STABILIZED FINITE ELEMENT METHODS BASED ON MULTISCALE ENRICHMENT FOR THE STOKES PROBLEM ´ ERIC ´ RODOLFO ARAYA, GABRIEL R. BARRENECHEA, AND FRED VALENTIN Abstract. This work concerns the development of stabilized finite element methods for the Stokes problem considering non stable different (or equal) order of velocity and pressure interpolations. The approach is based on the enrichment of the standard polynomial space for the velocity component with multiscale functions which no longer vanish on the element boundary. On the other hand, since the test function space is enriched with bubble-like functions, a Petrov-Galerkin approach is employed. We use such strategy to propose stable variational formulations for both continuous piecewise linear in velocity and pressure, and piecewise linear/piecewise constant interpolation pairs. Optimal order convergence results are derived and numerical tests validate the proposed methods.
1. Introduction Finite element solution of Stokes problem poses the basic problem of satisfying the discrete Babuska-Brezzi (or inf-sup) condition (see [24] and the references therein). This is indeed a restriction from the point of view of implementation since equal order velocity and pressure spaces do not satisfy this condition. On the other hand, the minimal space to imagine, namely continuous piecewise linear polynomials for the velocity and piecewise constant polynomials for the pressure does not satisfy this condition either. Several solutions have been proposed to overcome this restriction, starting with [11] and the first consistent method [28]. Moreover, in [27, 23, 29, 34] the possibility of considering discontinuous spaces for the pressure was considered and justified. On the other hand, in [14, 13], the idea from [16] has been used to propose a new kind of stabilized finite element methods, with stabilizing terms containing now only jump terms across the inter element boundaries. For an overview on stabilized finite element methods for the Stokes problem, see [19] and [5]. On the other hand, the theoretical justification of stabilized methods have become a subject of interest in the last decade. In [2, 31, 4, 3], the connection between stabilized finite element methods and Galerkin methods enriched with bubble functions has been used to Date: January 24, 2005. Key words and phrases. Stokes equation, multiscale functions, SIMPLEST element, bubble function. 1
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R. ARAYA, G.R. BARRENECHEA, AND F. VALENTIN
propose new stabilized finite element methods for Stokes-like and linearized Navier-Stokes problems. Also, in [22] macro bubbles where used to derive a method analogous to the locally stabilized method from [29] containing jump terms across the inter element boundaries of the macroelements. In the resulting method, the stabilizing terms are defined over the macro elements, and there is no error analysis nor numerical validation of the method. All these works used the so-called bubble condensation procedure, i.e., the elimination of the bubble function at the element level and the writing of the method as the Galerkin part plus a term coming from the influence of the bubble functions on the formulation. A particular kind of bubble enrichment of the velocity space is the so-called Residual-Free Bubble (RFB) method (cf. [12, 7, 8]), in which the bubble function is now the solution of a problem containing the residual of the continuous equation at the element level (see [9, 10, 32] for the a-priori error analysis). This bubble part may be analytically condensed or numerically computed. In the latter case this procedure leads to the two level finite element method. The imposition of a zero boundary condition on the element boundary for RFB has lead to some numerical problems. Solutions for these problems have been proposed by relaxing the zero boundary condition, such the discontinuous enrichment method [18] (for the Helmholtz equation), and, more recently, the multiscale finite element method [21] where the main idea may be found, and [20] where the a-priori error analysis is performed (for the reactiondiffusion equation, an a-posteriori error estimator based on this idea has been proposed and analyzed in [1]). A particularity of such methods is that a Petrov-Galerkin strategy is proposed, in which the test function space is enriched with bubble functions in order to have a local problem containing the residual of the momentum equation on the right hand side. A special boundary condition (related to the one used in [25, 26] and [15]) is imposed in order to solve these local problems analytically. The resulting method is of Petrov-Galerkin type, in which the trial function space is generated by a basis formed by the addition of usual polynomial basis functions and these enrichment functions coming from the solution of the differential problem in each element (which are now, unlike RFB, known analytically, and hence the method is not of a two level FEM type), and the test function space is the standard polynomial space. The purpose of this work is to use the multiscale approach from [21, 20], combined with the static condensation procedure, in order to propose new stabilized finite element methods for the Stokes problem. We proceed as in [21], defining an enrichment function for the trial space for the velocity that no longer vanishes on the element boundary (and hence it is not a bubble function), and then we split it in a bubble part and a function being an harmonic extension of the boundary condition. This boundary condition comes from the solution of an
STABILIZED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM
3
elliptic ODE containing a part of the differential operator at the boundary, and a jump term as right hand side. Depending on the jump term chosen, this procedure will lead to different methods. Both functions are condensed, and hence we obtain a method which includes the usual GLS stabilizing terms at the element level, plus a positive jump term on the inter element boundaries, each one with a proper stabilization parameter. One special feature of these new methods is that the previously mentioned ODE at the element boundary may be solved analytically, and hence the the stabilization parameter associated with the jump terms is known exactly. The plan of the paper is as follows: in Section 2 we present the general framework and derive a general form of the method. Afterwards, in Sections 3 and 4 this framework is applied to derive concrete stabilized finite element methods for two families of interpolation spaces, namely P1 /P0 and continuous P1 /P1 elements, respectively. For both cases optimal order a-priori error estimates are derived for the natural norms of the unknowns, plus some extra control on the norm of the jumps presented in the formulation. As we already mentioned if we change the right hand side on the boundary condition, we can derive a new method. This is done in Section 5, where we give an alternative enrichment strategy leading to another family of methods, whose analysis is analogous to the one from Sections 3 and 4, and who contain now a boundary term containing the residual of the Cauchy stress tensor on the internal edges of the triangulation. Numerical experiments confirming the theoretical results and comparing the performance of all the methods are presented in Section 6, and some final remarks and conclusions are given in Section 7. 2. The model problem and the general framework Let Ω be an open bounded domain in R2 with polygonal boundary, f ∈ L2 (Ω)2 and let us
consider the following Stokes problem: (1)
−ν ∆u + ∇p = f ,
∇· u = 0
in Ω ,
u = 0 on ∂Ω , where ν ∈ R+ is the fluid viscosity.
Let now {Th }h>0 be a family of regular triangulations of Ω, build up using triangles K with
boundary ∂K. Let also Eh be the set of internal edges of the triangulation, hK := diam(K)
and h := max{hK : K ∈ Th }. Let Vh be the usual finite element space of continuous
piecewise polynomials of degree k, 1 ≤ k ≤ 2 with zero trace on ∂Ω. Let also Qh be a space of piecewise polynomials of degree l, 0 ≤ l ≤ 1, which may be continuous or discontinuous
in Ω and who belong to L20 (Ω). Let H m (Th ) and H0m (Th ) (m ≥ 1) be the spaces of functions
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R. ARAYA, G.R. BARRENECHEA, AND F. VALENTIN
whose restriction to K ∈ Th belongs to H m (K) and H0m (K), respectively. Furthermore,
(· , · )D stands for the inner product in L2 (D) (or in L2 (D)2 or L2 (D)2×2 , when necessary),
and we denote by k· ks,D (|· |s,D ) the norm (seminorm) in H s (D) (or H s (D)2 , if necessary). As usual, H 0 (D) = L2 (D), and |· |0,D = k· k0,D .
In order to propose a Petrov-Galerkin method for Stokes problem (1), let Eh ⊂ H01 (Ω)
be a finite dimensional space, called multiscale space, such that Vh ∩ Eh = {θ}. Then, we
propose the following Petrov-Galerkin scheme for (1): Find u1 + ue ∈ [Vh ⊕ Eh ]2 and p ∈ Qh
such that
ν(∇(u1 + ue ), ∇v h )Ω − (p, ∇· v h )Ω + (q, ∇· (u1 + ue ))Ω = (f , v h )Ω , for all v h ∈ [Vh ⊕ H01 (Th )]2 and all q ∈ Qh . Now, this Petrov-Galerkin scheme is equivalent to the following system:
(2) ν(∇(u1 + ue ), ∇v 1 )Ω − (p, ∇· v 1 )Ω + (q, ∇· (u1 + ue ))Ω = (f , v 1 )Ω (3)
ν(∇(u1 + ue ), ∇v b )K − (p, ∇· v b )K = (f , v b )K
∀(v 1 , q) ∈ Vh2 × Qh ,
∀v b ∈ H01 (K)2 , ∀ K ∈ Th .
Equation (3) above is equivalent to (−ν∆ue , v b )K = (f + ν∆u1 − ∇p, v b )K
∀v b ∈ H01 (K)2 ,
which, in strong form, may be written as (4)
−ν∆ue = f + ν∆u1 − ∇p in K.
Now, this differential problem above must be completed with boundary conditions. For reasons that will become clear in the sequel, we will impose the following boundary condition on ue : (5)
ue = g e
on each Z ⊂ ∂K ,
where g e = 0 if Z ⊂ ∂Ω, and g e is the solution of (6)
1 Jν∂n u1 + pI· nK in Z , hZ = 0 at the nodes ,
−ν ∂ss g e = ge
on the internal edges, where hZ = |Z|, n is the normal outward vector on ∂K, ∂s and ∂n
are the tangential and normal derivative operators, respectively, JvK stands for the jump of v across Z, and I is the R2×2 identity matrix.
STABILIZED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM
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Remark. Both the shape of the jump term and the h−1 Z coefficient on the boundary condition have been suggested by the error analysis. On the other hand, if we impose as right hand side in (6) the residual of the Cauchy stress tensor on ∂K we have another class of methods. This alternative will be analyzed in Section 5. ∂K Now, on each K ∈ Th , we can write ue = uK e + ue , where K
−ν∆uK = f + ν∆u1 − ∇p in K , e
(7)
uK = 0 e
on ∂K ,
and −ν∆u∂K = 0 e
(8)
u∂K = ge e
in K , on ∂K ,
where g e is the solution of (6). Such differential problems are well posed, and (3) is immediately satisfied. In this way, we can define two operators MK : L2 (K)2 → H01 (K)2 and BK : L2 (∂K)2 →
H 1 (K)2 such that (9)
uK e =
1 MK (f + ν∆u1 − ∇p) ∀K ∈ Th , ν
u∂K = e
1 BK (Jν ∂n u1 + pI· nK) ν
and (10)
∀K ∈ Th .
Next, since the enriched part ue is fully identified through (9)-(10) (or, equivalently, by (7)-(8)), we can perform statical condensation to derive a stabilized finite element method for our problem (1). First, integrating by parts, we have, on each K ∈ Th , ν(∇ue , ∇v 1 )K = −ν(ue , ∆v 1 )K + (ue , ν∂n v 1 )∂K , (q, ∇· ue )K = −(ue , ∇q)K + (ue , qI· n)∂K . Using these identities we can rewrite (2) in the following way i X h ν(∇u1 , ∇v 1 )Ω + − (ue , ν∆v 1 )K + (ue , ν∂n v 1 )∂K − (p, ∇· v 1 )Ω K∈Th
(11)
+(q, ∇· u1 )Ω +
X h
K∈Th
i − (ue , ∇q)K + (ue , qI· n)∂K = (f , v 1 )Ω ,
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R. ARAYA, G.R. BARRENECHEA, AND F. VALENTIN
which implies
(12)
+
X h
K∈Th
ν(∇u1 , ∇v 1 )Ω − (p, ∇· v 1 )Ω + (q, ∇· u1 )Ω − (ue , ν∆v 1 + ∇q)K +
(u∂K e , ν∂n v 1
i
+ qI· n)∂K = (f , v 1 )Ω ,
which, applying characterizations (9)-(10), becomes
(13)
+
X h1
K∈Th
ν
ν(∇u1 , ∇v 1 )Ω − (p, ∇· v 1 )Ω + (q, ∇· u1 )Ω (MK (−ν∆u1 + ∇p) − BK (Jν∂n u1 + pI· nK), ν∆v 1 + ∇q)K
i X 1 1 (MK (f ), ν∆v 1 + ∇q)K . + (BK (Jν ∂n u1 + pI· nK) , ν∂n v 1 + qI· n)∂K = (f , v 1 )Ω + ν ν K∈T h
Using this form, in the next sections we will present concrete stabilized finite element methods for both the simplest possible pair (P1 /P0 elements) and equal order P1 /P1 continuous finite elements. 3. The simplest element P1 /P0 3.1. The method. For this case, the finite element spaces are given by Vh := {v ∈ C 0 (Ω)2 : v|K ∈ P1 (K)2 , ∀ K ∈ Th } ∩ H01 (Ω)2 , for the velocity, and Q0h := {q ∈ L20 (Ω) : q|K ∈ P0 (K) , ∀ K ∈ Th } , for the pressure. Using these spaces, we propose the following stabilized method: Find (u1 , p0 ) ∈ Vh × Q0h such that (14) where
(15)
B0 (u1 , p0 ), (v 1 , q0 ) = F0 v 1 , q0 ∀ (v 1 , q0 ) ∈ Vh × Q0h , B0 (u1 , p0 ), (v 1 , q0 ) := ν(∇u1 , ∇v 1 )Ω − (p0 , ∇· v 1 )Ω + (q0 , ∇· u1 )Ω X + τZ (Jν∂n u1 + p0 I· nK, Jν∂n v 1 + q0 I· nK)Z , Z∈Eh
(16) and τZ is given by (17)
F0 v 1 , q0 := (f , v 1 )Ω , τZ :=
hZ . 12ν
STABILIZED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM
7
Remark. This method has some differences with other existing stabilized finite element methods with discontinuous pressure spaces (see, for example, [23, 29, 34, 14]). First, since τ Z is known exactly, we have no free constants to set. Up to the authors’ knowledge, this is the first time that the stabilization parameter corresponding to jump terms is known exactly. Furthermore, and in contrast to [22], the jump terms are derived without the use of a macro-element technique. Finally, another difference is the nature of the jump terms, not only containing pressure jumps, but also the jump on the normal derivative of u. Remark. One of the drawbacks of RFB method for the Stokes problem is that, due to the zero boundary condition on the element boundary, there is not a bubble-based enrichment that makes stable the P1 /P0 element (see [6] for a discussion), and hence, the use of a different boundary condition makes possible to stabilize the P1 /P0 element. 3.1.1. Derivation of the method. First we note that using spaces Vh and Q0h , equation (13) reduces to: Find (u1 , p0 ) ∈ Vh × Q0h such that ν(∇u1 , ∇v 1 )Ω − (p0 , ∇· v 1 )Ω + (q0 , ∇· u1 )Ω
(18)
+
X 1 (BK (Jν∂n u1 + p0 I· nK) , Jν∂n v 1 + q0 I· nK)Z = (f , v 1 )Ω , ν Z∈E h
for all (v 1 , q0 ) ∈ Vh × Q0h . Remark. Since BK is the inverse of an elliptic operator, we have, for all g ∈ L2 (∂K)2 , by denoting v = BK (g),
(BK (g), g)∂K = − (v, ∂ss v)∂K = (∂s v, ∂s v)∂K ≥ 0 , and hence we are adding a positive term to the formulation. We exploit next the fact that J∂n u1 + p0 I· nK is a constant function. To do it, we define Z
the (matrix) function buK := (BK (e1 )|BK (e2 )), where e1 , e2 are the canonical vectors in R2 , and we remark that, from it’s definition, buK = buK I, where buK is the solution of (19)
−∆buK = 0 in K,
buK = g(s) on each Z ⊂ ∂K,
where g = 0 if Z ⊂ ∂Ω, and g satisfies (20) in the internal edges.
− ∂ss g(s) =
1 hZ
in Z ,
g = 0 at the nodes ,
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R. ARAYA, G.R. BARRENECHEA, AND F. VALENTIN
Remark. The solution of (20) may be calculated explicitly and it is not difficult to realize that hZ (buK , 1)Z = . |Z| 12
(21)
Finally, since J∂n u1 + p0 I· nK is a constant function we obtain Z
(BK (Jν∂n u1 + p0 I· nK), Jν∂n v 1 + q0 I· nK)Z = =
(buK , 1)Z |Z|
Z
Z
buK
Jν∂n u1 + p0 I· nK · Jν∂n v 1 + q0 I· nK Z
Z
(Jν∂n u1 + p0 I· nK, Jν∂n v 1 + q0 I· nK)Z ,
and hence replacing this into (18) and using previous remark, we obtain method (14). 3.2. Error Analysis. From now on, C will denote a positive constant independent of h and ν, and that may change it’s value whenever it is written in two different places. Next result states the consistency of the proposed method. Lemma 1. Let (u, p) ∈ [H 2 (Ω) ∩ H01 (Ω)]2 × [H 1 (Ω) ∩ L20 (Ω)] be the weak solution of (1) and (u1 , p0 ) the solution of (14). Then,
(22)
B0 (u − u1 , p − p0 ), (v 1 , q0 ) = 0 for all (v 1 , q0 ) ∈ Vh × Q0h .
Proof. The results follows by noting that Jν∂n u + pI· nK = 0 a.e. across all the internal edges.
Moreover, defining the mesh-dependent norm "
k(v, q)kh := ν |v|21,Ω +
(23)
X
Z∈Eh
τZ kJν∂n v + qI· nKk20,Z
# 12
,
we have the following continuity and coercivity results. Lemma 2. Let be (v, q), (w, r) ∈ [H 2 (Th ) ∩ H01 (Ω)]2 × [H 1 (Th ) ∩ L20 (Ω)]. Then, bilinear
form B0 satisfies (24) (25)
B0 (v, q), (w, r) B0 (v, q), (v, q)
≤ k(v, q)kh k(w, r)kh + (∇· v, r)Ω − (q, ∇· w)Ω , = k(v, q)k2h .
Proof. The result follows immediately from the definition of B0 .
STABILIZED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM
9
In order to perform the numerical analysis of this method, we will consider the Lagrange interpolation operator Ih : C 0 (Ω) → Vh (if v = (v1 , v2 ) ∈ C 0 (Ω)2 , we denote Ih (v) = (Ih (v1 ), Ih (v2 ))) to approximate the velocity. Then, it is well known (cf. [17]) that
(26) (27)
|v − Ih (v)|m,K ≤ C h2−m |v|2,K K 2−t−1/2
|v − Ih (v)|t,Z ≤ C hZ
|v|2,ωZ
∀v ∈ H 2 (K), ∀v ∈ H 2 (ωZ ),
for all K ∈ Th , Z ∈ Eh , where ωZ := ∪{K ∈ Th : Z ⊂ ∂K}, and m = 0, 1, 2, t = 0, 1. Let
us remark that in order to obtain second estimate above, we have used the following local trace theorem (for a proof, see [33]): There exists C > 0, independent of h, such that 1 2 2 2 kvk0,K + hK |v|1,K , (28) kvk0,∂K ≤ C hK for all v ∈ H 1 (K).
In order to approximate the pressure we will consider Πh : L2 (Ω) → Q0h the L2 (Ω)-
projection onto Q0h . This projection satisfies (cf. [17]) (29)
kq − Πh (q)k0,Ω ≤ C h |q|1,Ω ,
if q ∈ H 1 (Ω), and hence, using local trace theorem (28), we obtain i 12 hX 2 hZ kJq − Πh (q)Kk0,Z (30) ≤ C h |q|1,Ω , Z∈Eh
for all q ∈ H 1 (Ω).
Lemma 3. Suppose (v, q) ∈ H 2 (Ω)2 × H 1 (Ω). Then, √ 1 k(v − Ih (v), q − Πh (q))kh ≤ Ch ν |v|2,Ω + √ |q|1,Ω . (31) ν
Proof. The result follows immediately from the norm definition and (26),(27) and (30).
Using previous results we can establish the following convergence result. Theorem 4. Let (u, p) ∈ [H 2 (Ω) ∩ H01 (Ω)]2 × [H 1 (Ω) ∩ L20 (Ω)] be the solution of (1) and
(u1 , p0 ) the solution of (14). Then, the following error estimate holds √ 1 (32) k(u − u1 , p − p0 )kh ≤ Ch ν |u|2,Ω + √ |p|1,Ω . ν
˜ h , p˜h ) := (Ih (u), Πh (p)) ∈ Vh × Q0h . From Lemmata 1 and 2 we know that Proof. Let (u k(u − u1 , p − p0 )k2h = B0 (u − u1 , p − p0 ), (u − u1 , p − p0 ) ˜ h , p − p˜h ) = B0 (u − u1 , p − p0 ), (u − u
˜ h , p − p˜h )kh + (∇· (u − u1 ), p − p˜h )Ω − (∇· (u − u ˜ h ), p − p0 )Ω . ≤ C k(u − u1 , p − p0 )kh k(u − u
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R. ARAYA, G.R. BARRENECHEA, AND F. VALENTIN
Now, (33)
(∇· (u − u1 ), p − p˜h )Ω = − (∇· u1 , p − p˜h )Ω = 0 ,
since u is a solenoidal field and ∇· u1 ∈ Q0h . On the other hand i X h ˜ h ), p − p0 )Ω = ˜ h )K + (p − p0 , (u − u ˜ h )· n)∂K (∇· (u − u − (∇p, u − u K∈Th
˜ h )Ω + = − (∇p, u − u
K∈Th
˜ h k0,Ω + ≤ |p|1,Ω ku − u 2
≤ Ch |p|1,Ω |u|2,Ω ≤ Ch2 |p|1,Ω |u|2,Ω
X
X
˜ h )∂K ((p − p0 )I· n, u − u
Z∈Eh
˜ h )Z (J(p − p0 )I· nK, u − u 3
X h2 √ √Z kJ(p − p0 )I· nKk0,Z ν |u|2,ωZ +C ν Z∈E h
X 1 X hZ kJ(p − p0 )I· nKk20,Z + Cγ h2 ν |u|22,ωZ + γ Z∈E ν Z∈E h
h
1 1 X hZ ≤ Ch2 ((1 + γ) ν |u|22,Ω + |p|21,Ω ) + kJ(p − p0 )I· nKk20,Z , ν γ Z∈E ν h
where γ > 0. Now, using the local trace theorem (28) and the fact that Vh is constituted by linear polynomials we arrive at X hZ kJ(p − p0 )I· nKk20,Z ≤ ν Z∈E h
≤ 2 ≤ C
X hZ ν Z∈E h
kJν ∂n (u − u1 ) + (p − p0 )I· nKk20,Z + kJν ∂n (u − u1 )Kk20,Z
i X h hZ kJ∂n (u − u1 ) + (p − p0 )I· nKk20,Z + ν |u − u1 |21,Ω + ν h2 |u|22,Ω ν Z∈E h
!
≤ C˜ k(u − u1 , p − p0 )k2h + C ν h2 |u|22,Ω . Hence, choosing γ = 2C˜ we obtain (34)
1 1 k(u − u1 , p − p0 )k2h ≤ Ch2 (ν |u|22,Ω + |p|21,Ω ) , 2 ν
and the result follows by extracting square root.
Remark. Last result gives a convergence result for the velocity, plus a convergence result for the jump terms. More precisely, this result implies ν |u − u1 |1,Ω ≤ C h and 21 P 2 ≤ C h, which are both optimal in order and regularity. Z∈Eh hZ kJ∂n (u − u1 )Kk0,Z
STABILIZED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM
11
3.2.1. A convergence result for the pressure. Last result in previous section does not give convergence on the natural norm of the pressure. That is why a convergence result for the pressure in the L2 (Ω) norm is now given. In the proof of next result we will use the Cl´ement interpolation operator (cf. [17, 24]), Ch : H 1 (Ω) → Vh . This operator satisfies |v − Ch (v)|m,Ω ≤ C h1−m |v|1,Ω
(35)
∀v ∈ H 1 (Ω) ,
for m = 0, 1, with the obvious extension to vector-valued functions. Theorem 5. Let (u, p) ∈ [H 2 (Ω) ∩ H01 (Ω)]2 × [H 1 (Ω) ∩ L20 (Ω)] be the solution of (1) and
(u1 , p0 ) the solution of (14). Then, the following error estimate holds h i kp − p0 k0,Ω ≤ C h ν |u|2,Ω + |p|1,Ω . (36)
Proof. From the continuous inf-sup condition (see [24]), there exists w ∈ H01 (Ω)2 such that ∇· w = p − p0 in Ω and |w|1,Ω ≤ C kp − p0 k0,Ω . Let w h = Ch (w). Then, applying the
consistency of the method we obtain kp − p0 k20,Ω = (∇· w, p − p0 )Ω
= (∇· (w − w h ), p − p0 )Ω + (∇· w h , p − p0 )Ω X = [−(w − w h , ∇p)K + (w − w h , (p − p0 )I· n)∂K ] K∈Th
+ ν (∇(u − u1 ), ∇w h )Ω +
X
Z∈Eh
τZ (Jν∂n (u − u1 ) + (p − p0 )I· nK, Jν∂n wh K)Z
h X h2 X K τZ kJ(p − p0 )I· nKk20,Z + ν|u − u1 |21,Ω |p|21,K + ≤ ν Z∈E K∈T h
+
X
Z∈Eh
h
τZ kJν∂n (u − u1 ) + (p − p0 )I· nKk20,Z
i 21
·
i 12 h X ν X X −1 2 2 2 2 τ kJν∂ w Kk τ kw − w k + ν |w | + . kw − w k + Z n h h h h 0,Z 0,Z 1,Ω 0,K Z h2K K∈T Z∈E Z∈E h
h
h
Now, using the local trace theorem (28) and (35) we easily obtain i 12 h X ν X X −1 2 2 2 2 τZ kJν∂n wh Kk0,Z τZ kw − w h k0,Z + ν |w h |1,Ω + kw − wh k0,K + h2 Z∈Eh Z∈Eh K∈Th K √ √ ≤ C ν |w|1,Ω ≤ C ν kp − p0 k0,Ω .
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R. ARAYA, G.R. BARRENECHEA, AND F. VALENTIN
Hence, dividing by kp − p0 k0,Ω and using (28) again we have kp − p0 k0,Ω ≤ X √ h X h2K 2 ≤ C ν |p|1,K + ν |u − u1 |21,Ω + τZ kJν∂n (u − u1 ) + (p − p0 )I· nKk20,Z ν K∈T Z∈E h
h
i 12
+ τZ kJν∂n (u − u1 )Kk20,Z i 12 √ h h2 2 2 2 2 |p|1,Ω + k(u − u1 , p − p0 )kh + νh |u|2,Ω ≤ C ν ν i 21 √ h h2 2 2 2 ≤ C ν |p|1,Ω + νh |u|2,Ω , ν
and the result follows.
3.2.2. An error estimate for ku − u1 k0,Ω . Throughout this section we will suppose that the solution of the problem: Find (ϕ, π) such that:
(37)
−ν ∆ ϕ − ∇π = u − u1
,
∇· ϕ = 0
ϕ = 0
in Ω ,
on ∂Ω ,
where (u1 , p0 ) is the solution of (14), belongs to [H 2 (Ω) ∩ H01 (Ω)]2 × [H 1 (Ω) ∩ L20 (Ω)], and
that the following estimate holds (38)
ν kϕk2,Ω + kπk1,Ω ≤ C ku − u1 k0,Ω .
Theorem 6. Let (u, p) ∈ [H 2 (Ω) ∩ H01 (Ω)]2 × [H 1 (Ω) ∩ L20 (Ω)] be the solution of (1) and
(u1 , p0 ) the solution of (14). Then, the following error estimate holds
√ 1 1 ku − u1 k0,Ω ≤ C h2 max{1, √ } ( ν |u|2,Ω + √ |p|1,Ω ) . ν ν Proof. Let (ϕh , πh ) := (Ih (ϕ), Πh (π)) ∈ Vh × Q0h . Then, multiplying first equation in (37)
by u − u1 and second by −(p − p0 ), from the definition of bilinear form B0 , the regularity
of (ϕ, π) and the consistency of the method and Lemma 2, we obtain ku − u1 k20,Ω = = ν (∇ϕ, ∇(u − u1 ))Ω + (π, ∇· (u − u1 ))Ω − (p − p0 , ∇· ϕ)Ω = B0 ((u − u1 , p − p0 ), (ϕ, π)) = B0 ((u − u1 , p − p0 ), (ϕ − ϕh , π − πh ))
≤ k(u − u1 , p − p0 )kh k(ϕ − ϕh , π − πh )kh − (p − p0 , ∇· (ϕ − ϕh ))Ω + (π − πh , ∇· (u − u1 ))Ω .
STABILIZED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM
13
Now, using (33) we see that (π − πh , ∇· (u − u1 ))Ω = 0, and hence, using interpolation
inequalities (26), Lemma 3 and Theorems 4 and 5, we arrive at ku − u1 k20,Ω ≤
≤ k(u − u1 , p − p0 )kh k(ϕ − ϕh , π − πh )kh + kp − p0 k0,Ω k∇· (ϕ − ϕh )k0,Ω h i 12 h i 12 1 ≤ k(u − u1 , p − p0 )k2h + kp − p0 k20,Ω k(ϕ − ϕh , π − πh )k2h + ν k∇· (ϕ − ϕh )k20,Ω ν 1 h i i 12 h 1 1 2 ≤ Ch2 ν |u|22,Ω + |p|21,Ω ν |ϕ|22,Ω + |π|21,Ω ν ν √ 1 1 ≤ C max{1, √ } h2 ( ν |u|2,Ω + √ |p|1,Ω ) ku − u1 k0,Ω , ν ν and the result follows.
4. The method using P1 /P1 continuous elements 4.1. The method. For this case, the finite element space for the velocity is the same one as in previous section, but the pressure space is now given by Q1h := {q ∈ C 0 (Ω) : q|K ∈ P1 (K) , ∀ K ∈ Th } ∩ L20 (Ω) . As we will see in next section, the method coming directly from (13) is given by: Find (˜ u1 , p˜1 ) ∈ Vh × Q1h such that: (39)
B1 ((˜ u1 , p˜1 ), (v 1 , q1 )) = F(v 1 , q1 )
∀(v 1 , q1 ) ∈ Vh × Q1h ,
where (40)
B1 ((u1 , p1 ), (v 1 , q1 )) := B((u1 , p1 ), (v 1 , q1 )) −
X 1 (BK (Jν∂n u1 K), ∇q1 )K , ν K∈T h
with
(41)
B (u1 , p1 ), (v 1 , q1 ) := ν(∇u1 , ∇v 1 )Ω − (p1 , ∇· v 1 )Ω + (q1 , ∇· u1 )Ω X X + τK (−ν∆u1 + ∇p1 , ν∆v 1 + ∇q1 )K + τZ (Jν∂n u1 K, Jν∂n v 1 K)Z , K∈Th
(42) (43)
Z∈Eh
F v 1 , q1 := (f , v 1 )Ω +
X
K∈Th
τK := C1
τK (f , ν∆v 1 + ∇q1 )K ,
h2K , ν
and τZ is given by (17), and C1 = 81 . The value C1 = analysis of original method (39) (see Appendix A).
1 8
has been suggested by the error
14
R. ARAYA, G.R. BARRENECHEA, AND F. VALENTIN
Now, for reasons that we will justify later (see Theorem 7 below), we will drop the term −
X 1 ( BK (Jν∂n u1 K), ∇q1 )K , ν K∈T h
and analyze (and implement) the following simplified version of (39): Find (u 1 , p1 ) ∈ Vh ×Q1h
such that (44)
∀ (v 1 , q1 ) ∈ Vh × Q1h .
B((u1 , p1 ), (v 1 , q1 )) = F(v 1 , q1 )
Remark. We see that method (44) has the form of a stabilized method of the GLS class, plus a non-standard jump term formed by the residual of the Cauchy stress tensor on the edges of the triangulation. This will give us a control on this residual, which is exclusive to continuous pressure spaces, since in that case pressure jumps vanish. Remark. The writing of the method as the restriction of a consistent method to P 1 /P1 elements is just in order to avoid some technical difficulties. A non-consistent presentation may be given and in that case we can prove that the consistency error does not imply a loose of precision. As we said before, we will perform the error analysis of method (44). This is due to the fact that the error of method (39) is bounded by the one of (44), as stated in the following result, whose proof may be found in Appendix A. Theorem 7. Let (u, p) ∈ H 2 (Ω)2 × H 1 (Ω) be the solution of (1). Then, method (39) is
˜ 1 , p˜1 ) ∈ Vh × Q1h , and the following error consistent. Moreover, (39) has a unique solution (u
estimate holds
˜ 1 k2h + kp − p˜1 k2h ≤ C (ku − u1 k2h + kp − p1 k2h ) , ku − u where (u1 , p1 ) ∈ Vh × Q1h is the solution of (44). 4.1.1. Derivation of the method. Using spaces Vh and Q1h , equation (13) reduces to: Find (u1 , p1 ) ∈ Vh × Q1h such that ν(∇u1 , ∇v 1 )Ω − (p1 , ∇· v 1 )Ω + (q1 , ∇· u1 )Ω + (45)
X 1 (MK (∇p1 ) − BK (Jν∂n u1 K), ∇q1 )K ν K∈T h
X 1 X 1 (BK (Jν∂n u1 K) , Jν∂n v 1 K)Z = (f , v 1 )Ω + (MK (f ), ∇q1 )K , + ν ν K∈T Z∈E h
h
STABILIZED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM
15
for all (v 1 , q1 ) ∈ Vh ×Q1h . Since ∇p1 ∈ R2 , we have MK (∇p1 ) = (MK (e1 ), MK (e2 ))∇p1 =: K
bpK ∇p1 . As in previous section, we see that bpK = bpK I, where bpK is the solution of −∆bpK = 1 in K,
(46)
bpK = 0 on ∂K .
Hence (MK (∇p1 ), ∇q1 )K =
Z
K
bpK
(bpK , 1)K ∇p1 · ∇q1 = (∇p1 , ∇q1 )K . |K| K K
On the other hand, from previous section we know that
(BK (Jν∂n u1 K), Jν∂n v 1 K)Z = τZ (Jν∂n u1 K, Jν∂n v 1 K)Z , where τZ has been defined in (17). Moreover, if we suppose that f is piecewise constant, we have MK (f ) = bpK f , and hence, in the same way as before (MK (f ), ∇q1 )K
(bpK , 1)K (f , ∇q1 )K . = |K|
Summing all this up, we arrive to the following expression for (45): Find (u 1 , p1 ) ∈ Vh × Q1h
such that
ν(∇u1 , ∇v 1 )Ω − (p1 , ∇· v 1 )Ω + (q1 , ∇· u1 )Ω +
X (bp , 1)K K (∇p1 , ∇q1 )K |K|ν K∈T h
X (bu , 1)Z X 1 K (BK (Jν∂n u1 K), ∇q1 )K + (Jν∂n u1 K, Jν∂n v 1 K)Z − ν |Z|ν Z∈E K∈T h
(47)
h
X (bp , 1)K K = (f , v 1 )Ω + (f , ∇q1 )K , |K|ν K∈T h
for all (v 1 , q1 ) ∈ Vh × Q1h . Finally, since the mesh is regular by a scaling argument (cf. [31])
we have that (48)
1 p (b , 1)K ∼ C1 h2K , |K| K
where C1 is a positive constant independent of h and ν. Hence, replacing (48) in (47) and defining τK appropriately, we obtain method (39). Remark. The assumption of the piecewise constant right hand side f it is just done in order to derive the method, but it does not affect the precision of it. Indeed, if we consider a general f ∈ H 1 (Ω)2 and take it’s projection onto the space of piecewise constant functions
we keep the same order of convergence of the method (see Appendix B).
16
R. ARAYA, G.R. BARRENECHEA, AND F. VALENTIN
4.2. Error analysis. Let us consider the mesh-dependent norms X (49) |||v|||2h := ν |v|21,Ω + τZ kJν∂n vKk20,Z , Z∈Eh
kqk2h :=
(50)
X
K∈Th
τK |q|21,K .
The first results concern the consistency and well-posedeness of stabilized method (44). Lemma 8. Let (u, p) ∈ [H 2 (Ω) ∩ H01 (Ω)]2 × [H 1 (Ω) ∩ L20 (Ω)] be the solution of (1) and (u1 , p1 ) the solution of (44). Then,
B (u − u1 , p − p1 ), (v 1 , q1 ) = 0 for all (v 1 , q1 ) ∈ Vh × Q1h .
Proof. The results follows from the definition of B and the fact that Jν∂n uK = 0 a.e. on the internal edges.
Lemma 9. Let be (v 1 , q1 ) ∈ Vh × Q1h . Then B (v 1 , q1 ), (v 1 , q1 ) = |||v 1 |||2h + kq1 k2h .
Proof. The result follows from the definition of B and the fact that ∆v 1 = 0 in each K ∈
Th .
Now, in order to approximate the velocity we will consider the Lagrange interpolation operator as in previous section and for the pressure interpolation we will use the Cl´ement interpolation operator Ch satisfying (35).
The following approximation result will be useful in the sequel.
Lemma 10. Let (v, q) ∈ [H 2 (Ω) ∩ H01 (Ω)]2 × [H 1 (Ω) ∩ L20 (Ω)] and q˜h := Ch (q) −
(Ch (q),1)Ω . |Ω|
Then,
(51) (52)
|||v −
Ih (v)|||2h
+
X h
K∈Th
τK−1 kv
−
Ih (v)k20,K
+
νh2K k∆(v
−
Ih (v))k20,K
1 h kq − q˜h kh + √ kq − q˜h k0,Ω ≤ C √ |q|1,Ω . ν ν
i
≤ C h2 ν |v|22,Ω ,
Proof. The result follows from the norm definition and using kq − q˜h k0,Ω ≤ kq − Ch (q)k0,Ω combined with (26),(27) and (35).
Using Lemmata 8-10 we can establish the following convergence result.
STABILIZED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM
17
Theorem 11. Let (u, p) ∈ [H 2 (Ω) ∩ H01 (Ω)]2 × [H 1 (Ω) ∩ L20 (Ω)] be the solution of (1) and
(u1 , p1 ) the solution of (44). Then, the following error estimate holds h√ i 1 |||u − u1 |||h + kp − p1 kh ≤ C h ν |u|2,Ω + √ |p|1,Ω . (53) ν ˜ h := Ih (u), p˜h := Ch (p) − Proof. Let u
(Ch (p),1)Ω |Ω|
˜ h , p − p˜h ). Applying and (η u , η p ) := (u − u
Lemma 9, the consistency of the method and integrating by parts we have ˜ h |||2h + kp1 − p˜h k2h = B((u1 − u ˜ h , p1 − p˜h ), (u1 − u ˜ h , p1 − p˜h )) |||u1 − u ˜ h , p1 − p˜h )) = B((η u , η p ), (u1 − u ˜ h ))Ω − (η p , ∇· (u1 − u ˜ h ))Ω − (η u , ∇(p1 − p˜h ))Ω = ν (∇η u , ∇(u1 − u X X + τK (−ν∆η u , ∇(p1 − p˜h ))K + τK (∇η p , ∇(p1 − p˜h ))K K∈Th
+
X
Z∈Eh
h
≤ ν
K∈Th
˜ h )K)Z τZ (Jν∂n η u K, Jν∂n (u1 − u
|η u |21,Ω
i 12 X X 1 p 2 −1 u 2 2 u 2 p 2 u 2 + kη k0,Ω + ( τK kη k0,K + ν τK k∆η k0,K ) + kη kh + τZ kJν∂n η Kk0,Z ν K∈T Z∈E h
h
˜ h |21,Ω + 3 · 3ν |u1 − u
h
X
τK k∇(p1 − p˜h )k20,K +
p˜h k2h
i 12
K∈Th
X
Z∈Eh
˜ h )Kk20,Z τZ kJν∂n (u1 − u
X √ h u 2 1 p 2 i 21 −1 u 2 2 u 2 p 2 ≤ 3 |||η |||h + [τK kη k0,K + νhK k∆η k0,K ] + kη kh + kη k0,Ω ν K∈T
i 12
h
h
· |||u1 −
˜ h |||2h u
+ kp1 −
.
Hence, dividing by the last term and applying Lemma 10 we arrive at (54)
h
˜ h |||h + kp1 − p˜h kh ≤ C νh |||u1 − u
2
|u|22,Ω
h2 2 i 12 + |p|1,Ω . ν
The result follows using triangular inequality and Lemma 10 once more.
Remark. In particular, from previous Theorem we have an O(h) convergence for |u − u 1 |1,Ω P 21 2 and , which are both optimal in order and regularity. Z∈Eh hZ kJ∂n (u − u1 )Kk0,Z
4.2.1. A convergence result for the pressure. In the last result of previous section we had an error estimate in the velocity, but, due to the norm definition, we did not guarantee the convergence of the pressure. Next result shows that we have an optimal error estimate in the natural norm of the pressure, which is independent of ν.
18
R. ARAYA, G.R. BARRENECHEA, AND F. VALENTIN
Theorem 12. Let (u, p) ∈ [H 2 (Ω) ∩ H01 (Ω)]2 × [H 1 (Ω) ∩ L20 (Ω)] be the solution of (1) and
(u1 , p1 ) the solution of (44). Then, the following error estimate holds h√ i kp − p1 k0,Ω ≤ C h ν |u|2,Ω + |p|1,Ω . (55)
Proof. From the continuous inf-sup condition (see [24]), there exists w ∈ H01 (Ω)2 such that ∇· w = p − p1 in Ω and |w|1,Ω ≤ C kp − p1 k0,Ω . Let wh = Ch (w) ∈ Vh . Then, applying the consistency of the method, (35) and previous theorem we obtain
kp − p1 k20,Ω = (∇· w, p − p1 )Ω = (∇· (w − w h ), p − p1 )Ω + (∇· w h , p − p1 )Ω X =− (w − w h , ∇(p − p1 ))K + ν (∇(u − u1 ), ∇w h )Ω K∈Th
+
X
Z∈Eh
X
≤
K∈Th
+
τZ (Jν∂n (u − u1 )K, Jν∂n wh K)Z
kw − wh k0,K |p − p1 |1,K + ν |u − u1 |1,Ω |wh |1,Ω
X
Z∈Eh
τZ kJν∂n (u − u1 )Kk0,Z kJν∂n wh Kk0,Z
≤
h X
τK |p −
·
h X
τK−1 kw
K∈Th
K∈Th
p1 |21,K
−
+ ν |u −
wh k20,K
+ν
u1 |21,Ω
|wh |21,Ω
+
X
τZ kJν∂n (u −
X
τZ kJν∂n wh Kk20,Z
Z∈Eh
+
Z∈Eh
ih i 21 √ h 2 2 ≤ C ν |||u − u1 |||h + kp − p1 kh |w|1,Ω + |w h |1,Ω
u1 )Kk20,Z i 12
i 21
√
1 νh (|u|2,Ω + √ |p|1,Ω ) kp − p1 k0,Ω , ν P where, in order to bound the term Z∈Eh τZ kJν∂n wh Kk20,Z we have used the local trace result ≤C
(28) and w h |K ∈ P1 (K)2 . The result follows then by dividing by the last term.
4.2.2. An error estimate for ku − u1 k0,Ω . Throughout this section we will suppose that the solution of the problem: Find (ϕ, π) such that:
(56)
−ν ∆ ϕ − ∇π = u − u1
,
∇· ϕ = 0
ϕ = 0
in Ω ,
on ∂Ω ,
where (u1 , p1 ) is the solution of (44), belongs to [H 2 (Ω) ∩ H01 (Ω)]2 × [H 1 (Ω) ∩ L20 (Ω)], and
that the following estimate holds (57)
ν kϕk2,Ω + kπk1,Ω ≤ C ku − u1 k0,Ω .
STABILIZED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM
19
Theorem 13. Under the hypothesis of Theorem 12 the following error estimate holds √ 1 1 ku − u1 k0,Ω ≤ C h2 max{1, √ } ( ν |u|2,Ω + √ |p|1,Ω ) . ν ν Ω Proof. Let (ϕh , πh ) := (Ih (ϕ), Ch (π) − (Ch (π),1) ) ∈ Vh × Q1h . Then, multiplying first equation |Ω|
in (56) by u − u1 and second by −(p − p1 ), from the definition of bilinear form B, the consistency of the method, the fact that J∂n ϕK = 0 a.e. on the internal edges, interpolation
inequalities (26), (27) and (35), and Theorems 11 and 12, we obtain ku − u1 k20,Ω = = ν (∇ϕ, ∇(u − u1 ))Ω + (π, ∇· (u − u1 ))Ω − (p − p1 , ∇· ϕ)Ω X = B((u − u1 , p − p1 ), (ϕ, π)) − τK (−ν∆(u − u1 ) + ∇(p − p1 ), ν∆ϕ + ∇π)K K∈Th
= B((u − u1 , p − p1 ), (ϕ − ϕh , π − πh )) −
X
K∈Th
τK (−ν∆(u − u1 ) + ∇(p − p1 ), ν∆ϕ + ∇π)K
h 1 ≤ ν |u − u1 |21,Ω + ν k∇· (u − u1 )k20,Ω + kp − p1 k20,Ω ν i 21 X X τK k − ν∆u + ∇(p − p1 )k20,K · τZ kJν∂n (u − u1 )Kk20,Z + 2 + K∈Th
Z∈Eh
h
ν |ϕ − ϕh |21,Ω +
+
X
Z∈Eh
h
X 1 τK k∇(π − πh )k20,K kπ − πh k20,Ω + ν k∇· (ϕ − ϕh )k20,Ω + ν K∈T h
τZ kJν∂n (ϕ − ϕh )Kk20,Z +
≤ C |||u −
u1 |||2h
+ νh
2
|u|22,Ω
X
K∈Th
+ kp −
τK kν∆ϕ + ∇πk20,K
p1 k2h
i 21
i 12 h 1 h2 2 i 21 2 2 2 + kp − p1 k0,Ω ν h |ϕ|2,Ω + |π|1,Ω ν ν
√ 1 1 ≤ C max{1, √ } h2 ( ν |u|2,Ω + √ |p|1,Ω ) ku − u1 k0,Ω , ν ν and the result follows.
Remark. As we claimed before, the error analysis is independent of the nature of right hand side f , and hence, we have actually justified method (44) for a general f ∈ L2 (Ω)2 . In
Appendix B we will show that if f ∈ H 1 (Ω)2 , then the difference between implementing P method (44) and the right hand side (f , v 1 )Ω + K∈Th ν −1 (MK (f ), ν∆v 1 + ∇q)K is smaller than the order of the method. On the other hand, method (44) has been justified for any constant C1 > 0, even if it has been presented with C1 = 81 .
20
R. ARAYA, G.R. BARRENECHEA, AND F. VALENTIN
5. An alternative formulation including the residual on the boundary In this section we propose another class of methods arising from a different choice of enrichment functions. We will denote by Rh the pressure space according to the choice of elements, i.e., Rh = Q1h for P1 /P1 elements and Rh = Q0h for P1 /P0 elements. The proposed method reads: Find (ur , pr ) ∈ Vh × Rh such that: Br ((ur , pr ), (v 1 , q)) = F(v 1 , q)
(58)
∀ (v 1 , q) ∈ Vh × Rh ,
where (59) +
X
K∈Th
Br ((u1 , p), (v 1 , q)) = ν(∇u1 , ∇v 1 )Ω − (p, ∇· v 1 )Ω + (q, ∇· u1 )Ω X τK (−ν∆u1 + ∇p, ν∆v 1 + ∇q)K + τ˜Z (J − ν∂n u1 + pI· nK, Jν∂n v 1 + qI· nK)Z , Z∈Eh
F is given by (42), τK by (43), and τ˜Z :=
(60)
hZ , 12αν
where α > 0 will be fixed in order to have a well posed problem. This method may be obtained in the same way as method (14) and (44) taking the enrichment function ue as being the solution of (4), together with the boundary conditions (61)
−ν∂ss ue =
1 J − ν ∂n ur1 + pr I· nK on each Z ⊂ ∂K, αhZ
ue = 0 at the nodes ,
on the internal edges, and ue = 0 on ∂K ∩ ∂Ω. In fact, using this choice of enrichment we
can perform the same derivation from Sections 3 and 4, neglecting once more a cross term appearing in P1 /P1 discretization. Remark. This method is different from (14) and (44) from two viewpoints. First, the boundary term contains the residual of the Cauchy stress tensor on the trial function. This fact comes from the choice of the enriched part as being a corrector for the residual inside the element and on the boundary. The other difference is the stabilization parameter on the edges. Now, this parameter contains a constant to set. Now, let the following mesh-dependent norm # 12 " X X τ˜Z kJqKk20,Z τK |q|21,K + |||(v 1 , q)|||h := ν |v 1 |21,Ω + (62) . K∈Th
Then, we have the following coercivity result.
Z∈Eh
STABILIZED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM
21
Lemma 14. Let us suppose that α > Ct /3, where Ct > 0 is the constant from local trace result (28). Then, for all (v 1 , q) ∈ Vh × Rh there holds Br ((v 1 , q), (v 1 , q)) ≥
1 |||(v 1 , q)|||2h . 2
Proof. Let (v 1 , q) ∈ Vh × Rh . Then, since ∆v 1 = 0 on each K ∈ Th , applying local trace
result (28) and the definition of τ˜Z we obtain X X Br ((v 1 , q), (v 1 , q)) = ν |v 1 |21,Ω + τK k∇qk20,K + τ˜Z (−ν 2 kJ∂n v 1 Kk20,Z + kJqKk20,Z ) K∈Th
≥ ν |v 1 |21,Ω − ≥
Z∈Eh
X X Ct X ν |v 1 |21,K + τK |q|21,K + τ˜Z kJqKk20,Z 6α K∈T K∈T Z∈E h
h
h
1 |||(v 1 , q)|||2h , 2
an the result follows.
Once this method has been proved to be stable, following an absolutely analogous procedure than those from Sections 3 and 4 we can prove the consistency of (58) and perform a complete error analysis of (58), obtaining the same results as in previous sections. 6. Numerical validations 6.1. An analytical Solution: convergence validation. For this test case, the domain is taken as the square Ω = (0, 1) × (0, 1), ν = 1, and f is set such as the exact solution of our
Stokes problem is given by
u1 (x, y) = −256x2 (x − 1)2 y(y − 1)(2y − 1) , u2 (x, y) = −u1 (y, x) , p(x, y) = 150(x − 0.5)(y − 0.5) . We perform convergence analysis for methods (14),(44) and (58) using continuous P 1 /P1 and P1 /P0 elements. 6.1.1. The P1 /P1 case. For this case we first depict in Figures 1-2 the convergence history for method (44). The results reproduce our theoretical results showing an O(h) order of 12 P 2 convergence for |u − u1 |1,Ω , |J∂n (u − u1 )K|h := and kp − Z∈Eh hZ kJ∂n (u − u1 )Kk0,Z p1 k0,Ω , and an O(h2 ) convergence for ku − u1 k0,Ω .
Method (58) is tested next. The results are depicted in Figures 3-4 using α = 1, where
the results are in perfect accordance with the theoretical results. The justification of this choice for α may be found in Figure 4 (on the right) where we have depicted the behaviour
22
R. ARAYA, G.R. BARRENECHEA, AND F. VALENTIN
of the error in terms of α (using a mesh of around 2500 elements) and we see that for α ≥ 1 the error is almost independent of α, showing that the restriction of Lemma 14 is not only
theoretical, but at the same time showing that, once we are inside the region predicted by the theory, the performance of the method is independent of α.
1
10
kp − p1k0,Ω h
error
1
error
0.1
|u − u1|1,Ω h
0.01
0.001 0.001
0.1
0.01
0.1
0.01 0.001
1
0.01
h
0.1
1
h
Figure 1. Method (44): convergence history for kp − p1 k0,Ω and |u − u1 |1,Ω .
1
|J∂n(u − u1)K|h h
1
0.01
error
error
0.1
10
ku − u1k0,Ω h2
0.001
0.1
0.0001 1e-05 0.001
0.01
0.1 h
1
0.01 0.001
0.01
0.1 h
Figure 2. Method (44): convergence history for ku − u1 k0,Ω and |J∂n (u − u1 )K|h .
1
STABILIZED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM
10
10
kp − p1k0,Ω h
23
|u − u1|1,Ω h
1 error
error
1 0.1
0.1 0.01
0.001 0.01
0.1 h
0.01 0.01
1
0.1 h
1
Figure 3. Method (58): convergence history for kp − p1 k0,Ω and |u − u1 |1,Ω .
1
100
ku − u1k0,Ω h2
10
error
error
0.1
0.01
0.001
0.0001 0.01
kp − p1k0,Ω |u − u1|1,Ω ku − u1k0,Ω
1 0.1 0.01
0.1 h
1
0.001
0.1
1
10
100
α
Figure 4. Method (58): convergence history for ku − u1 k0,Ω , and sensibility of (58) with respect to α
6.1.2. The P1 /P0 case. For this case we first depict in Figures 5-6 the convergence history for method (14). The results reproduce our theoretical results showing an O(h) order of
24
R. ARAYA, G.R. BARRENECHEA, AND F. VALENTIN
convergence for |u − u1 |1,Ω , |J∂n (u − u1 ) + (p − p0 )nK|h and kp − p0 k0,Ω , and an O(h2 )
convergence for ku − u1 k0,Ω .
Method (58) is tested next. The results are depicted in Figures 7-8 using α = 1, where the results are in perfect accordance with the theoretical results, giving an O(h) for |u − u 1 |1,Ω ,
|Jp − p0 K|h and kp − p0 k0,Ω , and an O(h2 ) convergence for ku − u1 k0,Ω . Concerning the choice of α, the situation now is quite different from the one in previous section. As a matter of
fact, since we only control the pressure via the jump terms governed by α, we can expect the error to grow as α grows, as it is shown in Figure 9 (for a mesh of 2500 elements) where we see that all the errors attain a minimum at α = 1 (i.e., using τ˜Z = τZ ), and then they present a growing behaviour. Values larger than 10 have been tested and the beahaviour is growing in all the errors. Related experiments have been performed using the GLS method (cf. [27]), obtaining similar results.
10
kp − p0k0,Ω h
error
error
10
1
0.1 0.01
0.1 h
1
|u − u1|1,Ω h
1
0.1 0.01
0.1 h
Figure 5. Method (14): convergence history for kp − p0 k0,Ω and |u − u1 |1,Ω .
1
STABILIZED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM
1
100
ku − u1k0,Ω h2
25
|J∂n(u − u1) + (p − p0)nK|h h
0.1 error
error
10 0.01
1 0.001
0.0001 0.01
0.1 h
Figure 6. Method (14): |J∂n (u − u1 ) + (p − p0 )nK|h .
10
kp − p0k0,Ω h
1
0.1 0.01
0.1 h
0.1 h
1
convergence history for ku − u1 k0,Ω and
error
error
10
0.1 0.01
1
|u − u1|1,Ω h
1
0.1 0.01
0.1 h
Figure 7. Method (58): convergence history for kp − p0 k0,Ω and |u − u1 |1,Ω .
26
R. ARAYA, G.R. BARRENECHEA, AND F. VALENTIN
0.1
10
ku − u1k0,Ω h2
|Jp − p0K|h h
error
error
0.01 1
0.001
0.0001 0.01
0.1
0.1 0.01
0.1
h
h
Figure 8. Method (58): convergence history for ku − u1 k0,Ω and kp − p0 k0,Ω .
100
kp − p0 k0,Ω |u − u1 |1,Ω ku − u1 k0,Ω
|Jp − p0 K|h
error
10 1 0.1 0.01 0.001
0.1
1 α Figure 9. Sensibility of method (58) to α.
10
STABILIZED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM
27
6.2. The lid-driven cavity problem. For this case we use the same domain as in previous section, we set f = 0, and the boundary conditions u = 0 on [{0} × (0, 1)] ∪ [(0, 1) × {0}] ∪
[{1} × (0, 1)] and u = (1, 0)t on (0, 1) × {1}. In Figure 10 we depict the pressure isovalues for both P1 /P0 and P1 /P1 approximations (using a mesh of around 1000 elements), showing, in both cases, the absence of oscillations.
Figure 10. Pressure isovalues for P1 /P0 (left) and P1 /P1 (right) approximations.
7. Concluding remarks In this paper we have analyzed and tested new stabilized finite element methods for the Stokes problem. These new methods arise from multiscale enrichment of the trial space for the velocity coupled with a Petrov-Galerkin strategy. This Petrov-Galerkin strategy makes possible to perform statical condensation both at the element level and at the inter element boundary level, making the method take the form of a classical stabilized finite element method, containing jump terms on the interior edges of the triangulation, and with the corresponding stabilization parameter known exactly. Optimal order error estimates were derived using the natural norms, results that were confirmed by the numerical experiments. Our belief is that our general methodology may be applied to other mixed problems, namely the Darcy and Brinkman flow problems, and to the advection-diffusion equation. This will be the subject of future works.
28
R. ARAYA, G.R. BARRENECHEA, AND F. VALENTIN
Appendix A. Proof of Theorem 7 The consistency of the method is immediate from the fact that Jν∂n uK = 0 a.e. on ∂K. To prove the well-posedeness of (39), we prove that B1 is an elliptic bilinear form. Let (v 1 , q1 ) ∈ Vh × Q1h , then from Lemma 9 we have that X 1 (BK (Jν∂n v 1 K), ∇q1 )K . B1 ((v 1 , q1 ), (v 1 , q1 )) = |||v 1 |||2h + kq1 k2h − ν K∈T h
Now, in order to treat last term above, let us denote by Z1 , Z2 , Z3 the sides of K, and let, for i = 1, 2, 3, bZKi be the solution of − ∆bZKi = 0 in K , bZKi = gi on each Z ⊂ ∂K , where gi = 0 if Zi ⊆ ∂Ω, and gi is the solution of −∂ss gi =
1 h Zi
in Zi , gi = 0 on ∂K − Zi , h
otherwise. First, we remark that from the maximum principle, we have that 0 ≤ b ZKi ≤ 8Zi P 3 Zi in K. On the other hand, it is easy to see that BK (Jν∂n v 1 K) = i=1 bK Jν∂n v 1 K|Zi , and
then, using that |K| ≤
h2K 2
2
and the inequality ab ≤ γ −1 a4 + γb2 (γ > 0) (denoting k· kR2 the
Euclidean norm on R2 ) we arrive at
3 X X X 1 1 Zi (BK (Jν∂n v 1 K), ∇q1 )K = (bK Jν∂n v 1 K|Zi , ∇q1 )K ν ν K∈T i=1 K∈T h
h
=
X X (bZ , 1)K K Jν∂n v 1 K|Z · ∇q1 |K ν Z∈E K⊂ω Z
h
X X hZ |K| ≤ kJν∂n v 1 K|Z kR2 k∇q1 |K kR2 8ν Z∈E K⊂ω Z
h
≤ γ −1 ≤ γ −1
(63) Hence, choosing γ =
14 16
X X |K|k∇q1 k20,K X X |Z| kJν∂n v 1 Kk20,Z +γ 32ν 8ν Z∈E K⊂ω Z∈E K⊂ω Z
h
X
16ν
Z∈Eh
Z
h
hZ kJν∂n v 1 Kk20,Z
+γ
X
K∈Th
h2K k∇q1 k20,K 8ν
.
< 1 we arrive at
B1 ((v 1 , q1 ), (v 1 , q1 )) ≥ |||v 1 |||2h + kq1 k2h − ≥
X hZ X h2 K kJν∂n v 1 Kk20,Z − γ |q1 |21,K 14ν 8ν Z∈E K∈T
C∗ (|||v 1 |||2h
h
+
h
kq1 k2h ) ,
STABILIZED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM
29
where C∗ is a positive constant not depending on h or ν. Now, for the error estimate, applying the coercivity result and the consistency of the method we arrive at ˜ 1 |||2h + kp1 − p˜1 k2h ) ≤ B1 ((u1 − u ˜ 1 , p1 − p˜1 ), (u1 − u ˜ 1 , p1 − p˜1 )) C∗ (|||u1 − u ˜ 1 , p1 − p˜1 )) = B1 ((u1 − u, p1 − p), (u1 − u X = − (BK (Jν∂n (u1 − u)K), ∇(p1 − p˜1 ))K .
(64)
K∈Th
Finally, proceeding as in (63) it is not difficult to see that X
K∈Th
(BK (Jν∂n (u1 − u)K), ∇(p1 − p˜1 ))K ≤ C ≤ C (|||u − u1 |||2h + kp − p1 k2h ) +
X
Z∈Eh
τZ kJν∂n (u1 − u)Kk20,Z +
C∗ X τK |p1 − p˜1 |21,K 2 K∈T h
C∗ ˜ 1 |||2h + kp1 − p˜1 k2h ) , (|||u1 − u 2
and hence, there exists C > 0, independent of h and ν, such that ˜ 1 k2h + kp1 − p˜1 k2h ≤ C (ku − u1 k2h + kp − p1 k2h ) , ku1 − u and the result follows by triangular inequality. Remark. We have proved that the error of method (39) is bounded by the error of method (44). The same analysis of Theorems 12 and 13 may be carried out to prove error estimates ˜ 1 k0,Ω . on kp − p˜1 k0,Ω and ku − u Appendix B. The error if f is not piecewise constant As we claimed before, we have just supposed that f is piecewise constant in order to derive (44), but this assumption does not affect the convergence of the method, and hence (44) may be implemented as it is presented for a general function f ∈ L2 (Ω)2 . Now, if we
do not suppose that f is piecewise constant in the derivation, then method (44) becomes: Find (uh , ph ) ∈ Vh × Q1h such that (65)
B((uh , ph ), (v 1 , q1 )) = Fh (v 1 , q1 ) ,
for all (v 1 , q1 ) ∈ Vh × Q1h , where B is defined in (41) and Fh is given by (66)
Fh (v 1 , q1 ) := (f , v 1 )Ω +
X 1 (MK (f ), ∇q1 )K . ν K∈T h
Clearly, (65) has a unique solution (uh , ph ) ∈ Vh × Q1h . Moreover, following result holds.
30
R. ARAYA, G.R. BARRENECHEA, AND F. VALENTIN
Theorem 15. Let us suppose that f ∈ H 1 (Ω)2 . Then, under the hypothesis of Theorems 11,12 and 13, the following error estimate holds
(67) (68) (69)
√ 1 1 |||u − uh |||h + kp − ph kh ≤ Ch ( ν |u|2,Ω + √ |p|1,Ω + √ kf k1,Ω ) , ν ν √ 1 1 kp − ph k0,Ω ≤ Ch ( ν |u|2,Ω + √ |p|1,Ω + √ kf k1,Ω ) , ν ν √ 1 1 1 ku − uh k0,Ω ≤ Ch2 max{1, √ } ( ν |u|2,Ω + √ |p|1,Ω + √ kf k1,Ω ) . ν ν ν
Proof. Let (u1 , p1 ) be the solution of (44). First, applying [30], Lemma 5.3.1, we see that F(v 1 , q1 ) − Fh (v 1 , q1 ) kv 1 kh + kq1 kh (v 1 ,q1 )∈Vh ×Q1h −{θ} P 1 K∈Th (τK f − ν MK (f ), ∇q1 )K = sup kv 1 kh + kq1 kh (v 1 ,q1 )∈Vh ×Q1h −{θ} P 1 K∈Th kτK f − ν MK (f )k0,K |q1 |1,K ≤ sup . kv 1 kh + kq1 kh (v 1 ,q1 )∈Vh ×Q1h −{θ}
ku1 − uh kh + kp1 − ph kh ≤
(70)
sup
Now, let be f h the piecewise constant function given by Z 1 f. f h = |K| K K
This function, which is the (local) projection on the space of piecewise constant functions,
satisfies (cf. [17]) kf − f h k0,K ≤ C hK |f |1,K . Then, applying triangular inequality we arrive at
(71) 1 1 kτK f − MK (f )k0,K ≤ kτK (f − f h )k0,K + kτK f h − MK (f h )k0,K + kMK (f h − f )k0,K . ν ν First term is easily bounded using the approximation properties of f h and the definition of τK . Next, since MK (f h ) = bpK f h in each K ∈ Th , second term is bounded in the following way
kbpK k0,K 1 kf h kR2 kτK f h − MK (f h )k0,K ≤ |τK |kf h k0,K + ν ν CK |bpK |1,K ≤ |τK |kf h k0,K + kf h k0,K , 1 ν |K| 2 where CK > 0 is the constant such that kvk0,K ≤ CK |v|1,K , for all v ∈ H01 (K). Furthermore,
looking carefully at the behaviour of the Poincar´e constant CK we can see (cf. [30],Thm.
STABILIZED FINITE ELEMENT METHODS FOR THE STOKES PROBLEM
31
1.2.5) that CK ≤ hK . On the other hand, from the definition of bpK we have |bpK |21,K =
(bpK , 1)K , and then, applying (48) we arrive at p hK (bpK , 1)K h2K CK |bpK |1,K (72) kf h k0,K . kf k ≤ kf k ≤ C 1 1 h 0,K h 0,K ν ν |K| 2 ν |K| 2
To bound third term in (71) we remark that function e := MK (f − f h ) satisfies −∆e =
f − f h in K, e = 0 on ∂K, and hence (73)
2 kek0,K ≤ CK kf − f h k0,K ≤ h2K kf − f h k0,K ≤ C h3K |f |1,K .
Hence, applying (70)-(73) (and supposing h ≤ 1), we arrive at P h2K h K∈Th ν kf k1,K |q1 |1,K ku1 − uh kh + kp1 − ph kh ≤ C sup ≤ C √ kf k1,Ω , kv 1 kh + kq1 kh ν (v 1 ,q1 )∈Vh ×Q1h −{θ} and hence (67) follows by triangular inequality and Theorem 11. Estimates (68) and (69) are proved proceeding as in Theorems 12 and 13, and using (67). Acknowledgments.
A part of this work was done during the stay of F. Valentin at the Departamento de
Ingenier´ıa Matem´atica of Universidad de Concepci´on and the stay of R. Araya and G. Barrenechea at LNCC, Petr´opolis, Brazil, in the framework of the joint Chile(CONICYT)-Brazil(CNPq) project No. 2003-4-173 (Chile)-470180/01-3 (Brazil). Rodolfo Araya is partially supported by FONDECYT Project No. 1040595. Gabriel Barrenechea is partially supported by CONICYT-Chile through FONDECYT Project No. 1030674 and FONDAP Program on Applied Mathematics.
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[email protected] ´ tica, Universidad de Concepcio ´ n, Casilla 160-C, ConDepartamento de Ingenier´ıa Matema ´ cepcion, Chile E-mail address:
[email protected] ´ tica Aplicada, Laborato ´ rio Nacional de Computac ˜ o Cient´ıfica, Departamento de Matema ¸a ´ lio Vargas, 333, 25651-070 Petro ´ polis - RJ, Brazil Av. Getu E-mail address:
[email protected]
