Appl. Math. Mech. -Engl. Ed., 32(10), 1269–1286 (2011) DOI 10.1007/s10483-011-1499-6 c Shanghai University and Springer-Verlag Berlin Heidelberg 2011
Applied Mathematics and Mechanics (English Edition)
Adaptive mixed least squares Galerkin/Petrov finite element method for stationary conduction convection problems∗ Yun-zhang ZHANG ()1,2 ,
Yan-ren HOU ()1 ,
Hong-bo WEI ()1
(1. School of Science, Xi’an Jiaotong University, Xi’an 710049, P. R. China; 2. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471003, Henan Province, P. R. China)
Abstract An adaptive mixed least squares Galerkin/Petrov finite element method (FEM) is developed for stationary conduction convection problems. The mixed least squares Galerkin/Petrov FEM is consistent and stable for any combination of discrete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition. Using the general theory of Verf¨ urth, the posteriori error estimates of the residual type are derived. Finally, numerical tests are presented to illustrate the effectiveness of the method. Key words conduction convection problem, posteriori error analysis, mixed finite element, adaptive finite element, least squares Galerkin/Petrov method Chinese Library Classification O175, O24 2010 Mathematics Subject Classification 65N15, 65N30, 65N50, 65Z05, 76M10
1
Introduction
Let Ω ⊂ R2 be a bounded and connected polygonal domain with a Lipschitz continuous boundary Γ = ∂Ω. Consider the stationary conduction-convection problems, whose coupled equations governing viscous incompressible flow and heat transfer for the incompressible fluid are Boussinesq approximations to the stationary Navier-Stokes equations. Find u = (u1 , u2 ), p, and T such that ⎧ − νΔu + (u · ∇)u + ∇p = λjT, x ∈ Ω, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ div u = 0, x ∈ Ω, (1) ⎪ − ΔT + λu · ∇T = 0, x ∈ Ω, ⎪ ⎪ ⎪ ⎪ ⎩ u = 0, T = T0 , x ∈ Γ, where u represents the velocity vector, p is the pressure, T is the temperature, λ > 0 is the Groshoff number, j = (0, 1) is the unit vector, and ν > 0 is the viscosity. When solving the stationary conduction convection problems by mixed finite element methods (FEMs), it is an important convergence stability condition that the Babuˇska-Brezzi inequality (LBB condition) holds for the combination of the velocity and pressure finite element ∗ Received Feb. 21, 2011 / Revised Jun. 10, 2011 Project supported by the National Natural Science Foundation of China (Nos. 10871156 and 11171269) and the Fund of Xi’an Jiaotong University (No. 2009xjtujc30) Corresponding author Yan-ren HOU, Professor, Ph. D., E-mail:
[email protected]
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subspaces. Numerical experiments show that the violation of the LBB condition often leads to unphysical pressure oscillations. However, some kinds of mixed finite elements, which violate the LBB condition, are very attractive and usable on many occasions. In particular, the equalorder mixed finite elements are of practical importance in scientific computation because it is computationally convenient in a parallel processing and multi-grid context. Therefore, a lot of works focus on the stabilization of the lowest equal-order pairs. For example, the so-called least squares Galerkin/Petrov FEM[1–3] was developed in an attempt to circumvent this constraint, which was motivated by the the streamline upwind Petrov-Galerkin (SUPG) methods[4–5] . The method is consistent and stable for any combination of velocity and pressure finite element subspaces (without requiring the Babuˇska-Brezzi stability condition). Luo and Lu[6] , Sun et al.[7] , and Luo et al.[8] applied the least squares Galerkin/Petrov FEM to conduction convection problems, and obtained optimal order error estimates. There are many works devoting to the development of efficient numerical schemes for these equations, e.g., [9–15]. Adaptive methods often lead to efficient discretization to the problems with solutions, which are singular or have large variations in small scales. The posteriori error estimators are key ingredients in the design of adaptive methods[16–17] . There are many literatures on deriving such estimators for the finite element approximation of linear and nonlinear variational problems and for standard Galerkin and mixed finite element formulations, e.g., [15, 18–28]. Verf¨ urth[23, 25] have developed a general theory to derive the posteriori error estimates for nonlinear equations, which has been successefully applied in [15, 22, 26–27]. In this paper, we use the general theory of Verf¨ urth[23, 25] to derive the posteriori error estimates for the least squares Galerkin/Petrov FEM of stationary conduction convection problems. We anticipate that the adaptivity based on the effective error estimators can make the numerical simulation more efficient. This is indeed confirmed by the present work. The residual type posteriori error estimates are derived. Finally, we present three numerical examples to illustrate the efficiency of the method. The first two numerical examples are problems with known solutions, and the third numerical one is a benchmark problem. As the mixed least squares Galerkin/Petrov FEM does not need to satisfy the discrete LBB condition between the velocity subspace and the pressure subspace, we adopt two types of finite element spaces to implement these numerical examples. One is the pairs of P1 -P1 -P1 elements for (u, p, T ) spaces, and the other is the pairs of P2 -P1 -P2 elements for (u, p, T ) spaces. Numerical results of using these two pairs of finite element spaces are in accordance with the theoretical analysis. The numerical results reveal that the great potential of using the stabilized adaptive FEM can significantly reduce the computational cost of the conduction convection problems. Detailed descriptions, analysis, and numerical examples of our adaptive FEM approach are presented in the rest of the paper as follows. In Section 2, some function spaces and preliminary results are introduced, and the mixed least squares Galerkin/Petrov FEM and the priori error estimates for the stationary conduction convection problems are presented. In Section 3, by the general theory of Verf¨ urth[23, 25] , the posteriori error estimates are derived for the problem. In Section 4, some numerical tests are presented to illustrate the efficiency of the method. Finally, a simple summary is presented in Section 5.
2
Functional setting and finite element approximation tools For the mathematical setting of the problem (1), we introduce the following Hilbert spaces: (i) velocity space M = H 1 (Ω)2 = {v ∈ H 1 (Ω)2 : v = 0
on Γ},
(ii) pressure space Q = L20 (Ω) = {p ∈ L2 (Ω), (p, 1)Ω = 0},
Adaptive mixed least squares Galerkin/Petrov finite element method
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(iii) temperature space W = H 1 (Ω),
W0 = H01 (Ω).
The space L2 (Ω)d (d = 1, 2, 2 × 2) is equipped with the usual L2 -scalar product (·, ·) and L -norm · L2 or · 0 . The spaces W0 and M are equipped with their usual scalar product ((u, v)) and equivalent norm 2
uH01 = ∇uL2 = ∇u0 . For the problem (1), the following assumptions are recalled (see [12, 29–32]). (A1 ) There exists a constant C0 , which only depends on Ω and satisfies (i) u0 C0 ∇u0 , u0,4 C0 ∇u0 , ∀u ∈ H01 (Ω)2 (or H01 (Ω)), (ii) u0,4 C0 u1 , ∀u ∈ H 1 (Ω)2 (or H01 (Ω)), 1
1
1
(iii) u0,4 2 2 ∇u02 u02 , ∀u ∈ H01 (Ω)2 (or H01 (Ω)). (A2 ) Assume ∂Ω ∈ C k,α (k 0, α > 0). Then, for T0 ∈ C k,α (∂Ω), there exists an extension T0 ∈ C0k,α (R2 ) such that T0 k,q ε,
k 0,
1 q ∞,
where ε is an arbitrary small positive constant number. (A3 ) For all u ∈ M, v, w ∈ M (or T, ϕ ∈ H01 (Ω)), there hold ⎧ ⎨ |b(u, v, w)| N ∇u0 ∇v0 ∇w0 , ⎩ |¯b(u, T, ϕ)| N ∇u ∇T ∇ϕ , 0 0 0 where
⎧ ⎪ ⎪ ⎪ b(u, v, w) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¯b(u, T, ϕ) = ⎪ ⎨
1 2
1 2
2
2
Ω i,k=1
∂vk ui wk dx − ∂xi
∂T ui ϕdx − ∂xi Ω i=1
2
Ω i,k=1
2 Ω i=1
ui
ui
∂wk vk dx , ∂xi
∀u, v, w ∈ M,
∂ϕ T dx , ∀u ∈ M, T, ϕ ∈ W0 , ∂xi
⎪ ⎪ ⎪ |b(u, v, w)| ⎪ ⎪ , N = sup ⎪ ⎪ ⎪ u,v,w∈M ∇u0 ∇v0 ∇w0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ |¯b(u, T, ϕ)| ⎪ ⎪ ⎪ sup . ⎩N = u∈M,(T,ϕ)∈W ×W ∇u0 ∇T 0 ∇ϕ0 Under the assumptions ((A1 ), (A2 ), and (A3 )) and other conditions, Luo and Lu[6] and Luo[12] established the problem (1) with a unique solution (u, p, T ) ∈ M × Q × W satisfying ∇u0 A, where
∇T 0 B,
⎧ ⎨ A = 2ν −1 λ(3C0 + 1)T0 1 , ⎩ B = 2∇T + C −2 (3C + 1)T . 0 0 0 0 1 0
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Yun-zhang ZHANG, Yan-ren HOU, and Hong-bo WEI
Let Ωh be uniformly regular triangulations of the domain Ω. Define the mesh parameter h = max {hK ; hK = diam(K)}. K∈Ωh
Assume that Ωh satisfies the following conditions (see [32]): (I) Any two triangles in Ωh either are disjoint or share a complete smooth submanifold of their boundaries; (II) The ratio hK /K < is bounded from the above independently of K ∈ Ωh and h > 0, where K and hK denote the diameter of the largest ball inscribed into K and the diameter of K, respectively. We note that the condition (II) allows the use of locally refined meshes, and the ratio hK /K , for all K ∈ Ωh and all edges E of K, is bounded from the above and the below by constants which are independent of h, K, and E. We introduce the finite element subspaces Mh ⊂ M , Qh ⊂ Q, Wh ⊂ W , and W0,h ⊂ W0 as follows: ⎧ Mh = {v ∈ M ∩ C 0 (Ω)2 ; v|K ∈ Pl (K)2 , ∀K ∈ Ωh }, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Qh = {q ∈ Q ∩ C 0 (Ω); q|K ∈ Pk (K), ∀K ∈ Ωh }, ⎪ ⎪ Wh = {t ∈ W ∩ C 0 (Ω); t|K ∈ Pm (K), ∀K ∈ Ωh }, ⎪ ⎪ ⎪ ⎩ W0h = Wh ∩ H01 , where Pr (K) (r = l, k, or m) denote the spaces of the polynomials with the degree r on K ∈ Ωh . Here, r is a positive integer. The mixed least squares Galerkin/Petrov FEM for the conduction convection problems can be represented as follows: Find (uh , ph , Th ) ∈ Mh × Qh × Wh such that
⎧ ⎪ ⎪ Th |Γ = T0 , ⎪ ⎪ ⎪ ⎪ νa(uh , v) + b(uh , uh , v) − (ph , div v) + (q, div uh ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ δK (−νΔuh + uh · ∇uh + ∇ph , −νΔv + uh · ∇v + ∇q)K ⎨+ K∈Ωh
⎪ ⎪ ⎪ ⎪ ⎪ = λ(jTh , v) + λ δK (jTh , −νΔv + uh · ∇v + ∇q)K , ⎪ ⎪ ⎪ ⎪ K∈Ω h ⎪ ⎪ ⎪ ⎪ ⎩ a ¯(Th , S) + λ¯b(uh , Th , S) = 0
(2)
for all (v, q, S) ∈ Mh × Qh × W0h , where ⎧ δ = αh2K , α > 0, ⎪ ⎪ ⎨ K a(u, v) = (∇u, ∇v), ⎪ ⎪ ⎩ a ¯(T, S) = (∇T, ∇S). Note that the scheme (2) is a standard Galerkin finite element scheme if the parameter δK = 0.
Adaptive mixed least squares Galerkin/Petrov finite element method
Theorem 2.1[6, 12]
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Under the assumptions (A1 ), (A2 ), and (A3 ), if 2
AN + C02 λ2 (1 + 16B 2 N ) < 1 and (u, p, T ) ∈ (W01,∞ (Ω) ∩ H l+1 (Ω))2 × H k+1 (Ω) × H m+1 (Ω) is a solution to (1), then there exists a constant h∗ > 0 for all solutions (uh , ph , Th ) to (2) such that for all h h∗ , 1
δ 2 (−νΔ(u − uh ) + uh · ∇(u − uh ) + ∇(p − ph ))0,h 1
+ (νu − uh 21 + ∇(T − Th )20 ) 2 C(hm + hl + hk+1 ),
(3)
where δ|K = δK , ∀K ∈ Ωh , the constant C is independent of h, and the discrete norm
12 · 20,K . · 0,h = K
In the following, (1) and (2) will be written in abstract forms. Let us define ⎧ X = M × Q × W, ⎪ ⎪ ⎪ ⎨ Y = M × Q × W0 , ⎪ ⎪ ⎪ 1 ⎩ · X = · Y = ( · 21,2 + · 20,2 + · 21,2 ) 2 . Define the operator F : X → L(Y, R) as
F (u, p, T ), (v, q, S) = (ν∇u∇v + (u · ∇)uv − p∇ · v + q∇ · u − λjT · v) Ω
+ Ω
(∇T ∇S + λu · ∇T S).
(4)
Now, the problem (1) becomes to solve F (u, p, T ) = 0
(5)
in an abstract form together with boundary conditions. Basing on the definition of (Mh , Qh , Wh , W0h ), we define Xh = Mh × Qh × Wh , Yh = Mh × Qh × W0h and the discrete version of F as follows:
Fh (uh , ph , Th ), (vh , qh , Sh ) = F (uh , ph , Th ), (vh , qh , Sh ) δK (−νΔuh + uh · ∇uh + K∈Ωh
+ ∇ph − λjTh − νΔvh + uh · ∇vh + ∇qh )K .
(6)
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Yun-zhang ZHANG, Yan-ren HOU, and Hong-bo WEI
Then, the discrete problem (2) becomes to solve Fh (uh , ph , Th ) = 0.
(7)
Though the priori error analysis can offer theoretical assurance on the convergence of the FEMs as the mesh size gets smaller and smaller, to implement an adaptive strategy for the finite element approximations, we need posteriori error estimators. We choose to work with the residual-type estimators, which are derived in the next section.
3
Posteriori error estimates
In this section, we will derive a residual-based posteriori error estimator for the problem (2). We first present an abstract theory of the posteriori error estimates of Verf¨ urth[23, 25] . Let u ˜∗ = (u∗ , p∗ , T∗ ) be a solution to a nonlinear operator equation F (˜ u∗ ) = 0. We call u ˜∗ a regular solution if the Fr´echet derivative DF (˜ u∗ ) is well defined and is a linear homeomorphism, i.e., DF (˜ u∗ ) and its inverse are bijective and continuous linear operators. From [23,25], we have the following abstract results. Theorem 3.1[23, 25] Let ⎧ ⎨ Y ∗ = L(Y, R), ⎩ F ∈ C 1 (X, Y ∗ ). Let u ˜∗ be a regular solution to F (u∗ ) = 0 with ⎧ u∗ )L(Y,Y ∗ ) , ⎨ Z = DF (˜ ⎩ Z = DF (˜ u∗ )−1 L(Y ∗ ,Y ) . Assume, in addition, that DF is Lipschitz continuous at u ˜∗ with a constant γ > 0, i.e., there is a constant R0 > 0 such that γ=
sup ˜ u−˜ u∗ X
