Enhancing eigenvalue approximation by gradient recovery on adaptive meshes

Transcription

1 IMA Journal of Numerical Analysis Advance Access published October 29, 2008 IMA Journal of Numerical Analysis Page 1 of 15 doi: /imanum/drn050 Enhancing eigenvalue approximation by gradient recovery on adaptive meshes HAIJUN WU Department of Mathematics, Nanjing University, Jiangsu , People s Republic of China AND ZHIMIN ZHANG Department of Mathematics, Wayne State University, Detroit, MI 48202, USA [Received on 16 September 2007; revised on 14 June 2008] We utilize the recovered gradient by the polynomial-preserving recovery to enhance the eigenvalue approximation of the Laplace operator under adaptive meshes. Superconvergence rate is established and numerical tests on benchmark problems support our theoretical findings. Keywords: adaptive finite-element method; eigenvalue; superconvergence; gradient recovery. 1. Introduction Gradient recovery has been widely used for a posteriori error estimates (see Ainsworth & Oden, 2000; Babuška & Strouboulis, 2001; Chen & Xu, 2007; Fierro & Veeser, 2006; Zhang, 2007; Zienkiewicz et al., 2005; Zienkiewicz & Zhu, 1987, 1992a,b). Recently, it has been employed to enhance the eigenvalue approximations by the finite-element method under certain mesh conditions (see Naga et al., 2006; Shen & Zhou, 2006). However, it was unclear whether the gradient recovery is effective for adaptive meshes. In this paper we prove that under adaptive meshes the recovered gradient still helps to improve the convergence rate of the eigenvalue approximation. Our proof relies on the superconvergence result for adaptive meshes established in Wu & Zhang (2007). Furthermore, we demonstrate the effectiveness of the gradient recovery in enhancing eigenvalue approximation by two typical numerical examples: the Laplace operator on the L-shaped domain and the cracked domain. In both cases, we observe superconvergence rates for eigenvalue approximation. We would like to indicate that the enhancement can be applied to more general problems, see, e.g., Shen & Zhou (2006). Likewise, our results on adaptive meshes in this paper can be generalized. Due to the nonquasi-uniform nature of adaptive meshes, error estimates depend on the total degrees of freedom N, rather than the mesh size h (see Bangerth & Rannacher, 2003; Binev et al., 2004; Dörfler, 1996; Morin et al., 2002; Verfürth, 1995). Therefore, the standard analysis for h-version finite-element eigenvalue approximation cannot be used directly and some modifications are needed. Corresponding author. hjw@nju.edu.cn ag7761@wayne.edu c The author Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

2 2 of 15 H. WU AND Z. ZHANG A main message we would like to deliver here is that any finite-element eigenvalue code could use recovery techniques to improve the accuracy of eigenvalue approximation. In addition, the recovery procedure needs only the computed data from a finite-element solver and the cost is of order O(N). As for a posteriori error estimates for eigenvalue problems, there have been some recent works based on residual-type error estimators, see e.g. Duran et al. (1999, 2003) and Larson (2000); also see Noël (2004) for the eigenvalue problem associated with the Schrödinger operator and Walsh et al. (2007) for heterogeneous elastic structures. We consider a model eigenvalue problem: find (u, λ) H0 1(Ω) R with u L 2 (Ω) = 1 such that a(u, v) := u v = λ uv = λ(u, v) v H0 1 (Ω), (1.1) Ω Ω where Ω R 2 is a bounded polygonal domain with boundary Ω. It is well known that (1.1) has countably infinitely many positive eigenvalues (see Kuttler & Sigillito, 1984): 0 < λ 1 λ 2 with no finite accumulation point. Furthermore, the associated eigenfunctions u 1, u 2,... form a complete orthonormal basis for L 2 (Ω), i.e. (u i, u j ) = δ i j, where δ i j is the Kronecker delta symbol. At a corner of Ω with interior angle π/κ, for κ 1/2, it is well known that there exists a constant β j such that (see Kuttler & Sigillito, 1984) u j (r, θ) = β j r κ sin κθ + o(r κ ) as r 0. We want to approximate the first l eigenvalues λ 1 λ 2 λ l. Although it is possible that the eigenfunctions have singularities at more than one vertex, in this paper we consider only the case of one singular point. Suppose that the origin O is a vertex of Ω, then each of the first l eigenfunctions u 1, u 2,..., u l may have a singularity at the origin O and can be decomposed as a sum of a singular part and a smooth part (Grisvard, 1992): where m v j δ m x i y m i r and u j = v j + w j, j = 1, 2,..., l, (1.2) m w j x i y m i 1, m = 1,..., k + 2, i = 0,..., m, (1.3) and δ is a constant that depends on the interior angle of the corner. Here k = 1 for the linear finite element and k = 2 for the quadratic finite element. We assume that 0 < δ < k + 1, which means that for higher-order elements (larger k) we need higher regularity assumptions on the regular part w j in the decomposition (1.2) and, consequently, we need to include more terms in the singular part v j. Therefore the condition 0 < δ < k + 1 by no means enforces any restriction on the corner angles in terms of k. Let M h be a regular triangulation of Ω, E h be the set of all interior edges and N h be the set of all nodal points. Assume that the origin O N h and any triangle τ M h is considered as closed. Let Vh k = {v h: v h H0 1(Ω), v h τ P k (τ)}, k = 1, 2, be the conforming finite-element space associated with M h. Here P k denotes the set of polynomials with degree k. The finite-element method for (1.1) reads as follows: find (u h, λ h ) U h R with u h L 2 (Ω) = 1 such that a(u h, v h ) = λ h u h v h v h Vh k. (1.4) Ω

3 ENHANCING EIGENVALUE APPROXIMATION BY GRADIENT RECOVERY ON ADAPTIVE MESHES 3 of 15 The eigenvalues and eigenfunctions of the finite-element approximations (1.4) are λ 1h λ 2h λ Nh ; u 1h, u 2h,..., u Nh ; (u ih, u jh ) = δ i j. We consider an adaptive mesh and use the total degrees of freedom N (instead of the maximum mesh size h) to measure the rate of convergence. However, for notational convenience, we are still using h as an index. Throughout the paper, we use the notation A B to represent the inequality A constant B, where the constant depends only on the minimum angle of the triangles in M h, the constant δ and the domain Ω. The notation A B is equivalent to A B and B A. 2. Preliminary 2.1 Mesh quality We first introduce some notations (see Fig. 1). For an edge e E h, which is shared by two elements τ and τ, let Ω e = τ τ be the patch of e, h e denote the length of e and r e the distance from the origin O to the midpoint of e. For any τ M h, we denote by h τ its diameter and by r τ the distance from the origin to the barycentre of τ. Given an interior edge e E h, we say that Ω e is an ε approximate parallelogram if the lengths of any two opposite edges differ only by ε. DEFINITION A family of triangulations {M h } is said to satisfy the condition (α, σ, μ) if there exist constants α > 0, 0 σ < 1 and μ > 0 such that the interior edges can be separated into two parts E h = E 1,h E 2,h : Ω e forms an ε e parallelogram with ε e he 1+α /re α+μ(1 α) for e E 1,h and the number of edges in E 2,h satisfies #E 2,h N σ. REMARK 2.1 The meaning of condition (α, σ, μ) is the following. The edges can be grouped into good (E 1,h ) and bad (E 2,h ), where the number of bad edges are much smaller than that of good edges. The ratio is #E 2,h N σ #E 1,h N = 1 N 1 σ. When r e = O(1), i.e. an edge e is far away from the singular point O, more restrictions are put on the triangle pair with the common edge e. The mesh condition requires that the two triangles form an O(h 1+α e ) FIG. 1. Notation in the patch Ω e.

4 4 of 15 H. WU AND Z. ZHANG parallelogram, which is the same as in previous works (see Chen & Huang, 1995; Lakhany et al., 2000; Li & Zhang, 1999; Huang & Xu, 2008; Naga & Zhang, 2004, 2005; Naga et al., 2006; Wahlbin, 1995; Xu & Zhang, 2003; Zhu & Lin, 1989). When e is in a neighbourhood of O, if h e re 1+μ(1 α)/α then the condition (α, σ, μ) is fulfilled by those edges such that Ω e is an O(h e ) parallelogram, which means that there is no restriction at all. Roughly speaking, the number of edges in E 1,h that have no restriction imposed is O(N 1 α ) if h τ rτ 1 μ h μ for any τ M h. Here h and μ are positive constants. An explanation is given in Wu & Zhang (2007). We see from the above discussion that the closer we are to the singular point, the less the restriction that is imposed on the mesh. As a matter of fact, for an adaptively refined mesh, the closer we are to the singular point, the worse the mesh quality is, in the sense of forming parallelograms. In Section 4 we demonstrate that the aforementioned mesh condition is satisfied by actual adaptive meshes near the origin on two benchmark problems: the eigenvalue problem on a unit sector domain with angle 3π/2 and the unit disk domain with a crack. The following lemma provides an estimate for the total degrees of freedom N when the mesh M h satisfies h τ rτ 1 μ h μ. The proof can be found in Wu & Zhang (2007). LEMMA 2.2 Assume that h τ rτ 1 μ h μ for any τ M h, where h and μ are positive constants. Then the number of degrees of freedom N of the finite-element equation (1.4) satisfies N 1. (2.1) h2μ REMARK 2.3 The condition h τ rτ 1 μ h μ can be viewed as a discrete mesh density function. The positive number h min τ Mh h τ is the size of the minimum element. For an element τ in the neighbourhood h μ implies that h τ h. Roughly speaking, the condition h τ rτ 1 μ h μ indicates that the triangles in M h are distributed according to the circles with radii m 1/μ h and common centre the origin, where m = 1, 2, 3,.... In the rest of the paper we choose μ = δ/2 for linear elements and μ = δ/3 for quadratic elements based on the results of Wu & Zhang (2007). of O, we have r τ h τ and the condition h τ r 1 μ τ 2.2 The gradient recovery operator G h Given a node z N h, we select n m = (k + 2)(k + 3)/2 sampling points z j N h, j = 1, 2,..., n, in an element patch ω z containing z (z is one of the z j ), and fit a polynomial of degree k + 1, in the least squares sense, with values of u h at those sampling points. In other words, we are looking for p k+1 P k+1 such that n (p k+1 u h ) 2 (z j ) = min j=1 q P k+1 j=1 The recovered gradient in the neighbourhood of z is then defined as n (q u h ) 2 (z j ). (2.2) G h u h = p k+1. (2.3) It was proved in Naga & Zhang (2004, 2005) that the above least squares fitting procedure has a unique solution under some mild geometric conditions. For linear elements, this condition is that n sampling

5 ENHANCING EIGENVALUE APPROXIMATION BY GRADIENT RECOVERY ON ADAPTIVE MESHES 5 of 15 points are not on the same conic curve. Furthermore, the gradient recovery operator G h : C(Ω) Vh k V h k, k = 1 or 2, has the following properties (see Naga & Zhang, 2004; Wu & Zhang, 2007; Zhang, 2004; Zhang & Naga, 2005): (i) G h v h L 2 (Ω) v h L 2 (Ω) v h V k h ; (ii) for any nodal point z, (G h p)(z) = p(z) if p P k+1 (ω z ); (iii) (G h φ)(z) 1 max φ(z ) for any node z in an element τ M h ; h τ z N h ω z (iv) G h φ = G h I k h φ. Here, Ih k: H 0 1(Ω) V h k is the standard Lagrange interpolation operator at element vertices and edge centres. We further introduce the elliptic projector P h : H0 1(Ω) V h k such that a(φ, v h ) = a(p h φ, v h ) v h V k h. (2.4) Let û jh = P h u j be the elliptic projection of the eigenfunction u j, j = 1, 2,..., l. The following superconvergences of G h û jh u j L 2 (Ω) and I h ku j û jh L 2 (Ω) are proved in Wu & Zhang (2007, Theorems 3.4, 4.5, 5.3, 5.5). THEOREM 2.4 Let û jh be the elliptic projection of u j onto Vh k, k = 1, 2, j = 1, 2,..., l. Assume that M h satisfies the condition (α, σ, δ/(k + 1)) with 0 < α 1 and 0 σ < 1 and that h τ h δ/(k+1) for any τ M h. Then, r 1 δ/(k+1) τ 1 + (ln N)1/2 G h û jh u j L 2 (Ω) N k/2+ρ, ( Ih k u 1 + (ln N)1/2 α j û jh L 2 (Ω) N k/2+ρ, ρ = min 2, 1 σ ). (2.5) Approximations of eigenvalues and eigenfunctions We first introduce the estimate of u j û jh L 2 (Ω) from Wu & Zhang (2007, Lemmas 5.7, 5.8). THEOREM 2.5 Let û jh be the elliptic projection of u j onto V k h, k = 1, 2. Assume that h τ r 1 δ/(k+1) τ h δ/(k+1) for any τ M h. Then, u j û jh L 2 (Ω) 1, j = 1, 2,..., l. N k/2 Henceforth, we assume that l < N. Next, we set U j = span{u 1, u 2,..., u j } and define S j as the unit sphere of U j in L 2 (Ω). By using Theorem 2.5 and the standard argument for finite-element approximations of eigenvalue problems (see, e.g., Ern & Guermond, 2004), we have the following error estimates. The proofs are omitted. THEOREM 2.6 Assume that h τ rτ 1 δ/(k+1) h δ/(k+1) for any τ M h, k = 1, 2. Then, there exists an N 0 such that, for all N > N 0, j = 1, 2,..., l, 0 λ jh λ j max v S j inf v h V k h v v h 2 L 2 (Ω), (2.6)

6 6 of 15 H. WU AND Z. ZHANG u j u jh L 2 (Ω) max v S j inf v h V k h v v h L 2 (Ω), (2.7) u j u jh L 2 (Ω) u j û jh L 2 (Ω). (2.8) 3. Asymptotically exact error estimates for λ jh λ j and eigenvalue enhancement The following identity is crucial for our method (see Strang & Fix, 1973, Lemma 6.3; Babuška & Osborn, 1991, Lemma 9.1): λ jh λ j = (u j u jh ) 2 L 2 (Ω) λ j u j u jh 2 L 2 (Ω). (3.1) We have the following error estimates for λ jh λ j and u j u jh. THEOREM 3.1 Assume that M h satisfies the condition (α, σ, δ/(k +1)) with 0 < α 1 and 0 σ < 1 and that h τ rτ 1 δ/(k+1) h δ/(k+1) for any τ M h, k = 1, 2. Then, there exists an N 0 such that, for all N > N 0, j = 1, 2,..., l, u j u jh L 2 (Ω) 0 λ jh λ j 1 N k, (3.2) u j u jh L 2 (Ω) 1, (3.3) N k/2 ( 1 + (ln N)1/2 α N k/2+ρ, ρ = min 2, 1 σ 2 ). (3.4) Proof. Estimates (3.2) and (3.3) can be proved by Theorems 2.5 and 2.6. We only need to prove (3.4). It is clear that u j û jh L 2 (Ω) u j I k h u j L 2 (Ω) + I k h u j û jh L 2 (Ω). Using the argument in the proof of Lemma 5.7 in Wu & Zhang (2007), we deduce that On the other hand, from the Poincaré inequality, Therefore, from Theorem 2.4, u j I k h u j L 2 (Ω) 1 N (k+1)/2. I k h u j û jh L 2 (Ω) (I k h u j û jh ) L 2 (Ω). u j û jh L 2 (Ω) 1 + (ln N)1/2 N k/2+ρ, (3.5) which implies (3.4) by using (2.8). This completes the proof of the theorem. Next, we prove the superconvergence between u j and G h u jh.

7 ENHANCING EIGENVALUE APPROXIMATION BY GRADIENT RECOVERY ON ADAPTIVE MESHES 7 of 15 THEOREM 3.2 Assume that M h satisfies the condition (α, σ, δ/(k +1)) with 0 < α 1 and 0 σ < 1 and that h τ rτ 1 δ/(k+1) h δ/(k+1) for any τ M h, k = 1, 2. Then there exists an N 0 such that, for all N > N 0, j = 1, 2,..., l, we have ( 1 + (ln N)1/2 α G h u jh u j L 2 (Ω) N k/2+ρ, ρ = min 2, 1 σ ). (3.6) 2 Proof. Recall that û jh = P h u j. It follows from property (i) and Theorem 2.4 that G h u jh u j L 2 (Ω) G hu jh G h û jh L 2 (Ω) + G hû jh u j L 2 (Ω) 1 + (ln N)1/2 (u jh û jh ) L 2 (Ω) + N k/2+ρ. (3.7) Let e h = u jh û jh. From (1.1), (1.4) and the definition of the elliptic projection, we have e h 2 L 2 (Ω) = a(u jh û jh, e h ) = a(u jh u j, e h ) = λ jh u jh e h λ j u j e h By (3.4) and (3.5), we have e h L 2 (Ω) = (λ jh λ j ) u jh e h + λ j (u jh u j )e h Ω Ω [(λ jh λ j ) + λ j u jh u j L 2 (Ω) ] e h L 2 (Ω). 1+(ln N)1/2. Hence it follows from Theorem 3.1 that N k/2+ρ (u jh û jh ) L 2 (Ω) = e 1 + (ln N)1/2 h L 2 (Ω) N k/2+ρ. (3.8) The proof is completed by combining (3.7) and (3.8). Next we define the error estimator for the jth eigenfunction: Then we derive from (3.1) that λ jh λ j η 2 jh (1 = (u j u jh ) 2 L 2 (Ω) η jh = G h u jh u jh L 2 (Ω). (3.9) η 2 jh ) Ω (u j u jh ) 2 L 2 (Ω) λ j u j u jh 2 L 2 (Ω). (3.10) The following theorem proves the asymptotic exactness of the error estimator. THEOREM 3.3 Assume that M h satisfies the condition (α, σ, δ/(k +1)) with 0 < α 1 and 0 σ < 1 and that h τ rτ 1 δ/(k+1) h δ/(k+1) for any τ M h, k = 1, 2. Let 1 N k/2 u j u jh L 2 (Ω). (3.11) Then there exists an N 0 such that, for all N > N 0, j = 1, 2,..., l, we have 1 η 2 jh 1 + (ln N)1/2 (u j u jh ) 2 N ρ, (3.12) L 2 (Ω) 1 λ jh λ j η 2 jh ( 1 + ln N α N ρ, ρ = min 2, 1 σ ). (3.13) 2 Ω

8 8 of 15 H. WU AND Z. ZHANG Proof. From Theorem 3.2 and the triangle inequality, we have η jh (u j u jh ) L 2 (Ω) G hu jh j L 2 (Ω) Therefore (3.12) follows from (3.11). Combining (3.10), (3.12) and Theorem 3.1, we have λ jh λ j η ln N jh N k+ρ, 1 + (ln N)1/2 N k/2+ρ. which implies (3.13) by using (3.11) and (3.12). The inequality (3.13) says that η 2 jh is an asymptotically exact error estimator for λ jh λ j and that λ jh = λ jh η 2 jh = λ jh G h u jh u jh 2 L 2 (Ω) (3.14) is a better approximation of λ j than λ jh under our mesh condition. 4. Implementation and numerical examples The implementation of the adaptive algorithm in this section is based on COMSOL Multiphysics. We define a local a posteriori error estimator on the element τ as η jτ := G h u jh u jh L 2 (τ) and a global error estimator for λ jh : η 2 jh = τ M h η 2 jτ. We use η τ = η 1τ as the indicators for mesh refinements. We now describe the adaptive algorithm used in this paper. Algorithm. Given a tolerance TOL > 0 and an integer l. Generate an initial mesh M h over Ω. While max 1 j l η 2 jh > TOL do the following. Choose a set of elements M h M h such that ητ 2 τ M h 1/2 > 0.7 τ M h η 2 τ Then refine the elements in M h. Denote the new mesh by M h also. Solve the discrete problem (1.4) on M h for λ jh (1 j l) and let λ jh = λ jh η 2 jh. Compute error estimators on M h. End while. The reason for using the indicator associated with the first eigenfunction η τ = η 1τ is that the singularity of u 1 usually dominates the others. 1/2.

9 ENHANCING EIGENVALUE APPROXIMATION BY GRADIENT RECOVERY ON ADAPTIVE MESHES 9 of 15 In the following examples quadratic finite elements are used in the computations. In order to access exact eigenvalues for convergence tests, we use circular domains instead of square domains. Note that our theory only covers polygonal domains. Nevertheless, the theory can be extended to curved domains with some more involved analysis taking into account the effect of curved boundaries. EXAMPLE 4.1 Consider the eigenvalue problem (1.1) on the L-shaped domain Ω = {(r, θ) R 2 : 0 < r < 1, 0 < θ < 3π/2}. The eigenvalues and eigenfunctions for this example are λ j = α 2 j, u j = v j / v j L 2 (Ω), v j = J 2m j /3(α j r) sin(2m j θ/3), where m j is some integer depending on j and α j is a zero of the Bessel function J 2m j /3. From the definition of the Bessel function (cf. Olver, 1974), J ν (z) z ν 2 ν Γ (ν+1) as z 0. Therefore v j = J 2m j /3(α j r) sin(2m j θ/3) has the same singularity as r 2m j /3 sin(2m j θ/3), where m j = 1, 2, 3, 4, 1, 5, 2, 6, 7, 3, 8, 4, 1, 9, 5, 2, 10, 6, 11, 3 for j = 1, 2,..., 20, respectively. Figure 2 plots the initial mesh and the adaptively refined mesh of 1844 elements after 11 adaptive iterations. The final mesh (after 21 adaptive iterations) has elements. The minimum area of the final mesh is Table 1 presents the first 20 exact eigenvalues λ j, 1 j 20, obtained by the secant method, the error between the exact eigenvalue λ j and the eigenvalue approximation λ jh, and the error between the exact eigenvalue λ j and the enhanced eigenvalue approximation λ jh for Example 4.1 after 21 adaptive iterations. We see that the enhanced eigenvalue approximations are accurate to one or two more decimal places than the original eigenvalue approximations. Figure 3 shows the error between the exact eigenvalue λ j and the eigenvalue approximation λ jh, and the error between the exact eigenvalue λ j and the enhanced eigenvalue approximation λ jh for Example 4.1 with j = 1, 10, 20. We observe that λ j λ jh O(N 2 ), j = 1, 10, 20, λ 1 λ 1h O(N 2.3 ), λ 10 λ 10h O(N 2.2 ), λ 20 λ 20h O(N 2.2 ). FIG. 2. The initial mesh (left) and the adaptively refined mesh (right) of 1844 elements after 11 adaptive iterations for Example 4.1.

10 10 of 15 H. WU AND Z. ZHANG TABLE 1 The eigenvalues λ j, 1 j 20, and the errors λ j λ jh and λ j λ jh for Example 4.1 after 21 adaptive iterations j λ j λ j λ jh λ j λ jh FIG. 3. The errors λ j λ jh (left) and λ j λ jh (right), j = 1, 10, 20, versus the degrees of freedom for Example 4.1. Note that the decays of λ j λ jh ( j = 1, 10, 20) are quasi-optimal and the decays of λ j λ jh ( j = 1, 10, 20) are faster with orders of O(N 2.3 ), O(N 2.2 ) and O(N 2.2 ), respectively. EXAMPLE 4.2 Consider the eigenvalue problem (1.1) on the domain with a crack Ω = {(r, θ) R 2 : 0 < r < 1, 0 < θ < 2π}.

11 ENHANCING EIGENVALUE APPROXIMATION BY GRADIENT RECOVERY ON ADAPTIVE MESHES 11 of 15 The eigenvalues and eigenfunctions for this example are λ j = α 2 j, u j = v j / v j L 2 (Ω), v j = J m j /2(α j r) sin(m j θ/2), where m j is some integer depending on j and α j is a zero of the Bessel function J m j /2. Here v j = J m j /2(α j r) sin(m j θ/2) has the same singularity as r m j /2 sin(m j θ/2), where m j = 1, 2, 3, 4, 5, 1, 6, 7, 2, 8, 3, 9, 4, 10, 5, 11, 1, 6, 12, 2 for j = 1, 2,..., 20, respectively. Figure 4 plots the initial mesh and the adaptively refined mesh of 1468 elements after 11 adaptive iterations. The final mesh (after 25 adaptive iterations) has elements. The minimum area of the final mesh is Table 2 presents the first 20 exact eigenvalues λ j, 1 j 20, obtained by the secant method, the error between the exact eigenvalue λ j and the eigenvalue approximation λ jh, and the error between the exact eigenvalue λ j and the enhanced eigenvalue approximation λ jh for Example 4.2 after 25 adaptive iterations. We see that the enhanced eigenvalue approximations are accurate to one or two more decimal places than the original eigenvalue approximations. Figure 5 shows the error between the exact eigenvalue λ j and the eigenvalue approximation λ jh, and the error between the exact eigenvalue λ j and the enhanced eigenvalue approximation λ jh for Example 4.2 with j = 1, 10, 20. We observe that λ j λ jh O(N 2 ), j = 1, 10, 20, λ 1 λ 1h O(N 2.5 ), λ 10 λ 10h O(N 2.3 ), λ 20 λ 20h O(N 2.3 ). Note that the decays of λ j λ jh ( j = 1, 10, 20) are quasi-optimal and the decays of λ j λ jh ( j = h μ for Examples 4.1 and 4.2. Here μ = 2/9 for Example 4.1 and μ = 1/6 for Example 4.2. Since our theoretical results cover only the case when Ω is a polygonal domain, we verify the condition (α, σ, μ) and the mesh density assumption for edges and triangles away from the circular part of the boundary. First, we verify the condition (α, σ, μ) for edges in the circle D 1 of radius 3/4 centred 1, 10, 20) are faster with orders of O(N 2.5 ), O(N 2.3 ) and O(N 2.3 ), respectively. The convergence rates of recovered eigenvalues on the cracked domain are slightly better than those on the L-shaped domain due to the better mesh quality in the former. Finally, we provide numerical verifications of the condition (α, σ, μ) and the mesh density assumption h τ r 1 μ τ FIG. 4. The initial mesh (left) and the adaptively refined mesh (right) of 1844 elements after 11 adaptive iterations for Example 4.1.

12 12 of 15 H. WU AND Z. ZHANG TABLE 2 λ j, 1 j 20, the errors λ j λ jh and λ j λ jh for Example 4.2 after 25 adaptive iterations j λ j λ j λ jh λ j λ jh FIG. 5. λ j λ jh (left) and λ j λ jh (right), j = 1, 10, 20, versus the degrees of freedom for Example 4.2. at the origin. Here, by an edge in a circle, we mean that the midpoint of the edge is in the circle. In our computations, the diameters of triangles are greater than For simplicity, we choose E 1,h to be the set of edges e E h in the circle D 1 such that the patch Ω e forms a approximate parallelogram and E 2,h other edges in D 1. By doing so, we actually select exact parallelograms for e E 1,h, i.e. we regard e as a good edge if Ω e is a parallelogram and a bad edge otherwise. Denote by N he the number of

13 ENHANCING EIGENVALUE APPROXIMATION BY GRADIENT RECOVERY ON ADAPTIVE MESHES 13 of 15 edges e E h in the circle D 1 and by N he2 the number of edges e E 2,h. Figure 6 plots N he2 versus N he for Examples 4.1 and 4.2. It is shown that N he2 (N he ) 0.82 for both examples. Therefore, the meshes in both examples satisfy the condition (α, 0.82, μ) for any α > 0. Next, we verify the mesh density assumption h τ rτ 1 μ h μ for triangles in the circle D 2 of radius 1/2 centred at the origin. Here, by a triangle in a circle, we mean that the centre of the triangle is in the circle. Let C h max and C h min be the maximum and minimum values of the set {h τ /(rτ 1 μ h μ ): τ M h and in D 2 }, respectively. Figure 7 depicts C h max and C h min versus the number of adaptive iterations for Examples 4.1 and 4.2. The maximum and minimum values of C h max /C h min are about 6.35 and 2.63 for Example 4.1 and about 8.15 and 3.39 for Example 4.2. Therefore, the mesh density assumption is satisfied. We remark that, for triangles in the circle D 1 of radius 3/4 centred at the origin, the maximum and minimum values of C h max /C h min are about 9.00 and 3.04 for Example 4.1 and about and 3.98 for Example 4.2. FIG. 6. N he2, the number of bad edges in the circle D 1 with radius 3/4 and centre O, versus N he, the total number of edges in D 1, for Examples 4.1 (left) and 4.2 (right). The dotted lines give the reference slope of FIG. 7. C h max and C h min versus the number of adaptive iterations for Examples 4.1 (left) and 4.2 (right).

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