio-Bio, Campus Fernando May, Casilla 447, Chillan-Chile. E-mail: [email protected]. Key words: Micropolar fluids, penalty, Finite Element, fully discrete Abstract An optimal error estimate of the numerical velocity, pressure and angular velocity, is proved for the fully discrete penalty finite element method of the micropolar equations, when the parameters ², ∆tand hare sufficiently small. In order to obtain above we present the time discretization of the penalty micropolar equation which is based on the backward Euler scheme; the spatial discretization of the time discretized penalty Micropolar equation is based on a finite elements space pair (Hh, Lh) which satisfies some approximate assumption. 1 Introduction The equations that describes the motion of a viscous incompressible micropolar fluids in a bounded domain Ω ⊂R3, with boundary ∂Ω smooth a time interval [0, T], 0 < T < +∞ are given by (see [4]) (P) =            ut−ν1∆u+ (u· ∇)u+∇p= 2µrrot w+f, div u= 0, wt+Lw+ (u· ∇)w+ 4µrw= 2µrrot u+g, u(x, t) = 0,w(x, t) = 0 on ∂Ω×(0, T), u(x, 0) = a(x),w(x, 0) = b(x) in Ω, where Lw=−ν2∆w−ν3∇div w, with ν1=µ+µr, ν2=ca+cd, ν3=c0+cd−ca. The functions u,wand pdenote the velocity vector, the angular velocity vector of rotation of particles, and the pressure of the fluid, respectively. The functions fand 1 E. Ortega-Torres, M. Rojas-Medar gdenote external sources of linear and angular momentum, respectively. The positive constants µ, µr, c0, caand cdare viscosities and c0+cd> ca. Also, a,bare given functions in Ω. The penalty method applied to (P) is to approximate (u, p, w) by (u², p²,w²) satisfying the following penalty micropolar equations: (P)²=              ∂tu²−ν1∆u²+B(u²,u²) + ∇p²= 2µrrot w²+f, div u²+² ν1 p²= 0, ∂tw²+Lw²+B(u²,w²) + 4µrw²= 2µrrot u²+g, u²(x, t) = 0,w²(x, t) = 0 on ST, u²(x, 0) = a(x),w²(x, 0) = b(x) in Ω, where B(u,w) = (u· ∇)w+1 2( div u)wis the modified bilinear term introduced by Temam [7] and it is well known that lim²→0(u², p²,w²) = (u, p, w) with error bound ku−u²kL2(0,T;H1)+kw−w²kL2(0,T;H1)+kp−p²kL2(0,T;L2)≤C ²1/2, where C > 0 is a general positive constant depending on the data ν1, ν2, ν3,a,b,f,g, T . 2 Preliminaries By simplicity we denote Lk=Lk(Ω),Hm=Hm(Ω) and Lk(Hm) = Lk(0, T;Hm(Ω)). For the mathematical setting of (P), to consider the following function spaces L2 0={q∈L2(Ω) ; ZΩ q dx = 0}, V ={v∈H1 0(Ω) ; div v= 0} We define Au=−4uand A²u=−4u−1 ²∇div u, which are positive self-adjoint operators associated with the micropolar and penalty micropolar fluid equations, and they are defined from D(A) = H2∩H1 0onto L2. Also, for all u,v,w∈H1 0are well defined (A1/2 ²u, A1/2 ²v)=(∇u,∇v) + 1 ²(div u,div v), (L1/2w, L1/2v) = ν2(∇w,∇v) + ν3(div w,div v). Moreover, by Shen [6] there exists a constant M0>0 such that if ²M0≤1, then kAvk ≤ M0kA²vk,k∇vk ≤ M0kA1/2 ²vk,k∇wk ≤ ν−1 2kL1/2 ²wk.(1) We define the following trilinear forms on H1 0×H1 0×H1 0 b(u,v,w) = hB(u,v),wiH1 0×H1 0=1 2((u· ∇)v,w)−1 2((u· ∇)w,v), ∀u,v,w∈H1 0, and satisfies the property b(u,v,w) = −b(u,w,v). We make the following assumption on the prescribed data a,b,fand g. (A1) The initial velocity a∈D(A)∩V, the initial angular velocity b∈H2∩H1 0, and the external forces f,g∈W1,∞(L2). 2 Error estimates for full discrete penalty micropolar equations With the above notation, the variational formulation of the micropolar equations (P) is given by: Find u,w∈L∞(L2)∩L2(H1 0) and p∈L2(L2 0) such that (PV ) =            (ut,v) + ν1(∇u,∇v)−(div v, p) + (div u, q) + b(u,u,v) = 2 µr(rot w,v)+(f,v),∀v∈H1 0, q ∈L2 0 (wt,z)+(L1/2w, L1/2z)+4µr(w,z) + b(u,w,z) = 2 µr(rot u,z)+(g,z),∀z∈H1 0, u(0) = a,w(0) = bin Ω, and the penalty micropolar variational formulation of (P)²is defined as follows: Find u²,w²∈L∞(L2)∩L2(H1 0) and p²∈L2(L2 0) such that (PV )²=              (∂tu²,v) + ν1(∇u²,∇v)−(div v, p²) + (div u², q) + ² ν1 (p², q) +b(u²,u²,v) = 2 µr(rot w²,v)+(f,v),∀v∈H1 0, q ∈L2 0 (∂tw²,z) + (L1/2w², L1/2z)+4µr(w²,z) + b(u²,w²,z) = 2 µr(rot u²,z)+(g,z),∀z∈H1 0, u²(0) = a,w²(0) = bin Ω. Rather than assuming that the data are small, we make the existence of the solution on some interval [0, T ) an assumption (A2) The solution (u, p, w) of problem (PV ) exists on [0, T ), and u,w∈L∞(H1). (A3) The solution (u², p²,w²) of problem (PV )²exists on [0, T), and u²,w²∈L∞(H1). 3 Time discretization and regularity For the problem (PV)²we consider the time discretization by the backward Euler scheme (PV )n ²=              (dtun ²,v) + ν1(∇un ²,∇v)−(div v, pn ²) + (div un ², q) + ² ν1 (pn ², q) +b(un ²,un ²,v) = 2 µr(rot wn ²,v)+(f(tn),v),∀v∈H1 0, q ∈L2 0, (dtwn ²,z)+(L1/2wn ², L1/2z) + 4µr(wn ²,z) + b(un ²,wn ²,z) = 2 µr(rot un ²,z)+(g(tn),z),∀z∈H1 0, u0 ²=a, p0 ²= 0,w0 ²=bin Ω, for all 1 ≤n≤N, where tn=n∆tand 0 <∆t < 1 is the time step size, tN=T, (un ², pn ²,wn ²) is an approximation of (u², p²,w²) at time tn, and dtun ²=1 ∆t(un ²−un−1 ²) for 1≤n≤N, with dtu0 ², dtw0 ²∈L2. We can refer to Shen [5] for the proof of the following result. Theorem 3.1 Suppose that (A1), (A2), (A3) and ²M0≤1are valid. The following error estimate holds τ2(tm)k∇ u(tm)− ∇ um ²k2+τ2(tm)k∇ w(tm)− ∇ wm ²k2 +∆t m X n=1 τ2(tn)kp(tn)−pn ²k2≤C(²2+ ∆t2) (2) for all 1≤m≤N. Where M0is given in (1), τ(t) = min{t, 1}, and C > 0,is a general positive constant depending on the data ν1, ν2, ν3,f,g,Ω, T . 3 E. Ortega-Torres, M. Rojas-Medar In order to obtain the error bound of the finite element solution related to the problem (PV)n ², we are going to provide with some regularity results for the solutions of (PV)n ². Theorem 3.2 Suppose that (A1), (A3) and ² M0≤1are valid. There is a constant M > 0such that if ∆t M ≤1, then kA1/2 ²um ²k2+kL1/2wm ²k2+ ∆t m X n=1 (kA²un ²k2+kLwn ²k2+kpn ²k2 H1)≤C, (3) kA²un ²k2+kLwn ²k2+kpn ²k2 H1+ ∆t m X n=1 (kA1/2 ²dtun ²k2+kL1/2dtwn ²k2)≤C, (4) ∆t m X n=1 τ(tn)(kA²dtun ²k2+kLdtwn ²k2)≤C, (5) for all 1≤m≤N. The proof of the Theorem 3.2 can be done without no difficulty. 4 Finite element penalty method To avoid technical difficulties, the bounded domain Ω is assumed to be a polyhedron. Let πh={K}be a discretization of mesh size h, 0< h < 1 of the polyhedral domain Ω into closed subsets K, and the family πhsatisfies the usual regularity assumptions. For each h, let Hhand Lhbe the finite dimensional spaces to be used for approximating the “velocity space” H1 0and the “pressure space” L2 0, respectively. The spaces Hhand Lhsatisfy several approximations properties and one compatibility condition (see Girault and Raviart [2], Ciarlet [1], Heywood and Rannacher [3]), thus the continuous and discrete spaces are relate by the following hypotheses: (S1) There exists a continuous mapping rh:H2∩H1 0−→ Hhsuch that (i) (qh,div (v−rhv)) = 0,∀qh∈Lh,∀v∈H2∩H1 0, (ii)kv−rhvk+hkv−rhvkH1≤C h2kvkH2,∀v∈H2∩H1 0. (S2) The orthogonal projection operator jh:L2 0(Ω) −→ Lhsatisfies kq−jhqk ≤ C h kqkH1,∀q∈H1∩L2 0. (S3) (Inf-sup condition) There exists a constant β > 0,independent of h, such that sup vh∈Hh (qh,div vh) k∇vhk≥βkqhk,∀qh∈Lh. Also, it is true the inverse inequality k∇ vhk ≤ C h−1kvhk,∀vh∈Hh. Now, we define (un ²h, pn ²h,wn ²h) as the finite element approximations of (un ², pn ²,wn ²), 4 Error estimates for full discrete penalty micropolar equations which satisfied the following penalty finite element system (PV )n ²h =                  (dtun ²h,vh) + ν1(∇un ²h,∇vh)−(div vh, pn ²h) +(div un ²h, qh) + ² ν1 (pn ²h, qh) + b(un ²h,un ²h,vh) = 2 µr(rot wn ²h,vh)+(f(tn),vh),∀vh∈Hh, qh∈Lh, (dtwn ²h,zh) + (L1/2wn ²h, L1/2zh) + 4µr(wn ²h,zh) + b(un ²h,wn ²h,zh) = 2 µr(rot un ²h,zh)+(g(tn),zh)∀zh∈Hh, u0 ²h =a, p0 ²h = 0,w0 ²h =bin Ω, We write the following theorem, in which we obtain similar results that in [8]. Theorem 4.1 Assume that (S1)-(S3) and that the hypotheses of the Theorem 3.2 are valid. Then it holds that kum ²hk2+kwm ²hk2+ ∆t m X n=1 (k∇ un ²hk2+k∇ wn ²hk2)≤C, (6) k∇ um ²hk2+kL1/2wm ²hk2+ ∆t m X n=1 (kdtun ²hk2+kdtwn ²hk2) ≤C+C h−3∆t m X n=1 [kun ²hk2(k∇˜enk2+k∇ ˜ θnk2) + k∇ un ²hk2(k˜enk2+k˜ θnk2)] (7) for all 1≤m≤Nand ˜en=un ²−un ²h,˜ θn=wn ²−wn ²h. 5 Optimal error analysis We establish the optimal error for u(tn)−un ²h,w(tn)−wn ²h and p(tn)−pn ²h. With this purpose, and since u(tn)−un ²h = (u(tn)−un ²) + (un ²−un ²h) , firstly we are going to do estimates for the error ˜en,˜ ξn=pn ²−pn ²h and ˜ θn. Lemma 5.1 Under the hypothesis of Theorem 4.1, it is holds the error estimate k˜emk2+k˜ θmk2+ ∆ t m X n=1 (k∇˜emk2+kL1/2˜ θmk2)≤C h2,(8) for all 1≤m≤N. Proof. Denoting en h= (I−rh)un ², θn h= (I−rh)wn ², ξn h= (I−ρh)pn ²,en=rhun ²− un ²h, ξn=ρhpn ²−pn ²h and θn=rhwn ²−wn ²h, we have ˜en= en h+en,˜ θn=θn h+θn,˜ ξn=ξn h+ξn. From (S1)-ii and considering (1) and (3), we obtain kem hk2+kθm hk2+C∆t m X n=1 (k∇ en hk2+kL1/2θn hk2)≤C h2.(9) Then, taking into account (9), to prove (8) it is enough to show kemk2+kθmk2+ ∆t m X n=1 (k∇ enk2+kL1/2θnk2)≤C h2.(10) 5 E. Ortega-Torres, M. Rojas-Medar Subtracting (PV)n ²h from (PV)n ²with v=vh= 2 en∆t, q =qh= 2 ξn∆tand z=zh= 2θn∆t, using (S1), (1), (4), definition of dtunand the H¨older and Young’s inequalities, we obtain kenk2− ken−1k2+kθnk2− kθn−1k2+ ∆t(µk∇ enk2+kL1/2θnk2) ≤∆t C h2(k∇ dtun ²k2+kL1/2dtwn ²k2)+∆t C h2(kAun ²k2+kpn ²k2 H1+kLwn ²k2) +∆t C h2(k∇ un ²hk2+k∇ un ²k2+kL1/2wn ²k2) + ∆t C dn(kenk2+kθnk2),(11) where dn=C(ν1, ν2)(k∇un ²k4+k∇ wn ²k4) and ∆ tis chosen such that 2∆tPm n=1 dn≤C. Then, summing (11) from 1 to m, by using Theorem 3.2 and Theorem 4.1, and the discrete Gronwall’s Lemma, is followed (10). Remark 1. From Theorem 3.2, Theorem 4.1 and Lemma 5.1, we can conclude k∇ um ²hk2+k∇˜emk2≤C h−1,k˜emk2 L3≤Ck˜emkk∇˜emk ≤ C h1/2.(12) Lemma 5.2 Under the hypotheses of Theorem 4.1 is true k∇(um ²−um ²h)k2+kL1/2(wm ²−wm ²h)k2+ ∆ t m X n=1 kpn ²−pn ²hk2≤C h1/2,(13) for all 1≤m≤N. Proof. We will consider to the notations done in the proof of the Lemma 5.1. Subtracting (PV)n ²h from (PV)n ²with v=vh,z=zhand q=qh, we obtain (dt˜en,vh) + ν1(∇˜en,∇vh)−(div vh,˜ ξn) + (div dt˜en, qh) + ² ν1 (dt˜ ξn, qh) +b(˜en,un ²,vh) + b(un ²,˜en,vh)−b(˜en,˜en,vh) = 2 µr(rot ˜ θn,vh),(14) (dt˜ θn,zh) + (L1/2˜ θn, L1/2zh) + 4µr(˜ θn,zh) + b(˜en,wn ²,zh) +b(un ²,˜ θn,zh)−b(˜en,˜ θn,zh) = 2 µr(rot˜en,zh).(15) Now, we observe that kdt˜ φnk2=kdtφn hk2+kdtφnk2+ 2(dtφn h, dtφn), and then 2 ∆ tkdt˜ φnk2≥∆tkdt˜ φnk2+ ∆ tkdtφnk2+ 2∆ t(dtφn h, dtφn).(16) Considering inequality (16) and setting vh= 2∆t dt˜en, qh= 2∆t˜ ξn,zh= 2∆t dt˜ θnin (14)-(15), we have ∆t(kdt˜enk2+kdtenk2)+∆t(kdt˜ θnk2+kdtθnk2) + ν1(k∇˜enk2− k∇˜en−1k2) +kL1/2˜ θnk2− kL1/2˜ θn−1k2+² ν1 (k˜ ξnk2− k˜ ξn−1k2)+4µr(k˜ θnk2− k˜ θn−1k2) ≤ −2∆ t(dten h, dten)−2∆ t(dtθn h, dtθn)−2∆t b(˜en,un ², dt˜en)−2∆t b(un ²,˜en, dt˜en) +2∆t b(˜en,˜en, dt˜en)−2∆t b(˜en,wn ², dt˜ θn)−2∆t b(un ²,˜ θn, dt˜ θn) +2∆t b(˜en,˜ θn, dt˜ θn) + 4 ∆t µr(rot ˜ θn, dt˜en) + 4∆t µr(rot˜en, dt˜ θn).(17) 6 Error estimates for full discrete penalty micropolar equations Now, by using the H¨older and Young’s inequalities, Theorem 3.2 and (12), we obtain the following inequalities 2 (dten h, dten) + 2(dtθn h, dtθn)≤Ckdten hk2+1 2kdtenk2+1 2kdtθnk2,(18) 2b(˜en,un ², dt˜en)+2b(un ²,˜en, dt˜en) + 4 µr(rot ˜ θn, dt˜en) ≤C(k∇˜enk2+k∇ ˜ θnk2) + 1 2kdt˜enk2,(19) 2b(˜en,˜en, dt˜en)≤Ck˜enkL3k∇˜enk(kdten hkL6+kdtenkL6) ≤C h1/4k∇˜enk(k∇ dten hk+h−1kdtenk) ≤C h−3/2k∇˜enk2+C hk∇ dten hk2+1 2kdtenk2,(20) 2b(˜en,wn ², dt˜ θn) + 2 b(un ²,˜ θn, dt˜ θn)+4µr(rot˜en, dt˜ θn) ≤C(k∇˜enk2+k∇ ˜ θnk2) + kdt˜ θnk2,(21) 2b(˜en,˜ θn, dt˜ θn)≤C(k˜enkL3k∇ ˜ θnk+k∇˜enkk˜ θnkL3)(kdtθn hkL6+kdtθnkL6) ≤C h1/4(k∇ ˜ θnk+k∇˜enk)(k∇ dtθn hk+h−1kdtθnk) ≤C h−3/2(k∇ ˜ θnk2+k∇˜enk2) + C hk∇ dtθn hk2+1 2kdtθnk2.(22) The fact that h2≤C∆t≤C τ(tn) together (S1)-(ii), implies k∇ dten hk2+k∇ dtθn hk2≤C τ(tn) (kA dtun ²k2+kL dtwn ²k2).(23) Then, carrying (18)-(22) in (17), and taking into account (S1)-(ii) and (23), we get ∆t 2kdt˜enk2+ν1(k∇˜enk2− k∇˜en−1k2) + kL1/2˜ θnk2− kL1/2˜ θn−1k2 +² ν1 (k˜ ξnk2− k˜ ξn−1k2)+4µr(k˜ θnk2− k˜ θn−1k2) ≤C h2∆t(k∇ dtun ²k2+kL1/2dtwn ²k2) + C∆t h−3/2(k∇ ˜ θnk2+k∇˜enk2) +C∆t h τ(tn) (kA dtun ²k2+kL dtwn ²k2).(24) Thus, summing (24) from 1 to mand by using Theorem 3.2 and Lemma 5.1, we obtain k∇˜emk2+kL1/2˜ θmk2+ ∆t m X n=1 kdt˜enk2≤C h1/2.(25) To pressure, again subtracting (PV)n ²h from (PV)n ²with v=vhand q=qh, and by inf-sup condition, we deduce kξnk ≤ (div vh, ξn) k∇ vhk≤Ckdt˜enk+Ck∇˜enk+Ckξn hk+CkL1/2˜ θnk +Ck∇˜enk(k∇ un ²k+k∇ un ²hk),(26) 7 E. Ortega-Torres, M. Rojas-Medar and by Remark 1, from (26) we have kξnk2≤C(kdt˜enk2+k∇˜enk2+kξn hk2+kL1/2˜ θnk2+h−1k∇˜enk2).(27) Then, since ∆tkpn ²−pn ²hk2≤C∆t(kξn hk2+kξnk2), by (S2) together Lemma 5.1, (27) and (25), we conclude ∆t m X n=1 kpn ²−pn ²hk2≤C h1/2 and Lemma 5.2 is complete. Finally, from Theorem 3.1, Lemma 5.2 and the triangles inequality, we establish the following result Theorem 5.3 Under the hypotheses of Theorem 4.1 is hold the following optimal error estimate τ2(tm)k∇ (u(tm)−um ²h)k2+τ2(tm)kL1/2(w(tm)−wm ²h)k2 +∆ t m X n=1 τ2(tn)kp(tn)−pn ²hk2≤C(²2+ ∆t2+h1/2), for all 1≤m≤N. Acknowledgments Fondo Nacional de desarrollo cient´ıfico y tecnol´ogico Fondecyt-Chile, grant N◦1040205 and N◦7060025 References [1] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [2] V. Girault, P.A. Raviart, Finite Elements Methods for Navier-Stokes Equation: theory and algorithms, Springer-Verlarg, Berlin Heidelberg, 1987. [3] J.G. Heywood, R. 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