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A posterior error estimator and lower bound of a nonconforming finite element method
Journal of Computational and Applied Mathematics 265 (2014) 243–254
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A posterior error estimator and lower bound of a nonconforming finite element method✩ Qun Lin a,∗ , Fusheng Luo a , Hehu Xie b a
LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China
b
LSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China
article
info
Article history: Received 30 September 2012 Received in revised form 9 June 2013 MSC: 65N30 65N25 65L15 65B99
abstract In this paper, we present an a posteriori error estimator and the lower bound for a nonconforming finite element approximation, i.e. the extended Crouzeix–Raviart element, of the Laplace eigenvalue problem. Under the guideline of the analysis to the Laplace source problem, we first give out an error indicator and prove it as the global upper and local lower bounds of the approximation error. We also give the lower-bound analysis for this type of nonconforming element on the adaptive meshes. Some numerical experiments are presented to verify our theoretical results. © 2013 Elsevier B.V. All rights reserved.
Keywords: A posteriori error estimate Lower bound Adaptive finite element method Nonconforming Laplace eigenvalue problem
1. Introduction One vital component of the adaptive finite element method is to find a reliable and efficient a posteriori error estimate as the indicator to help us mark out the elements to be refined or coarsened (see, e.g., [1–6]). The first object of this paper is to derive an a posteriori error estimate of a nonconforming finite element for the Laplace eigenvalue problem. There have been many works concerning the a posteriori error estimates of the eigenvalue problems by the nonconforming finite element methods (see, e.g., [7–9] and the references cited therein). Recently, researchers have found an interesting phenomenon that some nonconforming elements can provide lower bounds for many kinds of eigenvalue problems such as Laplace eigenvalue problem, Stokes eigenvalue problem, Steklov eigenvalue problem, biharmonic eigenvalue problem (see, e.g., [10–14,9,15–18]). Especially, the extended Crouzeix–Raviart (ECR) element, with an additional degree of freedom, requires less of the regularity of the eigenfunction, and provides lowerbound properties, see the theorems and numerical results in [19,13,14,20]. So the second object in this paper is to investigate the lower-bound property of the ECR element for the Laplace eigenvalue problem on the adaptive meshes. In this paper, we present a residual based a posteriori error estimate for the nonconforming finite element approximation of the Laplace eigenvalue problem and give the corresponding analysis of reliability and efficiency by following the guideline
✩ This work was supported in part by National Science Foundations of China NSFC (91330202, 11001259, 11371026, 11031006, 2011CB309703), Croucher Foundation of Hong Kong Baptist University and the National Center for Mathematics and Interdisciplinary Science, CAS and the President Foundation of AMSS-CAS. ∗ Corresponding author. Tel.: +86 10 62624806. E-mail addresses: [email protected] (Q. Lin), [email protected] (F. Luo), [email protected] (H. Xie).
0377-0427/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cam.2013.09.030
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of [7,21]. The main tools used in the analysis are the Helmholtz decomposition, the local interpolation and the approximation properties of the finite element space. Then we give the analysis of the lower bound of the ECR element for the Laplace eigenvalue problem on the adaptive meshes. An outline of the paper goes as follows. In Section 2, we introduce the Laplace eigenvalue problem and the corresponding nonconforming discretization. Section 3 is devoted to presenting the residual based error indicators and giving the analysis of their reliability and efficiency. Then the adaptive finite element method for the eigenvalue problem is presented in Section 4. The analysis of the lower bound for the ECR element is given in Section 5. Section 6 is devoted to presenting some numerical results to validate our theoretical analysis. Finally, some concluding remarks are provided in Section 7. 2. Model problem and its discretization In this paper, we are concerned with the following Laplace eigenvalue problem
−1u = λu, u = 0, u2 dΩ = 1,
in Ω , on ∂ Ω ,
(2.1)
Ω
where Ω is a bounded Lipschitz polygonal domain in R2 . We set V = H01 (Ω ). Then the variational form of (2.1) is: Find (λ, u) ∈ R × V such that Ω u2 dΩ = 1 and a(u, v) = λ(u, v),
∀v ∈ V ,
(2.2)
where a(u, v) =
Ω
∇ u · ∇v dΩ ,
(u, v) =
Ω
uv d Ω .
From [22], we know (2.2) has an eigenvalue series {λj }: 0 < λ1 ≤ λ2 ≤ · · · ≤ λk ≤ · · · ,
lim λk = ∞,
k→∞
and the associated eigenfunctions u1 , u2 , . . . , uk , . . . , with (ui , uj ) = δij . Throughout the paper, the standard notation of the Sobolev spaces and the corresponding norms, semi-norms, and inner products as those in [3,23] are used. For a subdomain D of Ω , the Sobolev space H m (D) := W m,2 (D) with norm ∥ · ∥m,D and semi-norm | · |m,D is used. Then, we introduce a face-to-face triangular partition Th := {K } on the computational domain Ω such that
¯ = Ω
K.
K ∈Th
Here h = maxK ∈Th hK and hK = diamK denote the global and local mesh size, respectively. A family of partitions Th is said to be regular if it satisfies (see [3,23])
∃σ > 0 such that ρK /hK > σ ∀K ∈ Th , where ρK is the diameter of the maximum circle contained in K . Denote the set of all interior edges of Th as Eh , the set of the edges on the boundary as E∂ Ω and E := Eh ∪ E∂ Ω . In the context, C , C˜ represent constants independent of the mesh sizes h and hK and may vary at its different occurrences. According to the partition Th , we build the nonconforming finite element space Vh ̸⊂ V . Especially, we consider in this paper the nonconforming ECR element [19,14]. The ECR element can be defined as follows:
• ECR element: E P1 := span 1, x, y, x2 + y2 ,
Vh =
v ∈ L2 (Ω ), v|K ∈ E P1 , ∀ K ∈ T , ∀ℓ = ∂ K1 ∩ ∂ K2 ∈ Eh and
ℓ
ℓ
v|K1 dℓ =
ℓ
v|K2 dℓ,
v|K1 dℓ = 0, if ℓ ∈ E∂ Ω ,
which has the following properties due to its continuity
K ∈Th
∇(u − uh ) · ∇ × wh dK = K
ℓ∈Eh
ℓ
tℓ · ∇wh [u − uh ]dℓ = 0,
∀ u, uh ∈ V + V h .
(2.3)
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The interpolation operator corresponding to ECR element can be defined in the following way:
ℓ
(u − Πh u)ds = 0,
∀ℓ ∈ Eh ,
(u − Πh u)dK = 0,
(2.4)
∀K ∈ Th .
(2.5)
K
For the analysis of the nonconforming element, we introduce the following piecewise type norm and semi-norm
21
∥ u∥ j , h =
∥u∥2j,K
K ∈Th
and
21
|u|j,h =
2 j ,K
for u ∈
|u|
H (K ) j
Vh ,
K ∈Th
K ∈Th
where j = 0, 1, 2, . . . . Here we also define the lowest order linear conforming space
Wh := v ∈ V : v|K ∈ span{1, x, y}, ∀K ∈ Th . From these definitions, we emphasize the following two important properties: Property 2.1. The finite element space Vh satisfies: 1. For all vh ∈ Vh , we have ℓ [vh ]dℓ = 0 for ℓ ∈ Eh and ℓ vh dℓ = 0 for ℓ ∈ E∂ Ω , where [v] := v|K + − v|K − denotes the jump on the edge ℓ = ∂ K + ∩ ∂ K − . 2. Wh ⊂ Vh and Wh ⊂ V .
For the nonconforming space, we denote the elementwise operator by
(∇h v)|K := ∇(v|K ),
∀ K ∈ Th .
The nonconforming finite element method for (2.2) is defined as follows: Find (λh , uh ) ∈ R × Vh such that ∥uh ∥0,h = 1 and ah (uh , vh ) = λh (uh , vh ),
∀vh ∈ Vh ,
(2.6)
where ah (uh , vh ) =
K ∈Th
∇ uh · ∇vh dK .
K
It is also known from [22] that the discrete Laplace eigenvalue problem (2.6) has eigenvalue series 0 < λ1,h ≤ λ2,h ≤ · · · ≤ λk,h ≤ · · · ≤ λN ,h , and the corresponding eigenfunctions u1,h , u2,h , . . . , uk,h , . . . , uN ,h , with (ui,h , uj,h ) = δij , 1 ≤ i, j ≤ N, where N denotes the dimension of the finite element space Vh . At the end of this section, we review some results related to the a priori error estimates of the Laplace eigenvalue problem (see, e.g., [22,24]). Denote the error of the eigenfunction approximation uh as below e h = u − uh . Thus we have eh ∈ V + Vh . Lemma 2.1. For each finite element approximation (λh , uh ), there exists an eigenpair (λ, u) of (2.2) such that the following error estimates hold
∥eh ∥1,h ≤ Chγ ∥u∥1+γ ,
(2.7)
2γ
∥eh ∥0,h ≤ Ch ∥u∥1+γ , 2γ
|λ − λh | ≤ Ch ∥u∥
2 1+γ
(2.8)
,
(2.9)
where γ ∈ (0, 1] depends on the shape of the domain Ω (γ = 1 when Ω is convex) (see [25]), the const C depends only on λ.
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3. A posterior error indicator This section is devoted to deriving a reliable and efficient indicator for the nonconforming finite element approximation of the Laplace eigenvalue problem. There exists the Helmholtz decomposition for the space V + Vh (see, e.g., [26,27]). Lemma 3.1 ([26]). For any w ∈ V + Vh , there exist r ∈ V and s ∈ H 1 (Ω ) such that
∇h w = ∇ r + ∇ × s.
(3.1)
The boundedness property holds:
∥r ∥1 + ∥s∥1 ≤ C ∥∇h w∥0 .
(3.2)
To get the a posteriori error estimate, we need to use the Clément interpolation which has the following approximation properties. Property 3.1 ([28]). The Clément interpolation operator Ih : V → Wh has the following properties
∥u − Ih u∥0,K ≤ ChK |u|1,ωK ,
∀K ∈ Th ,
1/2
∥u − Ih u∥0,ℓ ≤ Chℓ |u|1,ωℓ , |u − Ih u|1,K ≤ C |u|1,ωK ,
(3.3)
∀ℓ ∈ Eh ,
(3.4)
∀K ∈ Th ,
(3.5)
where ωK denotes the set of elements that share a common vertex with K and ωℓ denotes the set of elements that contain ℓ as one of its edges. The a posteriori error indicator is defined by
12
ηh (λh , uh ) :=
η
,
2 K
(3.6)
K ∈Th
where the local indicator ηK is defined as
ηK2 := h2K ∥λh uh + 1uh ∥20,K +
1 2 ℓ∈∂ K
hℓ ∥Jℓ,t ∥20,ℓ
(3.7)
with
Jℓ,t =
[tℓ · ∇ uh ]|ℓ , 2tℓ · ∇ uh ,
if ℓ ∈ Eh , if ℓ ∈ E∂ Ω .
(3.8)
3.1. Reliability Denote the normal jump by
Jℓ,n =
[nℓ · ∇ uh ]|ℓ , 0,
if ℓ ∈ Eh , if ℓ ∈ E∂ Ω .
(3.9)
The following lemma states the relationship of the normal jump term with the residual on the mesh. Lemma 3.2. Let ℓ = K1 ∩ K2 and set the normal derivative direction from K1 to K2 . The following inequality holds,
h2ℓ ∥Jℓ,n ∥20,ℓ ≤ C h2K1 ∥λh uh + 1uh ∥20,K1 + h2K2 ∥λh uh + 1uh ∥20,K2 .
(3.10)
Proof. Take φℓ as the basis function that equals to 1 at the middle point of ℓ and vanishes at the midpoints of all other edges. Substituting vh with functions v = φℓ and integrating by parts, we get
ωℓ
∇h uh · ∇h v dK + ∆h uh v dK ωℓ ωℓ = nℓ · ∇ uh v dℓ − nℓ · ∇ uh v dℓ = − Jℓ,n v dℓ.
(λh uh + ∆h uh )v dK =
∂ K1
∂ K2
ℓ
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By the definition of v , there is a constant C such that hℓ ∥Jℓ,n ∥0,ℓ
≤ C Jℓ,n v dℓ. ℓ
Thus we arrive the desired result (3.10) with hℓ ∥Jℓ,n ∥0,ℓ ≤ C ∥λh uh + 1uh ∥0,K1 ∥φℓ ∥0,K1 + ∥λh uh + 1uh ∥0,K2 ∥φℓ ∥0,K2
≤ C (hK1 ∥λh uh + 1uh ∥0,K1 + hK2 ∥λh uh + 1uh ∥0,K2 ), and the proof is complete.
Then, we show that the error eh can be controlled by the indicator up to a higher order term. Theorem 3.1 (Reliability). For the error of the nonconforming finite element approximation (λh , uh ) of (λ, u), we have the following estimate
∥∇h eh ∥0,h ≤ C ηh (λh , uh ) + ∥λh uh − λu∥0,h ,
(3.11)
where the constant C depends only on the minimum angle of Th . Proof. Since eh = u − uh ∈ V + Vh , by the Helmholtz decomposition (3.1), there exist r ∈ V and s ∈ H 1 (Ω ) such that
∇h eh = ∇ r + ∇ × s, where r and s satisfy
∥r ∥1 + ∥s∥1 ≤ C ∥∇h eh ∥0 .
(3.12)
So we have
∥∇h eh ∥20,h =
∇ eh · (∇ r + ∇ × s)dK .
(3.13)
K
K ∈Th
Combining (3.13) with (2.2), (2.6) and (3.13) leads to
∥∇h eh ∥20 =
∇ eh · ∇ r + ∇ eh · (∇ × s) dK K
K ∈Th
=
∇ eh · ∇(r − Ih r )dK +
K
K ∈Th
∇ eh · (∇ × (s − Ih s))dK + K
(λu − λh uh )Ih rdK ,
(3.14)
K
where we use (2.3) and the facts that Ih r ∈ V , Ih r ∈ Vh and
∇ eh · ∇ × Ih sdK = 0. K
K ∈Th
The first two terms in (3.14) can be estimated as follows: 1. Integrating by parts and using (2.2), (2.6) leads to
∇ eh · ∇(r − Ih r )dK = K
(λu + 1uh )(r − Ih r )dK −
K
∂K
(∇ uh · n)(r − Ih r )dℓ.
2. Integrating by parts leads to
∇ eh · (∇ × (s − Ih s))dK = K
∂K
(∇ uh · t )(s − Ih s)dℓ −
∂K
(∇ u · t )(s − Ih s)dℓ.
It is from summing all with respect to elements and canceling the term on the edges of any element that 2 h eh 0,h
∥∇
∥
=
(λu + 1uh )(r − Ih r )dK + K
K ∈Th
− ∂K
(∇ uh · n)(r − Ih r )dℓ −
K ∈Th
(λu − λh uh )Ih rdK K
∂K
(∇ uh · t )(s − Ih s)dℓ
∂K
t · ∇ u(s − Ih s)dℓ due to its continuity
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≤C
∥λu − λh uh ∥0,K ∥r ∥0,K + ∥λh uh + 1uh ∥0,K hK |r |1,ωK
K ∈Th
−+
1
1/2
1/2
∥Jℓ,n ∥0,ωℓ hℓ |r |1,ωℓ + ∥Jℓ,t ∥0,ωℓ hℓ |s|1,ωℓ
2 ℓ∈∂ K
.
(3.15)
From (3.15), Cauchy–Schwarz inequality and the stability of the Helmholtz decomposition (3.2), we have
2 h eh 0,h
∥∇
∥
≤C
h2K
∥λh uh +
1uh 20,K
∥
K ∈Th
+
1 2 ℓ∈∂ K
Jℓ,n 20,ωℓ
hℓ ∥
∥
+∥
Jℓ,t 20,ωℓ
∥
1/2 + ∥λu − λ
2 h uh 0,h
∥
∥∇h eh ∥0,h .
(3.16)
Canceling the term ∥∇h eh ∥0,h on both sides and replacing the jump term in (3.16) by (3.10), we get the desired result (3.11). To show that ∥λh uh − λu∥0,h is a higher order term, we rewrite it as
∥λh uh − λu∥0,h ≤ |λh − λ|∥uh ∥0,h + λ∥u − uh ∥0,h . Then, from a priori error estimates (2.7)–(2.9), it is easy to see that both |λh − λ| and ∥u − uh ∥0,h are higher order terms compared with ∥∇h eh ∥0,h . 3.2. Efficiency We turn to prove the efficiency of the indicator via bubble functions. First we define the bubble functions bK on the element K and bℓ on the edge ℓ of K by barycentric functions λi,K : bK := λ1,K λ2,K λ3,K , and
bℓ :=
λ1,K1 λ2,K1 , λ1,K2 λ2,K2 ,
on K1 , on K2 ,
where K1 and K2 are the elements sharing the edge ℓ with nodes 1, 2. Then, we estimate the indicator term by term to show its efficiency. Lemma 3.3. There exists a constant C such that
hK ∥λh uh + 1uh ∥0,K ≤ C hK ∥λh uh − λu∥0,K + ∥∇ eh ∥0,K ,
∀ K ∈ Th .
(3.17)
Proof. Denote φK = C |K |(λh uh + 1uh )bK and choose C to make the following equality hold C |K |∥λh uh + 1uh ∥20,K =
= = ≤ ≤
(λh uh + 1uh )φK dK λh uh φK dK − ∇ uh · ∇φK dK K K (λh uh − λu)φK dK + ∇ eh · ∇φK dK K K C ∥λh uh − λu∥0,K ∥φK ∥0,K + ∥∇ eh ∥0,K ∥∇φK ∥0,K C h2K ∥λh uh − λu∥0,K + hK ∥∇ eh ∥0,K ∥λh uh + 1uh ∥0,K , K
where we use the fact ∥φK ∥0,K = C |K |∥λh uh + 1uh ∥0,K and ∥∇φK ∥0,K = C˜ hK ∥λh uh + 1uh ∥0,K for some constants C and C˜ . Canceling the term hK ∥λh uh + 1uh ∥0,K on both sides leads to the desired result (3.17) and the proof is complete. Lemma 3.4. There exists a constant C such that 1/2
hℓ ∥Jℓ,t ∥0,ℓ ≤ C ∥∇h eh ∥0,ωℓ .
(3.18)
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Proof. We integrate the jump term Jℓ,t with φ2,ℓ := Chℓ bℓ Jℓ,t and choose C to make the following equality hold Chℓ Jℓ,t 20,ℓ
∥
∥
= ℓ
[tℓ · ∇h uh ] · φ2,ℓ dℓ =
= ωℓ
ℓ
[tℓ · ∇h eh ]φ2,ℓ dℓ
(∇h eh ) · (∇h × φ2,ℓ )dK ≤ C ∥∇h eh ∥0,ωℓ ∥∇h φ2,ℓ ∥0,ωℓ ,
(3.19)
where we use [tℓ · ∇ u] = 0. There is some constant C such that 1/2
∥∇h φ2,ℓ ∥0,ωℓ = Chℓ ∥Jℓ,t ∥0,ℓ .
(3.20)
From (3.19) and (3.20), we obtain (3.18).
We are now in the position to state the efficiency of the estimator. Theorem 3.2 (Efficiency). The local error indicator defined by (3.6) has the following efficiency property
ηK ≤ C ∥∇h eh ∥0,ωK +
1/2 , h2K ∥λh uh − λu∥20,K
(3.21)
K ∈ωK
where ωK is defined in Property 3.1 and C is a constant. 4. Adaptive finite element method In this section, the adaptive finite element method algorithm is briefly reviewed. Starting from an initial mesh Th0 , by repeating the common cycle of four steps of the adaptive finite element method: SOLVE → ESTIMATE → MARK → REFINE, we form a nested conforming triangulation Thi . In the SOLVE step, we assume that this step can be finished in an ideal way which reaches a sufficient high accuracy and optimal time complexity as we want. For example, on mesh Thi , firstly, (λhi , uhi ) is solved from (2.6). In the ESTIMATE step, we compute out the error indicators by (3.6) and (3.7). Then, in the MARK step, the bulk marking strategy proposed by Dörfler [4] is adopted to define the marking set Mhi according to the local indicator (3.6), such that
K ∈Mh
ηK2 ≥ θ i
K ∈Th
ηK2 ,
for some θ ∈ (0, 1).
i
We mark out a set of triangles of the mesh Thi to be refined. In the REFINE step, some refined method is used to generate the new finer mesh Thi+1 and keep the new mesh conforming and shape regular as defined in [3,23]. There are many refined methods, i.e., the red and green refinement [1], longest refinement [29,30], and newest vertex bisection [31]. In this paper, we use the newest vertex bisection method. For more details, please refer to [2,32,6]. The adaptive algorithm goes as follows:
Adaptive Algorithm 1. 2. 3. 4. 5. 6.
Choose a parameter θ ∈ (0, 1), pick an initial mesh Th0 and set i = 0. Solve the problem (2.6) to obtain the eigenpair (λhi , uhi ). Compute out the local indicators ηK . Construct the set Mhi ⊂ Thi by MARK procedure with the parameter θ . Refine Thi by REFINE procedure to get a new conforming mesh Thi+1 . Let i := i + 1 and go to Step 2.
5. Lower bound of the eigenvalue problem In this section, the lower-bound analysis of the eigenvalue problem by the ECR element is given on the adaptive meshes. For the eigenvalue problem, we have the following basic expansion which was first proved in its full term in [18] by the similar argument in [10] (where the expansion has only the first three terms) and has been extensively used in [19,13,33,17].
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Lemma 5.1. Suppose (λ, u) is the eigenpair of the original problem (2.2), (λh , uh ) ∈ R × Vh is the eigenpair of the discrete problem (2.6). We have the following expansion
λ − λh = ∥u − uh ∥21,h − λh ∥vh − uh ∥20,h + λh (∥vh ∥20,h − ∥u∥20 ) + 2ah (u − vh , uh ),
∀vh ∈ Vh .
(5.1)
Lemma 5.2. For the ECR element interpolation, the following property holds
∥u − Πh u∥0,h ≤ Ch∥u − Πh u∥1,h , ah u − Πh u, vh = 0, ∀vh ∈ Vh .
(5.2) (5.3)
Lemma 5.3. On the regular type of mesh Th , we have the following inequality for the eigenfunction approximation uh by the ECR element
∥uh − Π0 uh ∥0,h ≤ C ∥∇h (u − uh )∥0,h + h∥λu − λh uh ∥0,h ,
(5.4)
where Π0 denotes the piecewise constant interpolation. Proof. It is easy to know the following estimate holds
∥uh − Π0 uh ∥20,h ≤ C
h2K ∥∇ uh ∥20,K .
(5.5)
K ∈Th
Let ω ¯ K denote the set of elements that share a common edge with K . In the finite dimensional space Vh (ω¯ K ) := {vh |ω¯ K , ∀vh ∈ Vh }, we investigate the relationship of the two norms:
K ∈ω ¯T
h2K ∥∇vh ∥20,K
and
K ∈ω ¯T
h2K ∥λh vh + 1vh ∥20,K +
1 2 ℓ∈∂ T
hℓ ∥[∇vh ]∥20,ℓ .
1 2 2 2 It is easily known that K ∈ω ¯ T hK ∥λh vh + 1vh ∥0,K + 2 ℓ∈∂ T hℓ ∥[∇vh ]∥0,ℓ = 0 leads to compactness and scaling argument shows there exists a constant C such that
K ∈ω ¯T
h2K ∥∇vh ∥20,K ≤ C
K ∈ω ¯T
h2K ∥λh vh + 1vh ∥20,K +
1 2 ℓ∈∂ T
K ∈ω ¯T
h2K ∥∇vh ∥20,K = 0. A
,
hℓ ∥[∇vh ]∥20,ℓ
∀vh ∈ Vh .
(5.6)
From (5.6) and Lemma 3.2, we have
K ∈ω ¯T
h2K ∥∇ uh ∥20,K ≤ C
K ∈ω ¯T
h2K ∥λh uh + 1uh ∥20,K +
1 2 ℓ∈∂ T
hℓ ∥Jℓ,t ∥20,ℓ
.
(5.7)
Combining (5.5), (5.7) and Theorem 3.2 leads to
∥uh − Π0 uh ∥20,h ≤ C
T ∈Th K ∈ω ¯T
h2K ∥∇ uh ∥20,K
≤C
T ∈Th
K ∈ω ¯T
h2K
∥λh uh +
1uh 20,K
∥
≤C
h2K
∥λh uh +
K ∈Th
1uh 20,K
∥
+
+
1 2 ℓ∈∂ T
1 2 ℓ∈∂ K
hℓ Jℓ,t 20,ℓ
∥
∥
hℓ Jℓ,t 20,ℓ
∥
∥
≤ C ∥∇h eh ∥20,h + h2 ∥λu − λh uh ∥20,h . This is the desired result (5.4) and we complete the proof.
Theorem 5.1. Let λ and λh be an eigenvalue and its corresponding numerical approximation by ECR element. Assume u ∈ H 1+γ (Ω ) with 0 < γ ≤ 1. When h is small enough, we have
λh < λ.
(5.8)
Proof. We choose vh = Πh u in Lemma 5.1. First, from the results in [13], we have
∥u − uh ∥0,h ≤ Chγ ∥u − uh ∥1,h .
(5.9)
Since (5.3), the following inequality holds
∥∇h (u − Πh u)∥0,h ≤ ∥∇h (u − uh )∥0,h .
(5.10)
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Fig. 1. The initial triangulation and the ones after 12 adaptive iterations for Example 1 by the ECR element.
Then for the second term in (5.1), from Lemma 5.2, (5.9) and (5.10), we have
∥uh − Πh u∥20,h ≤ 2∥uh − u∥20,h + 2∥u − Πh u∥20,h ≤ Ch2γ ∥u − uh ∥21,h .
(5.11)
From Lemma 5.3, (5.9) and the result in [24], we have
∥u − Π0 u∥0,h ≤ C ∥u − uh ∥0,h + ∥uh − Π0 uh ∥0,h ≤ C ∥∇h (u − uh )∥0,h + h∥λu − λh uh ∥0,h + ∥u − uh ∥0,h ≤ C ∥∇h (u − uh )∥0,h + hγ ∥u − uh ∥1,h .
(5.12)
For the third term in (5.1), from (5.2)–(5.3), (5.10) and (5.12), we have
∥vh ∥20,h − ∥u∥20 = (Πh u − u, Πh u + u) = 2(Πh u − u, u) + ∥u − Πh u∥20,h = 2(Πh u − u, u − Π0 u) + ∥u − Πh u∥20,h ≤ Ch∥u − Πh u∥1,h ∥u − Π0 u∥0,h + Ch2 ∥u − Πh u∥21,h ≤ Ch∥u − Πh u∥21,h + Ch∥u − uh ∥21,h ≤ Ch∥u − uh ∥21,h . Combining (5.1) and (5.9), (5.11), (5.13), we know the first term in (5.1) is dominant and then (5.8) can be derived.
(5.13)
6. Numerical examples In this section we present two examples of the Laplace eigenvalue problems to show the efficiencies of the a posteriori error estimate (3.6) and the lower-bound result (5.8). Example 6.1. In this example, we solve the Laplace eigenvalue problem (2.1) with the ECR element on the L-shape domain
Ω = (−1, 1) × (−1, 1)/[0, 1) × (−1, 0]. Since Ω has a reentrant corner, eigenfunctions with singularities are expected.
The convergence order for eigenvalue approximations is less than 2 by the linear finite element method which is the order predicted by the theory for regular eigenfunctions. First, we investigate the numerical results for the first eigenvalue approximations. Since the exact eigenvalue is not known, we choose an adequately accurate approximation λ = 9.6397238440219 as the exact first eigenvalue for our numerical tests. We give the numerical results for the first eigenpair approximation of Adaptive Algorithm with the parameter θ = 0.4. Fig. 1 shows the initial triangulation and the triangulation after 12 adaptive iterations by the ECR element. Fig. 2 gives the corresponding numerical results. We also test Adaptive Algorithm for 5 smallest eigenvalue approximations and their associated eigenfunction approximations. Fig. 3 shows the corresponding eigenvalue errors and a posteriori error estimators ηh (λh , uh ) produced by Adaptive Algorithm with the ECR element. From Figs. 1–3, we can find the approximations of eigenvalues as well as eigenfunctions have the optimal convergence rate which validate the efficiency of the a posteriori error estimator (3.6). Also, it is observed that the ECR element can obtain the lower bound of the eigenvalue problem on the adaptive meshes.
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Fig. 2. The errors of the smallest eigenvalue approximations and the a posteriori errors of the associated eigenfunction approximations by Adaptive Algorithm with the ECR element for Example 1.
Fig. 3. The eigenvalue errors and the a posteriori error estimates of the eigenfunction approximations by Adaptive Algorithm with the ECR element for Example 1.
Fig. 4. The initial triangulation and the ones after 12 adaptive iterations for Example 2 by the ECR element.
Example 6.2. In this example, we solve the Laplace eigenvalue problem (2.1) with the ECR element on the crack domain (see Fig. 4). Since Ω has also a reentrant corner, eigenfunctions with singularities are also expected. Here, we also use Adaptive Algorithm to solve this Laplace eigenvalue problem.
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253
Fig. 5. The errors of the smallest eigenvalue approximations and the a posteriori errors of the associated eigenfunction approximations by Adaptive Algorithm with the ECR element for Example 2.
Fig. 6. The eigenvalue errors and the a posteriori error estimates of the eigenfunction approximations by Adaptive Algorithm with the ECR element for Example 2.
First, we investigate the numerical results for the first eigenvalue approximations. Since the exact eigenvalue is not known, we also choose an adequately accurate approximation with extrapolation method as the exact first eigenvalue for our numerical tests. We give the numerical results for the first eigenpair approximation of Adaptive Algorithm with the parameter θ = 0.4. Fig. 4 shows the initial triangulation and the triangulation after 12 adaptive iterations by the ECR element. Fig. 5 gives the corresponding numerical results. Similarly to Example 6.1, we also test Adaptive Algorithm for 5 smallest eigenvalue approximations and their associated eigenfunction approximations. Fig. 6 shows the corresponding eigenvalue errors and a posteriori error estimators ηh (λh , uh ) produced by Adaptive Algorithm with the ECR element. From Figs. 4–6, we can also find the approximations of eigenvalues as well as eigenfunctions have optimal convergence rate which validate the efficiency of the a posteriori error estimator (3.6). It is also observed that the ECR element can obtain the lower bound of the eigenvalues on the adaptive meshes. 7. Concluding remarks In this paper, we first provide an a posteriori error estimator for the Laplace eigenvalue problem by the nonconforming ECR element. Based on the a posteriori error estimator, the corresponding adaptive finite element method is also built. We also give the lower-bound analysis for the Laplace eigenvalue problem by the ECR element. The idea of the lowerbound analysis can also be used for other types of eigenvalue problems by the nonconforming finite element methods. Some numerical examples are provided to validate the theoretical results. References [1] R. Bank, A. Sherman, A. Weiser, Refinement algorithms and data structures for regular local mesh refinement, in: Scientific Computing, IMACS/NorthHolland Publishing Company, Amsterdam, 1983, pp. 3–17.
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Contents lists available at ScienceDirect
Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
A posterior error estimator and lower bound of a nonconforming finite element method✩ Qun Lin a,∗ , Fusheng Luo a , Hehu Xie b a
LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China
b
LSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China
article
info
Article history: Received 30 September 2012 Received in revised form 9 June 2013 MSC: 65N30 65N25 65L15 65B99
abstract In this paper, we present an a posteriori error estimator and the lower bound for a nonconforming finite element approximation, i.e. the extended Crouzeix–Raviart element, of the Laplace eigenvalue problem. Under the guideline of the analysis to the Laplace source problem, we first give out an error indicator and prove it as the global upper and local lower bounds of the approximation error. We also give the lower-bound analysis for this type of nonconforming element on the adaptive meshes. Some numerical experiments are presented to verify our theoretical results. © 2013 Elsevier B.V. All rights reserved.
Keywords: A posteriori error estimate Lower bound Adaptive finite element method Nonconforming Laplace eigenvalue problem
1. Introduction One vital component of the adaptive finite element method is to find a reliable and efficient a posteriori error estimate as the indicator to help us mark out the elements to be refined or coarsened (see, e.g., [1–6]). The first object of this paper is to derive an a posteriori error estimate of a nonconforming finite element for the Laplace eigenvalue problem. There have been many works concerning the a posteriori error estimates of the eigenvalue problems by the nonconforming finite element methods (see, e.g., [7–9] and the references cited therein). Recently, researchers have found an interesting phenomenon that some nonconforming elements can provide lower bounds for many kinds of eigenvalue problems such as Laplace eigenvalue problem, Stokes eigenvalue problem, Steklov eigenvalue problem, biharmonic eigenvalue problem (see, e.g., [10–14,9,15–18]). Especially, the extended Crouzeix–Raviart (ECR) element, with an additional degree of freedom, requires less of the regularity of the eigenfunction, and provides lowerbound properties, see the theorems and numerical results in [19,13,14,20]. So the second object in this paper is to investigate the lower-bound property of the ECR element for the Laplace eigenvalue problem on the adaptive meshes. In this paper, we present a residual based a posteriori error estimate for the nonconforming finite element approximation of the Laplace eigenvalue problem and give the corresponding analysis of reliability and efficiency by following the guideline
✩ This work was supported in part by National Science Foundations of China NSFC (91330202, 11001259, 11371026, 11031006, 2011CB309703), Croucher Foundation of Hong Kong Baptist University and the National Center for Mathematics and Interdisciplinary Science, CAS and the President Foundation of AMSS-CAS. ∗ Corresponding author. Tel.: +86 10 62624806. E-mail addresses: [email protected] (Q. Lin), [email protected] (F. Luo), [email protected] (H. Xie).
0377-0427/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cam.2013.09.030
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of [7,21]. The main tools used in the analysis are the Helmholtz decomposition, the local interpolation and the approximation properties of the finite element space. Then we give the analysis of the lower bound of the ECR element for the Laplace eigenvalue problem on the adaptive meshes. An outline of the paper goes as follows. In Section 2, we introduce the Laplace eigenvalue problem and the corresponding nonconforming discretization. Section 3 is devoted to presenting the residual based error indicators and giving the analysis of their reliability and efficiency. Then the adaptive finite element method for the eigenvalue problem is presented in Section 4. The analysis of the lower bound for the ECR element is given in Section 5. Section 6 is devoted to presenting some numerical results to validate our theoretical analysis. Finally, some concluding remarks are provided in Section 7. 2. Model problem and its discretization In this paper, we are concerned with the following Laplace eigenvalue problem
−1u = λu, u = 0, u2 dΩ = 1,
in Ω , on ∂ Ω ,
(2.1)
Ω
where Ω is a bounded Lipschitz polygonal domain in R2 . We set V = H01 (Ω ). Then the variational form of (2.1) is: Find (λ, u) ∈ R × V such that Ω u2 dΩ = 1 and a(u, v) = λ(u, v),
∀v ∈ V ,
(2.2)
where a(u, v) =
Ω
∇ u · ∇v dΩ ,
(u, v) =
Ω
uv d Ω .
From [22], we know (2.2) has an eigenvalue series {λj }: 0 < λ1 ≤ λ2 ≤ · · · ≤ λk ≤ · · · ,
lim λk = ∞,
k→∞
and the associated eigenfunctions u1 , u2 , . . . , uk , . . . , with (ui , uj ) = δij . Throughout the paper, the standard notation of the Sobolev spaces and the corresponding norms, semi-norms, and inner products as those in [3,23] are used. For a subdomain D of Ω , the Sobolev space H m (D) := W m,2 (D) with norm ∥ · ∥m,D and semi-norm | · |m,D is used. Then, we introduce a face-to-face triangular partition Th := {K } on the computational domain Ω such that
¯ = Ω
K.
K ∈Th
Here h = maxK ∈Th hK and hK = diamK denote the global and local mesh size, respectively. A family of partitions Th is said to be regular if it satisfies (see [3,23])
∃σ > 0 such that ρK /hK > σ ∀K ∈ Th , where ρK is the diameter of the maximum circle contained in K . Denote the set of all interior edges of Th as Eh , the set of the edges on the boundary as E∂ Ω and E := Eh ∪ E∂ Ω . In the context, C , C˜ represent constants independent of the mesh sizes h and hK and may vary at its different occurrences. According to the partition Th , we build the nonconforming finite element space Vh ̸⊂ V . Especially, we consider in this paper the nonconforming ECR element [19,14]. The ECR element can be defined as follows:
• ECR element: E P1 := span 1, x, y, x2 + y2 ,
Vh =
v ∈ L2 (Ω ), v|K ∈ E P1 , ∀ K ∈ T , ∀ℓ = ∂ K1 ∩ ∂ K2 ∈ Eh and
ℓ
ℓ
v|K1 dℓ =
ℓ
v|K2 dℓ,
v|K1 dℓ = 0, if ℓ ∈ E∂ Ω ,
which has the following properties due to its continuity
K ∈Th
∇(u − uh ) · ∇ × wh dK = K
ℓ∈Eh
ℓ
tℓ · ∇wh [u − uh ]dℓ = 0,
∀ u, uh ∈ V + V h .
(2.3)
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The interpolation operator corresponding to ECR element can be defined in the following way:
ℓ
(u − Πh u)ds = 0,
∀ℓ ∈ Eh ,
(u − Πh u)dK = 0,
(2.4)
∀K ∈ Th .
(2.5)
K
For the analysis of the nonconforming element, we introduce the following piecewise type norm and semi-norm
21
∥ u∥ j , h =
∥u∥2j,K
K ∈Th
and
21
|u|j,h =
2 j ,K
for u ∈
|u|
H (K ) j
Vh ,
K ∈Th
K ∈Th
where j = 0, 1, 2, . . . . Here we also define the lowest order linear conforming space
Wh := v ∈ V : v|K ∈ span{1, x, y}, ∀K ∈ Th . From these definitions, we emphasize the following two important properties: Property 2.1. The finite element space Vh satisfies: 1. For all vh ∈ Vh , we have ℓ [vh ]dℓ = 0 for ℓ ∈ Eh and ℓ vh dℓ = 0 for ℓ ∈ E∂ Ω , where [v] := v|K + − v|K − denotes the jump on the edge ℓ = ∂ K + ∩ ∂ K − . 2. Wh ⊂ Vh and Wh ⊂ V .
For the nonconforming space, we denote the elementwise operator by
(∇h v)|K := ∇(v|K ),
∀ K ∈ Th .
The nonconforming finite element method for (2.2) is defined as follows: Find (λh , uh ) ∈ R × Vh such that ∥uh ∥0,h = 1 and ah (uh , vh ) = λh (uh , vh ),
∀vh ∈ Vh ,
(2.6)
where ah (uh , vh ) =
K ∈Th
∇ uh · ∇vh dK .
K
It is also known from [22] that the discrete Laplace eigenvalue problem (2.6) has eigenvalue series 0 < λ1,h ≤ λ2,h ≤ · · · ≤ λk,h ≤ · · · ≤ λN ,h , and the corresponding eigenfunctions u1,h , u2,h , . . . , uk,h , . . . , uN ,h , with (ui,h , uj,h ) = δij , 1 ≤ i, j ≤ N, where N denotes the dimension of the finite element space Vh . At the end of this section, we review some results related to the a priori error estimates of the Laplace eigenvalue problem (see, e.g., [22,24]). Denote the error of the eigenfunction approximation uh as below e h = u − uh . Thus we have eh ∈ V + Vh . Lemma 2.1. For each finite element approximation (λh , uh ), there exists an eigenpair (λ, u) of (2.2) such that the following error estimates hold
∥eh ∥1,h ≤ Chγ ∥u∥1+γ ,
(2.7)
2γ
∥eh ∥0,h ≤ Ch ∥u∥1+γ , 2γ
|λ − λh | ≤ Ch ∥u∥
2 1+γ
(2.8)
,
(2.9)
where γ ∈ (0, 1] depends on the shape of the domain Ω (γ = 1 when Ω is convex) (see [25]), the const C depends only on λ.
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3. A posterior error indicator This section is devoted to deriving a reliable and efficient indicator for the nonconforming finite element approximation of the Laplace eigenvalue problem. There exists the Helmholtz decomposition for the space V + Vh (see, e.g., [26,27]). Lemma 3.1 ([26]). For any w ∈ V + Vh , there exist r ∈ V and s ∈ H 1 (Ω ) such that
∇h w = ∇ r + ∇ × s.
(3.1)
The boundedness property holds:
∥r ∥1 + ∥s∥1 ≤ C ∥∇h w∥0 .
(3.2)
To get the a posteriori error estimate, we need to use the Clément interpolation which has the following approximation properties. Property 3.1 ([28]). The Clément interpolation operator Ih : V → Wh has the following properties
∥u − Ih u∥0,K ≤ ChK |u|1,ωK ,
∀K ∈ Th ,
1/2
∥u − Ih u∥0,ℓ ≤ Chℓ |u|1,ωℓ , |u − Ih u|1,K ≤ C |u|1,ωK ,
(3.3)
∀ℓ ∈ Eh ,
(3.4)
∀K ∈ Th ,
(3.5)
where ωK denotes the set of elements that share a common vertex with K and ωℓ denotes the set of elements that contain ℓ as one of its edges. The a posteriori error indicator is defined by
12
ηh (λh , uh ) :=
η
,
2 K
(3.6)
K ∈Th
where the local indicator ηK is defined as
ηK2 := h2K ∥λh uh + 1uh ∥20,K +
1 2 ℓ∈∂ K
hℓ ∥Jℓ,t ∥20,ℓ
(3.7)
with
Jℓ,t =
[tℓ · ∇ uh ]|ℓ , 2tℓ · ∇ uh ,
if ℓ ∈ Eh , if ℓ ∈ E∂ Ω .
(3.8)
3.1. Reliability Denote the normal jump by
Jℓ,n =
[nℓ · ∇ uh ]|ℓ , 0,
if ℓ ∈ Eh , if ℓ ∈ E∂ Ω .
(3.9)
The following lemma states the relationship of the normal jump term with the residual on the mesh. Lemma 3.2. Let ℓ = K1 ∩ K2 and set the normal derivative direction from K1 to K2 . The following inequality holds,
h2ℓ ∥Jℓ,n ∥20,ℓ ≤ C h2K1 ∥λh uh + 1uh ∥20,K1 + h2K2 ∥λh uh + 1uh ∥20,K2 .
(3.10)
Proof. Take φℓ as the basis function that equals to 1 at the middle point of ℓ and vanishes at the midpoints of all other edges. Substituting vh with functions v = φℓ and integrating by parts, we get
ωℓ
∇h uh · ∇h v dK + ∆h uh v dK ωℓ ωℓ = nℓ · ∇ uh v dℓ − nℓ · ∇ uh v dℓ = − Jℓ,n v dℓ.
(λh uh + ∆h uh )v dK =
∂ K1
∂ K2
ℓ
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247
By the definition of v , there is a constant C such that hℓ ∥Jℓ,n ∥0,ℓ
≤ C Jℓ,n v dℓ. ℓ
Thus we arrive the desired result (3.10) with hℓ ∥Jℓ,n ∥0,ℓ ≤ C ∥λh uh + 1uh ∥0,K1 ∥φℓ ∥0,K1 + ∥λh uh + 1uh ∥0,K2 ∥φℓ ∥0,K2
≤ C (hK1 ∥λh uh + 1uh ∥0,K1 + hK2 ∥λh uh + 1uh ∥0,K2 ), and the proof is complete.
Then, we show that the error eh can be controlled by the indicator up to a higher order term. Theorem 3.1 (Reliability). For the error of the nonconforming finite element approximation (λh , uh ) of (λ, u), we have the following estimate
∥∇h eh ∥0,h ≤ C ηh (λh , uh ) + ∥λh uh − λu∥0,h ,
(3.11)
where the constant C depends only on the minimum angle of Th . Proof. Since eh = u − uh ∈ V + Vh , by the Helmholtz decomposition (3.1), there exist r ∈ V and s ∈ H 1 (Ω ) such that
∇h eh = ∇ r + ∇ × s, where r and s satisfy
∥r ∥1 + ∥s∥1 ≤ C ∥∇h eh ∥0 .
(3.12)
So we have
∥∇h eh ∥20,h =
∇ eh · (∇ r + ∇ × s)dK .
(3.13)
K
K ∈Th
Combining (3.13) with (2.2), (2.6) and (3.13) leads to
∥∇h eh ∥20 =
∇ eh · ∇ r + ∇ eh · (∇ × s) dK K
K ∈Th
=
∇ eh · ∇(r − Ih r )dK +
K
K ∈Th
∇ eh · (∇ × (s − Ih s))dK + K
(λu − λh uh )Ih rdK ,
(3.14)
K
where we use (2.3) and the facts that Ih r ∈ V , Ih r ∈ Vh and
∇ eh · ∇ × Ih sdK = 0. K
K ∈Th
The first two terms in (3.14) can be estimated as follows: 1. Integrating by parts and using (2.2), (2.6) leads to
∇ eh · ∇(r − Ih r )dK = K
(λu + 1uh )(r − Ih r )dK −
K
∂K
(∇ uh · n)(r − Ih r )dℓ.
2. Integrating by parts leads to
∇ eh · (∇ × (s − Ih s))dK = K
∂K
(∇ uh · t )(s − Ih s)dℓ −
∂K
(∇ u · t )(s − Ih s)dℓ.
It is from summing all with respect to elements and canceling the term on the edges of any element that 2 h eh 0,h
∥∇
∥
=
(λu + 1uh )(r − Ih r )dK + K
K ∈Th
− ∂K
(∇ uh · n)(r − Ih r )dℓ −
K ∈Th
(λu − λh uh )Ih rdK K
∂K
(∇ uh · t )(s − Ih s)dℓ
∂K
t · ∇ u(s − Ih s)dℓ due to its continuity
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≤C
∥λu − λh uh ∥0,K ∥r ∥0,K + ∥λh uh + 1uh ∥0,K hK |r |1,ωK
K ∈Th
−+
1
1/2
1/2
∥Jℓ,n ∥0,ωℓ hℓ |r |1,ωℓ + ∥Jℓ,t ∥0,ωℓ hℓ |s|1,ωℓ
2 ℓ∈∂ K
.
(3.15)
From (3.15), Cauchy–Schwarz inequality and the stability of the Helmholtz decomposition (3.2), we have
2 h eh 0,h
∥∇
∥
≤C
h2K
∥λh uh +
1uh 20,K
∥
K ∈Th
+
1 2 ℓ∈∂ K
Jℓ,n 20,ωℓ
hℓ ∥
∥
+∥
Jℓ,t 20,ωℓ
∥
1/2 + ∥λu − λ
2 h uh 0,h
∥
∥∇h eh ∥0,h .
(3.16)
Canceling the term ∥∇h eh ∥0,h on both sides and replacing the jump term in (3.16) by (3.10), we get the desired result (3.11). To show that ∥λh uh − λu∥0,h is a higher order term, we rewrite it as
∥λh uh − λu∥0,h ≤ |λh − λ|∥uh ∥0,h + λ∥u − uh ∥0,h . Then, from a priori error estimates (2.7)–(2.9), it is easy to see that both |λh − λ| and ∥u − uh ∥0,h are higher order terms compared with ∥∇h eh ∥0,h . 3.2. Efficiency We turn to prove the efficiency of the indicator via bubble functions. First we define the bubble functions bK on the element K and bℓ on the edge ℓ of K by barycentric functions λi,K : bK := λ1,K λ2,K λ3,K , and
bℓ :=
λ1,K1 λ2,K1 , λ1,K2 λ2,K2 ,
on K1 , on K2 ,
where K1 and K2 are the elements sharing the edge ℓ with nodes 1, 2. Then, we estimate the indicator term by term to show its efficiency. Lemma 3.3. There exists a constant C such that
hK ∥λh uh + 1uh ∥0,K ≤ C hK ∥λh uh − λu∥0,K + ∥∇ eh ∥0,K ,
∀ K ∈ Th .
(3.17)
Proof. Denote φK = C |K |(λh uh + 1uh )bK and choose C to make the following equality hold C |K |∥λh uh + 1uh ∥20,K =
= = ≤ ≤
(λh uh + 1uh )φK dK λh uh φK dK − ∇ uh · ∇φK dK K K (λh uh − λu)φK dK + ∇ eh · ∇φK dK K K C ∥λh uh − λu∥0,K ∥φK ∥0,K + ∥∇ eh ∥0,K ∥∇φK ∥0,K C h2K ∥λh uh − λu∥0,K + hK ∥∇ eh ∥0,K ∥λh uh + 1uh ∥0,K , K
where we use the fact ∥φK ∥0,K = C |K |∥λh uh + 1uh ∥0,K and ∥∇φK ∥0,K = C˜ hK ∥λh uh + 1uh ∥0,K for some constants C and C˜ . Canceling the term hK ∥λh uh + 1uh ∥0,K on both sides leads to the desired result (3.17) and the proof is complete. Lemma 3.4. There exists a constant C such that 1/2
hℓ ∥Jℓ,t ∥0,ℓ ≤ C ∥∇h eh ∥0,ωℓ .
(3.18)
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249
Proof. We integrate the jump term Jℓ,t with φ2,ℓ := Chℓ bℓ Jℓ,t and choose C to make the following equality hold Chℓ Jℓ,t 20,ℓ
∥
∥
= ℓ
[tℓ · ∇h uh ] · φ2,ℓ dℓ =
= ωℓ
ℓ
[tℓ · ∇h eh ]φ2,ℓ dℓ
(∇h eh ) · (∇h × φ2,ℓ )dK ≤ C ∥∇h eh ∥0,ωℓ ∥∇h φ2,ℓ ∥0,ωℓ ,
(3.19)
where we use [tℓ · ∇ u] = 0. There is some constant C such that 1/2
∥∇h φ2,ℓ ∥0,ωℓ = Chℓ ∥Jℓ,t ∥0,ℓ .
(3.20)
From (3.19) and (3.20), we obtain (3.18).
We are now in the position to state the efficiency of the estimator. Theorem 3.2 (Efficiency). The local error indicator defined by (3.6) has the following efficiency property
ηK ≤ C ∥∇h eh ∥0,ωK +
1/2 , h2K ∥λh uh − λu∥20,K
(3.21)
K ∈ωK
where ωK is defined in Property 3.1 and C is a constant. 4. Adaptive finite element method In this section, the adaptive finite element method algorithm is briefly reviewed. Starting from an initial mesh Th0 , by repeating the common cycle of four steps of the adaptive finite element method: SOLVE → ESTIMATE → MARK → REFINE, we form a nested conforming triangulation Thi . In the SOLVE step, we assume that this step can be finished in an ideal way which reaches a sufficient high accuracy and optimal time complexity as we want. For example, on mesh Thi , firstly, (λhi , uhi ) is solved from (2.6). In the ESTIMATE step, we compute out the error indicators by (3.6) and (3.7). Then, in the MARK step, the bulk marking strategy proposed by Dörfler [4] is adopted to define the marking set Mhi according to the local indicator (3.6), such that
K ∈Mh
ηK2 ≥ θ i
K ∈Th
ηK2 ,
for some θ ∈ (0, 1).
i
We mark out a set of triangles of the mesh Thi to be refined. In the REFINE step, some refined method is used to generate the new finer mesh Thi+1 and keep the new mesh conforming and shape regular as defined in [3,23]. There are many refined methods, i.e., the red and green refinement [1], longest refinement [29,30], and newest vertex bisection [31]. In this paper, we use the newest vertex bisection method. For more details, please refer to [2,32,6]. The adaptive algorithm goes as follows:
Adaptive Algorithm 1. 2. 3. 4. 5. 6.
Choose a parameter θ ∈ (0, 1), pick an initial mesh Th0 and set i = 0. Solve the problem (2.6) to obtain the eigenpair (λhi , uhi ). Compute out the local indicators ηK . Construct the set Mhi ⊂ Thi by MARK procedure with the parameter θ . Refine Thi by REFINE procedure to get a new conforming mesh Thi+1 . Let i := i + 1 and go to Step 2.
5. Lower bound of the eigenvalue problem In this section, the lower-bound analysis of the eigenvalue problem by the ECR element is given on the adaptive meshes. For the eigenvalue problem, we have the following basic expansion which was first proved in its full term in [18] by the similar argument in [10] (where the expansion has only the first three terms) and has been extensively used in [19,13,33,17].
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Lemma 5.1. Suppose (λ, u) is the eigenpair of the original problem (2.2), (λh , uh ) ∈ R × Vh is the eigenpair of the discrete problem (2.6). We have the following expansion
λ − λh = ∥u − uh ∥21,h − λh ∥vh − uh ∥20,h + λh (∥vh ∥20,h − ∥u∥20 ) + 2ah (u − vh , uh ),
∀vh ∈ Vh .
(5.1)
Lemma 5.2. For the ECR element interpolation, the following property holds
∥u − Πh u∥0,h ≤ Ch∥u − Πh u∥1,h , ah u − Πh u, vh = 0, ∀vh ∈ Vh .
(5.2) (5.3)
Lemma 5.3. On the regular type of mesh Th , we have the following inequality for the eigenfunction approximation uh by the ECR element
∥uh − Π0 uh ∥0,h ≤ C ∥∇h (u − uh )∥0,h + h∥λu − λh uh ∥0,h ,
(5.4)
where Π0 denotes the piecewise constant interpolation. Proof. It is easy to know the following estimate holds
∥uh − Π0 uh ∥20,h ≤ C
h2K ∥∇ uh ∥20,K .
(5.5)
K ∈Th
Let ω ¯ K denote the set of elements that share a common edge with K . In the finite dimensional space Vh (ω¯ K ) := {vh |ω¯ K , ∀vh ∈ Vh }, we investigate the relationship of the two norms:
K ∈ω ¯T
h2K ∥∇vh ∥20,K
and
K ∈ω ¯T
h2K ∥λh vh + 1vh ∥20,K +
1 2 ℓ∈∂ T
hℓ ∥[∇vh ]∥20,ℓ .
1 2 2 2 It is easily known that K ∈ω ¯ T hK ∥λh vh + 1vh ∥0,K + 2 ℓ∈∂ T hℓ ∥[∇vh ]∥0,ℓ = 0 leads to compactness and scaling argument shows there exists a constant C such that
K ∈ω ¯T
h2K ∥∇vh ∥20,K ≤ C
K ∈ω ¯T
h2K ∥λh vh + 1vh ∥20,K +
1 2 ℓ∈∂ T
K ∈ω ¯T
h2K ∥∇vh ∥20,K = 0. A
,
hℓ ∥[∇vh ]∥20,ℓ
∀vh ∈ Vh .
(5.6)
From (5.6) and Lemma 3.2, we have
K ∈ω ¯T
h2K ∥∇ uh ∥20,K ≤ C
K ∈ω ¯T
h2K ∥λh uh + 1uh ∥20,K +
1 2 ℓ∈∂ T
hℓ ∥Jℓ,t ∥20,ℓ
.
(5.7)
Combining (5.5), (5.7) and Theorem 3.2 leads to
∥uh − Π0 uh ∥20,h ≤ C
T ∈Th K ∈ω ¯T
h2K ∥∇ uh ∥20,K
≤C
T ∈Th
K ∈ω ¯T
h2K
∥λh uh +
1uh 20,K
∥
≤C
h2K
∥λh uh +
K ∈Th
1uh 20,K
∥
+
+
1 2 ℓ∈∂ T
1 2 ℓ∈∂ K
hℓ Jℓ,t 20,ℓ
∥
∥
hℓ Jℓ,t 20,ℓ
∥
∥
≤ C ∥∇h eh ∥20,h + h2 ∥λu − λh uh ∥20,h . This is the desired result (5.4) and we complete the proof.
Theorem 5.1. Let λ and λh be an eigenvalue and its corresponding numerical approximation by ECR element. Assume u ∈ H 1+γ (Ω ) with 0 < γ ≤ 1. When h is small enough, we have
λh < λ.
(5.8)
Proof. We choose vh = Πh u in Lemma 5.1. First, from the results in [13], we have
∥u − uh ∥0,h ≤ Chγ ∥u − uh ∥1,h .
(5.9)
Since (5.3), the following inequality holds
∥∇h (u − Πh u)∥0,h ≤ ∥∇h (u − uh )∥0,h .
(5.10)
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251
Fig. 1. The initial triangulation and the ones after 12 adaptive iterations for Example 1 by the ECR element.
Then for the second term in (5.1), from Lemma 5.2, (5.9) and (5.10), we have
∥uh − Πh u∥20,h ≤ 2∥uh − u∥20,h + 2∥u − Πh u∥20,h ≤ Ch2γ ∥u − uh ∥21,h .
(5.11)
From Lemma 5.3, (5.9) and the result in [24], we have
∥u − Π0 u∥0,h ≤ C ∥u − uh ∥0,h + ∥uh − Π0 uh ∥0,h ≤ C ∥∇h (u − uh )∥0,h + h∥λu − λh uh ∥0,h + ∥u − uh ∥0,h ≤ C ∥∇h (u − uh )∥0,h + hγ ∥u − uh ∥1,h .
(5.12)
For the third term in (5.1), from (5.2)–(5.3), (5.10) and (5.12), we have
∥vh ∥20,h − ∥u∥20 = (Πh u − u, Πh u + u) = 2(Πh u − u, u) + ∥u − Πh u∥20,h = 2(Πh u − u, u − Π0 u) + ∥u − Πh u∥20,h ≤ Ch∥u − Πh u∥1,h ∥u − Π0 u∥0,h + Ch2 ∥u − Πh u∥21,h ≤ Ch∥u − Πh u∥21,h + Ch∥u − uh ∥21,h ≤ Ch∥u − uh ∥21,h . Combining (5.1) and (5.9), (5.11), (5.13), we know the first term in (5.1) is dominant and then (5.8) can be derived.
(5.13)
6. Numerical examples In this section we present two examples of the Laplace eigenvalue problems to show the efficiencies of the a posteriori error estimate (3.6) and the lower-bound result (5.8). Example 6.1. In this example, we solve the Laplace eigenvalue problem (2.1) with the ECR element on the L-shape domain
Ω = (−1, 1) × (−1, 1)/[0, 1) × (−1, 0]. Since Ω has a reentrant corner, eigenfunctions with singularities are expected.
The convergence order for eigenvalue approximations is less than 2 by the linear finite element method which is the order predicted by the theory for regular eigenfunctions. First, we investigate the numerical results for the first eigenvalue approximations. Since the exact eigenvalue is not known, we choose an adequately accurate approximation λ = 9.6397238440219 as the exact first eigenvalue for our numerical tests. We give the numerical results for the first eigenpair approximation of Adaptive Algorithm with the parameter θ = 0.4. Fig. 1 shows the initial triangulation and the triangulation after 12 adaptive iterations by the ECR element. Fig. 2 gives the corresponding numerical results. We also test Adaptive Algorithm for 5 smallest eigenvalue approximations and their associated eigenfunction approximations. Fig. 3 shows the corresponding eigenvalue errors and a posteriori error estimators ηh (λh , uh ) produced by Adaptive Algorithm with the ECR element. From Figs. 1–3, we can find the approximations of eigenvalues as well as eigenfunctions have the optimal convergence rate which validate the efficiency of the a posteriori error estimator (3.6). Also, it is observed that the ECR element can obtain the lower bound of the eigenvalue problem on the adaptive meshes.
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Fig. 2. The errors of the smallest eigenvalue approximations and the a posteriori errors of the associated eigenfunction approximations by Adaptive Algorithm with the ECR element for Example 1.
Fig. 3. The eigenvalue errors and the a posteriori error estimates of the eigenfunction approximations by Adaptive Algorithm with the ECR element for Example 1.
Fig. 4. The initial triangulation and the ones after 12 adaptive iterations for Example 2 by the ECR element.
Example 6.2. In this example, we solve the Laplace eigenvalue problem (2.1) with the ECR element on the crack domain (see Fig. 4). Since Ω has also a reentrant corner, eigenfunctions with singularities are also expected. Here, we also use Adaptive Algorithm to solve this Laplace eigenvalue problem.
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253
Fig. 5. The errors of the smallest eigenvalue approximations and the a posteriori errors of the associated eigenfunction approximations by Adaptive Algorithm with the ECR element for Example 2.
Fig. 6. The eigenvalue errors and the a posteriori error estimates of the eigenfunction approximations by Adaptive Algorithm with the ECR element for Example 2.
First, we investigate the numerical results for the first eigenvalue approximations. Since the exact eigenvalue is not known, we also choose an adequately accurate approximation with extrapolation method as the exact first eigenvalue for our numerical tests. We give the numerical results for the first eigenpair approximation of Adaptive Algorithm with the parameter θ = 0.4. Fig. 4 shows the initial triangulation and the triangulation after 12 adaptive iterations by the ECR element. Fig. 5 gives the corresponding numerical results. Similarly to Example 6.1, we also test Adaptive Algorithm for 5 smallest eigenvalue approximations and their associated eigenfunction approximations. Fig. 6 shows the corresponding eigenvalue errors and a posteriori error estimators ηh (λh , uh ) produced by Adaptive Algorithm with the ECR element. From Figs. 4–6, we can also find the approximations of eigenvalues as well as eigenfunctions have optimal convergence rate which validate the efficiency of the a posteriori error estimator (3.6). It is also observed that the ECR element can obtain the lower bound of the eigenvalues on the adaptive meshes. 7. Concluding remarks In this paper, we first provide an a posteriori error estimator for the Laplace eigenvalue problem by the nonconforming ECR element. Based on the a posteriori error estimator, the corresponding adaptive finite element method is also built. We also give the lower-bound analysis for the Laplace eigenvalue problem by the ECR element. The idea of the lowerbound analysis can also be used for other types of eigenvalue problems by the nonconforming finite element methods. Some numerical examples are provided to validate the theoretical results. References [1] R. Bank, A. Sherman, A. Weiser, Refinement algorithms and data structures for regular local mesh refinement, in: Scientific Computing, IMACS/NorthHolland Publishing Company, Amsterdam, 1983, pp. 3–17.
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