A simple low-degree optimal finite element scheme for the elastic transmission eigenvalue problem
AA SIMPLE LOW-DEGREE OPTIMAL FINITE ELEMENT SCHEME FOR THEELASTIC TRANSMISSION EIGENVALUE PROBLEM
YINGXIA XI, XIA JI, AND SHUO ZHANGA bstract . The paper presents a finite element scheme for the elastic transmission eigen-value problem written as a fourth order eigenvalue problem. The scheme uses piecewisecubic polynomials and obtains optimal convergence rate. Compared with other low-degreeand nonconforming finite element schemes, the scheme inherits the continuous bilinearform which does not need extra stabilizations and is thus simple to implement.
1. I ntroduction
The transmission eigenvalue problem is important in the qualitative reconstruction inthe inverse scattering theory of inhomogeneous media. For example, the eigenvalues canbe used to obtain estimates for the physical characteristics of the hidden scatterer [6, 31]and have a multitude of applications in inverse problems for target identification and non-destructive testing [9, 17]. Besides, the transmission eigenvalues play a key role in theuniqueness and reconstruction in inverse scattering theory. Moreover, they can be usedto design invisibility materials [22]. There are di ff erent types of transmission eigenvalueproblems, such as the acoustic transmission eigenvalue problem, the electromagnetic trans-mission eigenvalue problem, and the elastic transmission eigenvalue problem, etc.Since 2010, e ff ective numerical methods for the acoustic transmission eigenvalues havebeen developed by many researchers [1, 8, 10, 14, 15, 19, 20, 23, 24, 26, 27, 30, 33, 37–39],while there are much fewer works for the electromagnetic transmission eigenvalue problemand the elastic transmission eigenvalue problem [18, 21, 28, 32, 36, 40]. In this paper, wetry to develop e ff ective numerical methods for transmission eigenvalue problem of elasticwaves.The non-self-adjointness and nonlinearity plaguing the numerical study of the elastictransmission eigenvalue problem are compounded by the tensorial structure of the elasticwave equation. In [21], the elastic transmission eigenvalue problem is reformulated assolving the roots of a nonlinear function of which the values correspond to the general-ized eigenvalues of a series of fourth order self-adjoint eigenvalue problems discretized by H -conforming finite element methods. The authors apply the secant iterative method tocompute the transmission eigenvalues. However, at each step, a fourth order self-adjoint Mathematics Subject Classification.
Key words and phrases. elastic transmission eigenvalue problem, nonconforming finite element method, highaccuracy.The research of Y. Xi is supported in part by the National Natural Science Foundation of China with GrantNo.11901295, Natural Science Foundation of Jiangsu Province under BK20190431.The research of X. Ji is partially supported by the National Natural Science Foundation of China with GrantNos.11971468 and 91630313.The research of S. Zhang is supported partially by the National Natural Science Foundation of China withGrant Nos.11471026 and 11871465, the Strategic Priority Research Program of CAS, Grand No. XDB 41000000and National Centre for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences. a r X i v : . [ m a t h . NA ] J a n YINGXIA XI, XIA JI, AND SHUO ZHANG eigenvalue problem needs to be solved and only real eigenvalues can be captured. Based ona fourth order variational formulation, Yang et al. [40] study the H -conforming methodsincluding the classical H -conforming finite element method and the H -conforming spec-tral element method. In [36], Xi et al. propose an interior penalty discontinuous Galerkinmethod using C Lagrange elements (C IP) for the elastic transmission eigenvalue problemwhich use less degrees of freedom than C elements and much easier to be implemented.There have also existed some mixed methods for this problem [35, 41]. The mixed schemein [35] has similarity to Ciarlet-Raviart discretization of biharmonic problem [13] and isone of the least expensive methods in terms of the amount of degrees of freedom. Theconvergence analysis is presented under the framework of spectral approximation theoryof compact operator [3, 29] and the error analysis of a mixed finite element method for theStokes problem [16]. Although the existence of elastic transmission eigenvalues is beyondour concern, we want to remark that there exist only a few studies on the existence of theelasticity transmission eigenvalues [4, 5, 11, 12]. We hope that the numerical results cangive some hints on the analysis of the elasticity transmission eigenvalue problem.In this paper, we study the H -nonconforming finite element method B h for the elastictransmission eigenvalue problem. The method is first introduced in [42] for the biharmonicequation with optimal convergence rate. The method does not correspond to a locally de-fined finite element with Ciarlet’s triple but admits a set of local basis functions. It isapplied in the Helmholtz transmission eigenvalue problem [37] obtaining the optimal con-vergence rate. For the elastic transmission eigenvalue problem, the tensorial structure isthe main di ffi culty. The connection between bi-elastic eigenvalue problem and biharmoniceigenvalue problem is established which make the theoretical analysis possible. The prop-erty can be inherited by the B h element at the discrete level, and we design schemesaccordingly. In this paper, by a careful analysis, we reveal that when λ is not big, thebi-elastic problem takes the same essence as that of a tensorial biharmonic equation. Thisobservation can hint a stabilization formulation for general low-degree finite element suchas the Morley element. We give a new Morley element in section 5.3 and the idea can beapplied to the other low-degree finite element.The rest of this paper is organized as follows. In section 2, we introduce the problemand a piecewise cubic finite element. Section 3 presents the numerical scheme and errorestimate for the bi-elastic problem. Section 4 gives the error estimate of the elasticitytransmission eigenvalue problem. Numerical examples are presented in the last section.We also compare our method with Morley element scheme.2. P reliminaries Elastic transmission eigenvalue problem.
Before introducing the elastic transmis-sion eigenvalue problem, we present some notations first. Let Ω ⊂ R be a boundedconvex Lipschitz domain and x = ( x , x ) (cid:62) ∈ R . And all vectors will be denoted inbold script in the subsequent. We denote the displacement vector of the wave field by u ( x ) = ( u ( x ) , u ( x )) (cid:62) and the displacement gradient tensor by ∇ u = (cid:34) ∂ x u ∂ x u ∂ x u ∂ x u (cid:35) . The strain tensor ε ( u ) is given by ε ( u ) =
12 ( ∇ u + ( ∇ u ) (cid:62) ) , SIMPLE LOW-DEGREE OPTIMAL FINITE ELEMENT SCHEME FOR THE ELASTIC TRANSMISSION EIGENVALUE PROBLEM3 and the stress tensor σ ( u ) is given by the generalized Hooke law σ ( u ) = µε ( u ) + λ tr( ε ( u ))I , where the Lamé parameters µ, λ are two positive constants and I ∈ R × is the identitymatrix. Writing the above equation out, we have(1) σ ( u ) = (cid:34) ( λ + µ ) ∂ x u + λ∂ x u µ ( ∂ x u + ∂ x u ) µ ( ∂ x u + ∂ x u ) λ∂ x u + ( λ + µ ) ∂ x u (cid:35) . The two-dimensional elastic wave problem is described by the reduced Navier equation:Find u with zero trace on ∂ Ω , such that(2) ∇ · σ ( u ) + ω ρ u = , in Ω ⊂ R , where ω > ρ is the mass density.The elasticity transmission eigenvalue problem is to find ω (cid:44) u , v satisfying(3) ∇ · σ ( u ) + ω ρ u = in Ω , ∇ · σ ( v ) + ω ρ v = in Ω , u = v on ∂ Ω ,σ ( u ) n = σ ( v ) n on ∂ Ω , where σ n denotes the matrix multiplication of the stress tensor σ and the unit outwardnormal n . We assume that the mass density distributions satisfy the following inequalities(4) q ≤ ρ ( x ) ≤ Q , q ∗ ≤ ρ ( x ) ≤ Q ∗ , x ∈ Ω , where q , q ∗ and Q , Q ∗ are positive constants. We also assume that the two density distribu-tions are "non-intersecting" [4], i.e. Q ≤ ≤ q ∗ or Q ∗ ≤ ≤ q . Define the Sobolev space(5) V = { φ ∈ ( H ( Ω )) : φ = and σ ( φ ) ν = on ∂ Ω } . Introducing a new variable w = u − v ∈ V , following the same procedure in [21], the system(3) can be written as follows(6) ( ∇ · σ + ω ρ )( ρ − ρ ) − ( ∇ · σ + ω ρ ) w = . Let τ = ω . The corresponding weak formulation is to find τ and w (cid:44) ∈ V such that(7) (cid:16) ( ρ − ρ ) − ( ∇ · σ + τρ ) w , ( ∇ · σ + τρ ) ϕ (cid:17) = , ∀ ϕ ∈ V . A piecewise cubic finite element.
Before introducing the finite element, we givesome notations. Assume T h a shape regular mesh over Ω with mesh size h . Denote X h , X ih , X bh , E h , E ih , E bh the vertices, interior vertices, boundary vertices, the set of edges, interioredges and boundary edges, respectively. For any edge e ∈ E h , denote the unit normalvector of e by n e and denote the unit tangential vector of e by t e . For a triangle T ∈ T h ,we use P k ( T ) to denote the set of polynomials not higher than k and | T | means the areameasurement of element T . On an edge e , P k ( e ) and | e | are defined similarly.The cubic H -nonconforming finite element space B h [42] can be defined as follows: B h = (cid:8) v ∈ L ( Ω ) | v | T ∈ P ( T ) , v is continuous at vertices a ∈ X h and (cid:90) e (cid:126) v (cid:127) ds = , and (cid:90) e p e (cid:126) ∂ n v (cid:127) ds = , ∀ p e ∈ P ( e ) , ∀ e ∈ E ih , ∀ T ∈ T h (cid:9) , YINGXIA XI, XIA JI, AND SHUO ZHANG where (cid:126) v (cid:127) represents the jump of the scalar function v across e, and B h = (cid:8) v ∈ B h | v ( a ) = , a ∈ X bh ; (cid:90) e v ds = , and (cid:90) e p e ∂ n v ds = , ∀ p e ∈ P ( e ) , ∀ e ∈ E bh } . It should be noticed that B h does not correspond to a locally defined finite element withCiarlet (cid:48) s triple but admits a set of local basis functions. The explicit expression of basisfunctions can be referred to [37].Besides, we denote some finite element spaces which will be used in the subsequent by V kh = (cid:110) w ∈ H ( Ω ) : w | T ∈ P k ( T ) , ∀ T ∈ T h (cid:111) , V kh = V kh ∩ H ( Ω ) , k (cid:62) P kh = (cid:110) w ∈ L ( Ω ) : w | T ∈ P k ( T ) , ∀ T ∈ T h (cid:111) , P kh = P kh ∩ L ( Ω ) , k (cid:62) G kh ) = (cid:40) v ∈ ( L ( Ω )) : (cid:90) e p e (cid:126) v l (cid:127) ds = , ∀ p e ∈ P k − ( e ) , ∀ e ∈ E ih , l = , (cid:41) , k (cid:62) G kh ) = (cid:40) v ∈ ( G kh ) : (cid:90) e p e v l ds = , ∀ p e ∈ P k − ( e ) , ∀ e ∈ E bh , l = , (cid:41) , k (cid:62) . ( S kh ) = ( P kh ) ∩ ( H ( Ω )) , ( S kh ) = ( S kh ) ∩ ( H ( Ω )) , k (cid:62) . B h = (cid:110) φ h : φ h | T ∈ span { λ + λ + λ − / } , ∀ T ∈ T h (cid:111) , where λ i , i = , , B h space, we can verify the following results. Lemma 1. [43] For ∀ w h , v h ∈ B h , we have ( ∆ h w h , ∆ h v h ) = ( ∇ h w h , ∇ h v h ) . Lemma 2.
For ∀ w h , v h ∈ ( B h ) , it can be established that ( ∇ h div h w h , curl h rot h v h ) = .Proof. First, if w h , v h ∈ G kh , we can get(8) ( ∇ h w h , curl h v h ) = . Since G kh = S kh ⊕ B h [43], v h can be written as v h = v h + v h , v h ∈ S kh , v h ∈ B h . For v h , ( ∇ h w h , curl h v ) = (cid:88) T ( ∇ w h , curlv ) T = (cid:88) T (cid:90) ∂ T w h ∂ t v (9) = (cid:88) e (cid:90) e (cid:126) w h (cid:127) ∂ t e v h = . here we use ∂ t e v h ∈ P ( e ) and (cid:82) e p e (cid:126) w h (cid:127) = , ∀ p e ∈ P ( e ).For v h , ( ∇ h w h , curl h v ) = (cid:88) T ( ∇ w h , curlv ) T = (cid:88) T (cid:90) ∂ T ∂ t w h v = , (10)since ∂ t w h | ∂ T ∈ P ( ∂ T ) and v h is bubble on the boudary.The combination (9) with (10) leads to (8).If w h ∈ ( B h ) , then div h w h , rot h v h ∈ G kh0 , the proof is complete. (cid:3) SIMPLE LOW-DEGREE OPTIMAL FINITE ELEMENT SCHEME FOR THE ELASTIC TRANSMISSION EIGENVALUE PROBLEM5
3. A second order scheme for the bi - elastic problem Based on the above equation (2), we construct the bi-elastic eigenvalue problem: Find( K , w ) ∈ C × V satisfying(11) ( ∇ · σ ) ( β ∇ · σ ( w )) = K w , in Ω , w = , on ∂ Ω ,σ ( w ) ν = , on ∂ Ω , where σ ν denotes the matrix multiplication of the stress tensor σ and the unit outwardnormal ν . And β ( x ) is a bounded smooth non-constant function, β ( x ) (cid:62) β min > K , w ) ∈ C × V , such that(12) A β ( w , ϕ ) (cid:44) ( β ∇ · σ ( w ) , ∇ · σ ( ϕ )) = K ( w , ϕ ) , ∀ ϕ ∈ V . Theorem 3. (cid:107)∇ · σ ( w ) (cid:107) (cid:27) (cid:107) w (cid:107) , ∀ w ∈ V. (" (cid:27) " denotes " = " up to a constant.)Proof. Especially, we can obtain that (cid:16) ∇ · σ ( w ) , ∇ · σ ( ϕ ) (cid:17) = (cid:16) − µ ∆ w − λ ∇∇ · ( w ) , − µ ∆ ϕ − λ ∇∇ · ( ϕ ) (cid:17) = (cid:16) − ( λ + µ ) ∇∇ · w + µ curl rot w , − ( λ + µ ) ∇∇ · ϕ + µ curl rot ϕ (cid:17) = ( λ + µ ) (cid:16) − ∇∇ · w , −∇∇ · ϕ (cid:17) + µ (cid:16) curl rot w , curl rot ϕ (cid:17) = µ ( − ∆ w , − ∆ ϕ ) + [( λ + µ ) − µ ]( −∇∇ · ( w ) , −∇∇ · ( ϕ )) , ∀ w , ϕ ∈ V , (13) then we get µ (cid:107) − ∆ w (cid:107) ≤ (cid:107)∇ · σ ( w ) (cid:107) ≤ ( λ + µ ) (cid:107) − ∆ w (cid:107) . (14) Further, combining the above equation with (cid:107) − ∆ w (cid:107) = (cid:107)∇ w (cid:107) , w ∈ ( H ∩ H ) leads tothe desired result. (cid:3) Remark 4.
From the above derivations, we know that A β ( · , · ) is coercive on V. And it’seasy to verify the boundedness of A β ( · , · ) . The well-posedness of (12) follows by Lax-Milgram theorem. Remark 5.
The above theorem establishes the connection between bi-elastic eigenvalueproblem and biharmonic eigenvalue problem. The theoretical analysis can be put downto the biharmonic eigenvalue problem. We give some details to make the paper moreunderstandable.
The bi-elastic source problem for (11) is to find w ∈ V satisfying(15) ( ∇ · σ ) ( β ∇ · σ ( w )) = f , in Ω , w = , on ∂ Ω ,σ ( w ) ν = , on ∂ Ω . The cubic finite element scheme for (15) is defined as: find w h ∈ ( B h ) , such that(16) A β, h ( w h , ϕ h ) (cid:44) ( β ( ∇ · σ ) w h , ( ∇ · σ ) ϕ h ) = ( f , ϕ h ) , ∀ ϕ h ∈ ( B h ) . Define (cid:107) w h (cid:107) h = (cid:40) (cid:80) T ∈T h (cid:82) T |∇ · σ ( w h ) | dx dx (cid:41) . In the following, we prove that (cid:107) w h (cid:107) h is anorm on ( B h ) . Lemma 6. (cid:107) w h (cid:107) h is a norm on ( B h ) . YINGXIA XI, XIA JI, AND SHUO ZHANG
Proof.
Using Lemma 2, we obtain (cid:107) w h (cid:107) h = (cid:88) T ∈T h (cid:16) ∇ · σ ( w h ) , ∇ · σ ( w h ) (cid:17) = (cid:88) T ∈T h (cid:16) − ( λ + µ ) ∇∇ · w h + µ curl rot w h , − ( λ + µ ) ∇∇ · w h + µ curl rot w h (cid:17) = (cid:88) T ∈T h ( λ + µ ) (cid:16) − ∇∇ · w h , −∇∇ · w h (cid:17) + (cid:88) T ∈T h µ (cid:16) curl rot w h , curl rot w h (cid:17) = (cid:88) T ∈T h ( µ ( − ∆ w h , − ∆ w h ) + (cid:88) T ∈T h ([( λ + µ ) − µ ]( −∇∇ · ( w h ) , −∇∇ · ( w h )) ≥ µ (cid:107) ∆ h w h (cid:107) = µ (cid:107)∇ h w h (cid:107) , ∀ w h ∈ ( B h ) . (17) (cid:3) Lemma 7.
The finite element approximation (16) is well-posed.Proof.
By the definition of (cid:107) w h (cid:107) h , it’s easy to know that A β, h ( · , · ) is coercive and bounded.By the Lax-Milgram theorem, (16) is well-posed. The proof is complete. (cid:3) Lemma 8. (c.f. [42]) For ∀ w ∈ ( H ( Ω ) ∩ H k ( Ω )) ( k = , , there exists a positive constantC such that (18) inf v h ∈ ( B h ) | w − v h | , h ≤ ch k − | w | k , Ω . Lemma 9.
Assume T h be a shape regular mesh over Ω with mesh size h. There exists aconstant C > such that for ∀ w ∈ ( H ( Ω ) (cid:84) H k ( Ω )) ( k = , , it holds that (cid:88) T ∈T h (cid:90) ∂ T ( σ ( ∇ · σ ( w )) n ) · v h ds (cid:54) Ch k − | w | k , Ω | v h | , Ω . Proof.
The proof follows the same idea in Lemma 15 of [42], we ignore the details. (cid:3)
Lemma 10.
Under the assumption of Lemma 9, there exists a constant C > such that for ∀ w ∈ ( H ( Ω ) (cid:84) H k ( Ω )) ( k = , , it holds that (cid:88) T ∈T h µ (cid:90) ∂ T ( ∇· σ ( w )) · ( ε ( v h ) · n ) ds + (cid:88) T ∈T h λ (cid:90) ∂ T ( ∇ · σ ( w )) · ( ∇ · v h ) · n ds (cid:54) Ch k − | w | k , Ω | v h | , T . Proof.
The proof follows the same idea in Lemma 15 and 16 of [42], we ignore the details. (cid:3)
Theorem 11.
Let w ∈ ( H k ( Ω )) ∩ V ( k = , be the solution of (15) , and w h be thesolution of (16) , respectively. Then (cid:107)∇ · σ ( w − w h ) (cid:107) , Ω (cid:54) Ch k − | w | k , Ω . Proof.
By Strang lemma,(19) (cid:107)∇· σ ( w − w h ) (cid:107) , Ω (cid:54) C inf v h ∈ ( B h ) (cid:107)∇ · σ ( w − w h ) (cid:107) , Ω + sup v h (cid:44) ∈ ( B h ) ( ∇ · σ ( w ) , ∇ · σ ( v h )) − ( f , v h ) (cid:107)∇ · σ ( v h ) (cid:107) , D . The approximation error estimate follows by Lemma 8. Next we consider the consistencyerror. For ∀ v h ∈ ( B h ) , by (15),( f , v h ) = ( ∇ · σ ( ∇ · σ ( w )) , v h ) = Σ T ∈T h (cid:90) T ( ∇ · σ ( ∇ · σ ( w ))) · v h dxdy . (20) SIMPLE LOW-DEGREE OPTIMAL FINITE ELEMENT SCHEME FOR THE ELASTIC TRANSMISSION EIGENVALUE PROBLEM7
We focus on an element T ∈ T h and use the Green formula, then (cid:90) T ( ∇ · σ ( ∇ · σ ( w ))) · v h dxdy = (cid:90) ∂ T ( σ ( ∇ · σ ( w )) n ) · v h ds − (cid:90) T σ ( ∇ · σ ( w )) : ∇ v h dxdy = (cid:90) ∂ T ( σ ( ∇ · σ ( w )) n ) · v h ds − µ (cid:90) ∂ T ( ∇ · σ ( w )) · ( ε ( v h ) · n ) ds − λ (cid:90) ∂ T ( ∇ · σ ( w )) ( ∇ · v h ) · n ds + (cid:90) T ( ∇ · σ ( w )) · ( ∇ · σ ( v h )) dxdy . (21)Hence, the combination of (20), (21), Lemma 9 and Lemma 10 leads to the desired result. (cid:3) The finite element space B h leads immediately to a high-accuracy scheme for the eigen-value problem of bi-elastic problem (11).4. E rror estimate of the elasticity transmission eigenvalue problem Here we apply the B h scheme to the elastic transmission eigenvalue problem (7). Toanalyse the error estimate, we first define the sesquilinear forms on V × VA τ ( w , ϕ ) = (cid:16) ( ρ − ρ ) − ( ∇ · σ + τρ ) w , ( ∇ · σ + τρ ) ϕ (cid:17) + τ ( ρ w , ϕ )˜ A τ ( w , ϕ ) = (cid:16) ( ρ − ρ ) − ( ∇ · σ + τρ ) w , ( ∇ · σ + τρ ) ϕ (cid:17) + τ ( ρ w , ϕ ) B ( w , ϕ ) = ( σ ( w ) , ∇ ϕ ) . It’s easy to verify that A τ ( w , ϕ ) , ˜ A τ ( w , ϕ ) and B ( w , ϕ ) are symmetric. Using the Greenformula, the variational problem for (7) can be written as to find τ ∈ R and w ∈ V such that A τ ( w , ϕ ) = τ B ( w , ϕ ) , ∀ ϕ ∈ V . (22)for Q ≤ ≤ q ∗ and ˜ A τ ( w , ϕ ) = τ B ( w , ϕ ) , ∀ ϕ ∈ V . (23)for Q ∗ ≤ ≤ q . Lemma 12. [4, 21] Let ρ ( x ) , ρ ( x ) be smooth enough and assume that Q ≤ ≤ q ∗ .Then, A τ is a coercive sesquilinear form on V × V, i.e., there exists a constant γ > suchthat (24) A τ ( w , w ) (cid:62) γ (cid:107) w (cid:107) , ∀ w ∈ V . Under the assumption Q ∗ ≤ ≤ q, we can also obtain that ˜ A τ is a coercive sesquilinearform on V × V, i.e., there exists a constant γ > such that (25) ˜ A τ ( w , w ) (cid:62) γ (cid:107) w (cid:107) , ∀ w ∈ V . Besides, the bilinear form B ( · , · ) is symmetric and nonnegative on V × V. In the following, we take the case Q ≤ ≤ q ∗ for illustration. For the case Q ∗ ≤ ≤ q ,it follows similarly. For a fixed point τ , we define the following generalized eigenvalueproblem: Find ( λ ( τ ) , w ) ∈ R × V such that B ( w , w ) = A τ ( w , v ) = λ ( τ ) B ( w , v ) , ∀ v ∈ V . (26)The original eigenvalue problem (7) can be converted to solve the zero point of the nonlin-ear function(27) f ( τ ) = λ ( τ ) − τ. YINGXIA XI, XIA JI, AND SHUO ZHANG
Furthermore, from the definitions of A τ ( · , · ), f ( τ ) is continuous corresponding to τ basedon the eigenvalue perturbation theory (c.f. [2, 3]) and it’s easy to verify the existence ofzero points (c.f. [21]). In this paper, we consider the numerical method for (27).By B h scheme, the corresponding discretization form is to find ( τ h , w h ) ∈ R × ( B h ) such that B ( w h , w h ) = A τ h , h ( w h , ϕ h ) = τ h B h ( w h , ϕ h ) , ∀ ϕ h ∈ ( B h ) . (28)Similar to (26), we define the discretized generalized eigenvalue problem: Find ( λ h ( τ ) , w h ) ∈R × ( B h ) such that B h ( w h , w h ) = A τ, h ( w h , ϕ h ) = λ h ( τ ) B h ( w h , ϕ h ) , ∀ ϕ h ∈ ( B h ) . (29)The corresponding discretized nonlinear function is(30) f h ( τ ) = λ h ( τ ) − τ. Lemma 13.
In terms of (26) and (29), let w ∈ ( H ( Ω )) ∩ V, then we can obtain thefollowing approximate result (31) | λ ( τ ) − λ h ( τ ) | < Ch . Proof.
Based on the proof of theorem 11, we can prove this lemma similarly. (cid:3)
The following lemma states that the root of (30) approximates the root of (27) well ifthe mesh size is small enough.
Lemma 14. ( [21]) Let f ( τ ) and f h ( τ ) be two continuous functions. For small enough (cid:15) > , there exists some η > such that f (cid:48) ( τ ) (cid:54) − η < and | f ( τ ) − f h ( τ ) | < ε on [ x − (cid:15)δ , x + (cid:15)δ ] , for some < x < x , δ > is constant. If there exists (cid:101) τ ∈ [ x , x ] such thatf h ( (cid:101) τ ) = , then there exists a τ ∗ such that f ( τ ∗ ) = and holds the following approximateformula (32) | (cid:101) τ − τ ∗ | < (cid:15)δ . Further, combining Lemma 13 with Lemma 14, we can obtain(33) | (cid:101) τ − τ ∗ | < Ch , which implies that the convergence rate of elastic transmission eigenvalues using B h is 4.Assume that τ h is the approximation of the exact eigenvalue τ satisfying (22). Be-fore giving the convergence analysis of the eigenfunction approximation, similar to theHelmholtz transmission eigenvalue in [25], we introduce an auxiliary eigenvalue problem:Find (˜ τ, ˜ w ) ∈ R × V such that B ( ˜ w , ˜ w ) = A τ h ( ˜ w , ϕ ) = ˜ τ B ( ˜ w , ϕ ) , ∀ ϕ ∈ V . (34)For (22) and (34), using the standard theory of operator perturbation [25,37], we can obtainthe following approximate results. Lemma 15.
Assume that τ h and τ satisfy | τ − τ h | < Ch . Let ( τ, w ) , (˜ τ, ˜ w ) be the exactsolution of (22) and (34), respectively. Let w ∈ ( H ( Ω )) ∩ V, then, we can obtain thefollowing estimations (cid:107)∇ · σ ( w − ˜ w ) (cid:107) , Ω (cid:46) h , (35) | τ − ˜ τ | (cid:46) h . (36) SIMPLE LOW-DEGREE OPTIMAL FINITE ELEMENT SCHEME FOR THE ELASTIC TRANSMISSION EIGENVALUE PROBLEM9
Using Lemma 15 and the triangle inequality, we can obtain the following error esti-mates.
Theorem 16.
Let ( τ, w ) , ( τ h , w h ) be the solution of (22) and (28), respectively. Let w ∈ ( H ( Ω )) ∩ V and the domain Ω be convex. Then, there exist the following results (cid:107)∇ · σ ( w − w h ) (cid:107) , Ω (cid:46) h , (37) | τ − τ h | (cid:46) h . (38) Proof.
The estimation (38) follows from Lemma 14. Actually, we can regard (28) asthe discretized form of linear eigenvalue problem (34). Combining theorem 11 and theclassical theory of nonconforming finite element method (c.f. [3]), it’s easy to verify (cid:107)∇ · σ ( ˜ w − w h ) (cid:107) , Ω (cid:46) h , (39)Further, combining (35) with (39), we can obtain (37). (cid:3)
5. N umerical E xamples In this section, we give the numerical examples for the bi-elastic problem first, bothsource problem and eigenvalue problem are considered. What’s more, the comparisonwith Morley element are presented. Then we discuss the numerical examples for the elas-tic transmission eigenvalue problem and also its comparison with Morley element. Allexamples are done using Matlab 2016a on a laptop with 16G memory and 2.9GHz IntelCore i7-7500U processor.5.1.
The bi-elastic source problem.
In subsection 5.1 and 5.2, we use the initial meshsize h = . Five levels of uniformly refined triangular meshes are generated for thenumerical experiment and h k = h k − , k = , , , ,
5, and u h k represents the numericalsolution on mesh h k . Example 1 : We consider the boundary problem (15) on the unit square domain Ω = [0 , with the constant coe ffi cient β ( x ) = λ = , µ = . Theright-hand item f in (15) is assumed as f = π sin ( π x )256 (663 cos ( π x ) cos ( π x ) − cos ( π x ) cos ( π x ) + cos ( π x ) cos ( π x ) − cos ( π x ) − cos ( π x ) + , f = π sin ( π x )256 (663 cos ( π x ) cos ( π x ) − cos ( π x ) cos ( π x ) + cos ( π x ) cos ( π x ) − cos ( π x ) − cos ( π x ) + . The exact solution is w = ( sin ( π x ) sin ( π x ) , sin ( π x ) sin ( π x ) ) T .The finest degrees of freedom on h are 50182. On each mesh level, we compute theerror between the numerical solution and the exact solution measured by L , H , H normsrespectively. And the convergence rate is computed by log (cid:32) (cid:107) u − u h k (cid:107)(cid:107) u − u h k − (cid:107) (cid:33) , k = , , , , The numerical results are showed in the left graph of Figure 1. We can observe that(1) The convergence rate for source problem measured by H norm is O ( h );(2) The convergence rate for source problem measured by H norm is O ( h );(3) The convergence rate for source problem measured by L norm is O ( h );which are optimal and consistent with the theoretical results. -3 -2 -1 Size of mesh -8 -6 -4 -2 E rr o r s Convergence rates for source problem on square domain by L normby h normby h normslope=4slope=3slope=2 -3 -2 -1 Size of mesh -12 -10 -8 -6 -4 -2 E rr o r s Convergence rates for source problem on triangle by L normby h normby h normslope=4slope=3slope=2 F igure
1. The convergence rate for the bi-elastic source problem. Left:for
Example 1 . Right: for
Example 2 . Example 2 : We consider (15) on the triangle domain Ω whose vertices are (0 , , (1 , , ffi cient β = + x − x and Lamé parameters λ = , µ = . Theright-hand item f is taken as f = (49 x ) / + (289 x x ) / + x + (123 x x ) / + x x − (345 x ) / − (149 x x ) / + x x − (1425 x x ) / − x + x − x + x , f = x + (149 x x ) / + x − (123 x x ) / + x x − x − (289 x x ) / + x x − (1551 x x ) / + x − (49 x ) / + x − (327 x ) / . The exact solution is w = ( x x ( x + x − , x x ( x + x − ) T .The finest degrees of freedom on h are 25350. The convergence behaviour is showedin the right graph of Figure 1. The convergence orders are also optimal.5.2. The bi-elastic eigenvalue problem.
In this part, we consider the bi-elastic eigenvalueproblem (11).
Example 3 : We test on the square domain D = [0 , and take the Lamé parameters λ = , µ = , the constant coe ffi cient β = Example 4 : We consider the bi-elastic eigenvalue problem on square domain D = [0 , with Lamé parameters λ = , µ = and the non-constant coe ffi cient β ( x ) = + x − x .The convergence rate for the lowest six eigenvalues is showed in Table 2. The theoreticalfourth-order convergence rate can be observed. Example 5 : We consider the bi-elastic eigenvalue problem on the triangle domainwhose vertices are (0 , , (1 , , ( , √ ) with Lamé parameters λ = , µ = and thenon-constant coe ffi cient β = + x + x . SIMPLE LOW-DEGREE OPTIMAL FINITE ELEMENT SCHEME FOR THE ELASTIC TRANSMISSION EIGENVALUE PROBLEM11 T able
1. The performance of B h for Example 3 .Mesh 1 2 3 4 5 Trend
Ord λ λ (cid:38) λ (cid:38) λ (cid:38) λ (cid:38) λ (cid:38) λ (cid:38) able
2. The performance of B h for Example 4 .Mesh 1 2 3 4 5 Trend
Ord λ λ (cid:38) λ (cid:38) λ (cid:38) λ (cid:38) λ (cid:38) λ (cid:38) able
3. The performance of B h for Example 5 . Mesh 1 2 3 4 5 Trend
Ord λ λ (cid:38) λ (cid:38) λ (cid:38) λ (cid:38) λ (cid:38) λ (cid:38) Comparison with the Morley Element Scheme.
We check the Morley elementscheme for the eigenvalue problem (11). Since ( β ∇ · σ ( w ) , ∇ · σ ( ϕ )) is not coercive onMorley element space. From (13), we can obtain that (cid:16) ∇ · σ ( w ) , ∇ · σ ( ϕ ) (cid:17) = ( λ + λµ ) (cid:16) ∇∇ · w , ∇∇ · ϕ (cid:17) + µ ( ∆ w , ∆ ϕ ) . Hence, under the assumption that 0 < ˆ α <
1, the higher order variational formulation itemcan be transfered into (cid:16) ∇ · σ ( w ) , ∇ · σ ( ϕ ) (cid:17) = (cid:16) ∇ · σ ( w ) , ∇ · σ ( ϕ ) (cid:17) − ˆ αµ ( ∇ w , ∇ ϕ ) + ˆ αµ ( ∇ w , ∇ ϕ ) = (1 − ˆ α ) (cid:16) ∇ · σ ( w ) , ∇ · σ ( ϕ ) (cid:17) + ˆ α (cid:16) ∇ · σ ( w ) , ∇ · σ ( ϕ ) (cid:17) − ˆ αµ ( ∇ w , ∇ ϕ ) + ˆ αµ ( ∇ w , ∇ ϕ ) = (1 − ˆ α ) (cid:16) ∇ · σ ( w ) , ∇ · σ ( ϕ ) (cid:17) + ˆ αµ ( ∆ w , ∆ ϕ ) + ˆ α ( λ + λµ )( ∇∇ · w , ∇∇ · ϕ ) − ˆ αµ ( ∇ w , ∇ ϕ ) + ˆ αµ ( ∇ w , ∇ ϕ ) = (1 − ˆ α ) (cid:16) ∇ · σ ( w ) , ∇ · σ ( ϕ ) (cid:17) + ˆ α ( λ + λµ )( ∇∇ · w , ∇∇ · ϕ ) + ˆ αµ ( ∇ w , ∇ ϕ ) , The sequence number of eigenvalues T he e i gen v a l ue s unit square with =1/4, =1/16, - =8+x -x h=0.05h=0.025h=0.0125 , =6 , =1 , =0.1 The sequence number of eigenvalues T he e i gen v a l ue s triangle domain with =1/4, =1/4, - =4+x +x h=1/16h=1/32 h=1/64 , =3.6 , =1 , =0.1 F igure
2. The numerical performance by Morley element for bi-Elasticeigenvalue problem with non-constant coe ffi cient. Left: for λ = , µ = with β ( x ) = + x − x on unit square; Right: for λ = µ = with β ( x ) = + x + x on triangle domian.where ( ∇ w , ∇ ϕ ) = (cid:90) Ω (cid:88) s , t = ∂ w ∂ x s ∂ x t ∂ ϕ ∂ x s ∂ x t dx + (cid:90) Ω (cid:88) s , t = ∂ w ∂ x s ∂ x t ∂ ϕ ∂ x s ∂ x t dx . For Morley element, we consider the following variational formulation: find ( K , w ) ∈ C× V ,for ∀ ϕ ∈ V , we have (40) (cid:16) β ( x ) ∇· σ ( w ) , ∇· σ ( ϕ ) (cid:17) = (cid:16) ( β ( x ) − ˜ α ) ∇· σ ( w ) , ∇· σ ( ϕ ) (cid:17) + ˜ αµ ( ∇ w , ∇ ϕ ) + ˜ α ( λ + λµ ) (cid:16) ∇∇· w , ∇∇· ϕ (cid:17) = K ( w , ϕ ) . where 0 < ˜ α < β min is constant.The items in the middle of (40) are used to guarantee the coercivity of variational prob-lem.The Morley element discretization space for H ( Ω ) is denoted by V Mh . The correspond-ing discretized variational formulation is: Find w h ∈ ( V Mh ) and K h ∈ C , such that (41) (cid:16) ( β ( x ) − α ) ∇· σ ( w h ) , ∇· σ ( ϕ h ) (cid:17) + αµ ( ∇ w h , ∇ ϕ h ) + α ( λ + λµ ) (cid:16) ∇∇· w h , ∇∇· ϕ h (cid:17) = K h ( w h , ϕ h ) , ∀ ϕ h ∈ ( V Mh ) , where 0 < α < β min is a constant. We test the Morley element method on Example 4 and
Example 5 . For
Example 4 , the lowest ten computed eigenvalues on three successive gridlevels are showed in the left figure of Figure 2. We can observe that the numerical resultsare sensitive to the parameter α . For Example 5 , the numerical results are showed in theright figure of Figure 2. For di ff erent parameter α , the computed eigenvalues are di ff erent.For di ff erent β ( x ), the optimal α is also di ff erent.5.4. The elastic transmission eigenvalue problem.
Here we focus on the case ρ ( x ) ≤ Q ≤ ≤ q ∗ ≤ ρ ( x ). For the case ρ ( x ) ≤ Q ∗ ≤ ≤ q ≤ ρ ( x ), it follows analogously.We present some numerical results using three domains: the unit square Ω = [0 , , atriangle domain Ω whose vertices are (0 , , (1 , , ( , √ ) and an L-shaped domain givenby Ω = (0 , × (0 , \ [1 / , × [1 / , h = . Fivelevels of uniformly refined triangular meshes are generated for numerical experiments and h k = h k − / k = , , , h k are denoted SIMPLE LOW-DEGREE OPTIMAL FINITE ELEMENT SCHEME FOR THE ELASTIC TRANSMISSION EIGENVALUE PROBLEM13
Eigenvalue
Example 6 Example 7 Example 8 Example 9 Λ Λ + Λ Λ Λ Λ Λ Λ Λ + Λ able
4. The first ten elastic transmission eigenvalues on the finest meshlevel of h ≈ . λ h k ( k = , , , , log ( | λ h l − λ h λ h l + − λ h | ) , l = , , . We consider the following examples.
Example 6.
The unit square domain Ω with Lamé parameters µ = / , λ = / ρ ( x ) = , ρ ( x ) = Example 7.
The unit square domain Ω with Lamé parameters µ = / λ = /
12 andthe mass density ρ ( x ) = , ρ ( x ) = + x − x . Example 8.
The triangle domain Ω with Lamé parameters µ = / , λ = /
16 and themass density ρ ( x ) = , ρ ( x ) = + x + x . Example 9.
The L-shaped domain Ω with Lamé parameters µ = / , λ = /
16 andthe mass density ρ ( x ) = , ρ ( x ) = ω in [36]. For Example 6 , Example 7 , Example 8 , the finest degreesof freedom are 23564. The convergence order of the lowest six eigenvalues are showed inFigure 3, Figure 4, Figure 5, respectively. The convergence rate on unit square or triangledomain is approximately 4 which is optimal and consistent with the theoretical analysis.For
Example 9 , the finest degrees of freedom are 47628. The convergence rate on L-shaped domain is showed in Figure 6 which is lower than 4 due to the fact that reentrantcorner leads to the low regularity of eigenfunctions which is consistent with the resultsin [21, 36].5.5.
Comparision with the Morley scheme.
In this subsection, we check the Morleyelement scheme for the elastic transmission eigenvalue problem (7). We take the case ρ ( x ) ≤ Q ≤ ≤ q ∗ ≤ ρ ( x ) for illustration. For the case ρ ( x ) ≤ Q ∗ ≤ ≤ q ≤ ρ ( x ), itcan be treated analogously. For Morley element, we consider the expansion of varitionalformulation (7): find τ and w (cid:44) ∈ V such that, for ∀ ϕ ∈ V (43) (cid:16) ( ρ − ρ ) − ∇ · σ ( w ) , ∇ · σ ( ϕ ) (cid:17) + τ (cid:32) ρ ρ − ρ w , ∇ · σ ( ϕ ) (cid:33) + τ (cid:32) ρ ρ − ρ ∇ · σ ( w ) , ϕ (cid:33) + τ (cid:32) ρ ρ ρ − ρ w , ϕ (cid:33) = . In order to guarantee the coerciveness on the Morley finite element space, for the non-constant coe ffi cient, we assume 0 < α ≤ ρ min (cid:44) min ρ ( x ) − ρ ( x ) and transform the higher -2 -1 Size of mesh -4 -3 -2 -1 E rr o r s o f e i gen v a l ue s Convergence rates for Example 6 F igure
3. The convergence rates for the lowest six real eigenvalues for
Example 6 by B h . Y-axis means the numerical error of eigenvalues;X-axis means the size of mesh. -2 -1 Size of mesh -4 -3 -2 -1 E rr o r s o f e i gen v a l ue s Convergence rates for Example 7 F igure
4. The convergence rates for the lowest six real eigenvalues for
Example 7 by B h . Y-axis means the numerical error of eigenvalues;X-axis means the size of mesh.order variational formulation to the following form: (44) (cid:16) ( ρ − ρ ) − ∇· σ ( w ) , ∇· σ ( ϕ ) (cid:17) = (cid:16) (( ρ − ρ ) − − α ) ∇· σ ( w ) , ∇· σ ( ϕ ) (cid:17) + αµ ( ∇ w , ∇ ϕ ) + α ( λ + λµ ) (cid:16) ∇∇· w , ∇∇· ϕ (cid:17) . The form on the right-hand side of (44) guarantees the coercivity of the variational formu-lation on V . However, in the practical computation, the numerical performance is sensitiveto the choice of α . The left figure in Figure 7 shows the numerical performance by Morleyelement for unit square domain Ω = [0 , with the co ffi cients λ = / , µ = / , ρ ( x ) = , ρ ( x ) =
3. For a fixed α , we present the lowest 10 computed eigenvalues on threesuccessive grid levels. It’s observed that the numerical results are greatly dependent on the SIMPLE LOW-DEGREE OPTIMAL FINITE ELEMENT SCHEME FOR THE ELASTIC TRANSMISSION EIGENVALUE PROBLEM15 -2 -1 Size of mesh -4 -3 -2 -1 E rr o r s o f e i gen v a l ue s Convergence rates for Example 8 F igure
5. The convergence rates for the lowest six real eigenvalues for
Example 8 by B h . Y-axis means the numerical error of eigenvalues;X-axis means the size of mesh. -2 -1 Size of mesh -3 -2 -1 E rr o r s o f e i gen v a l ue s Convergence rates for Example 9 F igure
6. The convergence rates for the lowest six real eigenvalues for
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Optimal piecewise cubic finite element schemes for the biharmonic equation on general triangu-lations , arXiv: 1903.04897.S chool of S cience , N anjing U niversity of S cience and T echnology , N anjing hina . Email address : [email protected] S chool of M athematics and S tatistics , B eijing I nstitute of T echnology , B eijing hina ., B eijing K ey L aboratory on MCAACI, B eijing I nstitute of T echnology , B eijing hina . Email address : [email protected] LSEC, I nstitute of C omputational M athematics and S cientific/ E ngineering C omputing , A cademy of M athe - matics and S ystem S ciences , C hinese A cademy of S ciences , B eijing eople ’ s R epublic of C hina Email address ::