Startseite The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
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The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering

  • Yu Zhang , Hai Bi und Yidu Yang EMAIL logo
Veröffentlicht/Copyright: 11. Mai 2020
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Open Mathematics
Aus der Zeitschrift Open Mathematics Band 18 Heft 1

Abstract

In this study, for the first time, we discuss the posteriori error estimates and adaptive algorithm for the non-self-adjoint Steklov eigenvalue problem in inverse scattering. The differential operator corresponding to this problem is non-self-adjoint and the associated weak formulation is not H 1-elliptic. Based on the study of Armentano et al. [Appl. Numer. Math. 58 (2008), 593–601], we first introduce error indicators for primal eigenfunction, dual eigenfunction, and eigenvalue. Second, we use Gårding’s inequality and duality technique to give the upper and lower bounds for energy norm of error of finite element eigenfunction, which shows that our indicators are reliable and efficient. Finally, we present numerical results to validate our theoretical analysis.

Keywords: Steklov eigenvalues in inverse scattering; a posteriori error estimates; reliability and efficiency; adaptive algorithm
MSC 2010: 65N25; 65N30

1 Introduction

Steklov eigenvalue problems are extensively applied to surface waves (see, e.g., [1,2]), mechanical oscillators immersed in a viscous fluid (see [3,4]), the vibration modes of a structure in contact with an incompressible fluid (see [5,6]), the inverse scattering theory to reconstruct the index of refraction of inhomogeneous media (see [7]), etc. Many studies have been carried out on numerical methods for the Steklov eigenvalue problems (see, e.g., [6,8,9,10,11,12,13,14,15,16,17,18,19]). In recent years, researchers have paid attention to the Steklov eigenvalue problem in inverse scattering (see, e.g., [7,20,21,22]).

Adaptive finite element methods are favored in the current scientific and engineering computations. For a given tolerance, adaptive finite element methods require less degrees of freedom. So far, many excellent works on the a posteriori error estimates and adaptive algorithm have been summarized in previous studies [23–25]. There are some studies on the a posteriori error estimates and adaptive finite element methods for self-adjoint Steklov eigenvalue problems (see, e.g., [11,12,13,17,26–28]). As far as we know, up to now, there does not exist any study on the a posteriori error estimates and adaptive finite methods for the non-self-adjoint Steklov eigenvalue problem considered in this study. Therefore, deriving the a posteriori error estimates for the problem is still a new topic.

In this study, we first propose error indicators for primal eigenfunction, dual eigenfunction, and eigenvalue of the non-self-adjoint Steklov eigenvalue problem in inverse scattering. The theoretical analysis in this study primarily refers to [11]. But different from what Armentano and Patra discussed in [11], the differential operator corresponding to this problem is non-self-adjoint and the associated weak formulation is not H 1-elliptic, which leads to the difficulty of theoretical analysis. We use Gårding’s inequality and duality technique to solve this difficulty. Second, we give upper and lower bounds for the energy norm of error of finite element solution, which shows that our indicators are reliable and efficient. Finally, based on the error indicators, we design an adaptive algorithm and present numerical examples to show the efficiency of our algorithm. Numerical results coincide with our theoretical analysis.

For the basic theory of finite element, we refer to [24,29,30,31,32,33]. In this study, without special explanation, C denotes the constant independent of mesh size, which may stand for different values at its different occurrences.

2 Preliminary

Let Ω ⊂ ℝ 2 be a bounded polygonal domain with Lipschitz boundary Γ = ∂Ω. We consider the following Steklov eigenvalue problem:

(2.1) Δ u + k 2 n ( x ) u = 0 , in Ω ,

(2.2) ∂ u ∂ ν + λ u = 0 , on Γ ,

where ν denotes the unit outward normal to Γ, k is the wavenumber, and n(x) is the index of refraction. Assume that n(x) is a bounded complex value function given by

n ( x ) = n 1 ( x ) + i n 2 ( x ) k ,

where i = − 1 , n 1(x) ≥ δ > 0, and n 2(x) ≥ 0 are bounded and properly smooth.

Here, we use the standard notation for Sobolev spaces H s (Ω) with norm ∥ ⋅ ∥ H s ( Ω ) and H 0 1 ( Ω ) = { v ∈ H 1 ( Ω ) : v | Γ = 0 } . Let a ( ⋅ , ⋅ ) and b ( ⋅ , ⋅ ) be defined as follows:

a ( u , v ) = ∫ Ω ( ∇ u ⋅ ∇ v ¯ − k 2 n ( x ) u v ¯ ),

b ( u , v ) = ∫ Γ u v ¯ .

The variational formulation associated with (2.1)–(2.2) is given by: find ( λ , u ) ∈ ℂ × H 1 ( Ω ) , u ≠ 0, such that

(2.3) a ( u , v ) = − λ b ( u , v ) , ∀ v ∈ H 1 ( Ω ) ,

(2.4) ∥ u ∥ L 2 ( Γ ) = 1 .

Let π h be a family of triangulations of Ω such that any two triangles in π h share at most a vertex or an edge. h E denotes the diameter of element E. Let h = max E ∈ π h { h E } be the diameter of π h . We assume that the family of triangulations π h satisfies a minimal angle condition, i.e., there exists a constant σ > 0 such that h E ρ E ≤ σ , where ρ E is the diameter of the largest circle contained in E.

V h = {v ∈ H 1(Ω) : v| E ∈ P 1, ∀ E ∈ π h }, where P 1 denotes the space of linear polynomials. Then, the discrete variational formulation associated with (2.3)–(2.4) is given by: find ( λ h , u h ) ∈ ℂ × V h , u h ≠ 0, such that

(2.5)   a ( u h , v ) = − λ h b ( u h , v ) , ∀ v ∈ V h ,

(2.6)   ∥ u h ∥ L 2 ( Γ ) = 1 .

Consider the dual problem (2.3)–(2.4): find ( λ ⁎ , u ⁎ ) ∈ ℂ × H 1 ( Ω ) , u ⁎ ≠ 0 , such that

(2.7)   a ( v , u ⁎ ) = − λ ⁎ ¯ b ( v , u ⁎ ) , ∀ v ∈ H 1 ( Ω ) ,

(2.8)   ∥ u ⁎ ∥ L 2 ( Γ ) = 1 .

Note that the primal and dual eigenvalues are connected via λ = λ ⁎ ¯ .

The discrete variational formulation associated with (2.7)–(2.8) is given by: find ( λ h ⁎ , u h ⁎ ) ∈ ℂ × V h , u h ⁎ ≠ 0 , such that

(2.9)   a ( v , u h ⁎ ) = − λ h ⁎ ¯ b ( v , u h ⁎ ) , ∀ v ∈ V h ,

(2.10) ∥ u h ⁎ ∥ L 2 ( Γ ) = 1 .

Note that the primal and dual eigenvalues are connected via λ h = λ h ⁎ ¯ .

For convenience of reading, we introduce the following notation. Let λ be the jth eigenvalue of (2.3)–(2.4) with the ascent α = 1 and the algebraic multiplicity q, i.e., λ = λ i = ⋯ = λ i+q−1. Then, there are q eigenvalues λ j,h (j = i, i + 1, … , i + q − 1) of (2.5)–(2.6) converging to λ. Let M(λ) be the space spanned by all eigenfunctions corresponding to the eigenvalue λ. Let M h (λ) be the space spanned by all eigenfunctions of (2.5)–(2.6) corresponding to the eigenvalues λ j,h (j = i, i + 1, … , i + q − 1). For the dual problems (2.7)–(2.8) and (2.9)–(2.10), the definitions of M*(λ*) and M h ⁎ ( λ ⁎ ) are made similar to M(λ) and M h (λ), respectively.

Thanks to Theorem 3.2 in [20] and Theorem 2.1 in [21], the following lemmas are valid.

Lemma 2.1

Let M(λ) ⊂ H 1+r (Ω). Assume that λ and λ h are the jth eigenvalue of (2.3)–(2.4) and (2.5)–(2.6), respectively. There exists h 0 ∈ (0,1) such that, if h ≤ h 0 , then

(2.11) | λ − λ h | ≤ C h 2 r .

Let u h ∈ M h (λ), ∥ u h ∥ L 2 ( Γ ) = 1 , then there exists u ∈ M(λ) such that

(2.12) ∥ u − u h ∥ H 1 ( Ω ) ≤ C h r ,

(2.13) ∥ u − u h ∥ L 2 ( Ω ) ≤ C h 3 r / 2 ,

(2.14) ∥ u − u h ∥ L 2 ( Γ ) ≤ C h 3 r / 2 ,

where r < π/ω (with ω being the largest inner angle of Ω) and r = 1 if Ω is convex.

Similar conclusions hold for the finite element approximation of dual problem (2.7)–(2.8).

Lemma 2.2

Let M*(λ*) ⊂ H 1+r (Ω). Assume that λ* and λ h ⁎ are the jth eigenvalue of (2.7)–(2.8) and (2.9)–(2.10), respectively. There exists h 0 ∈ (0,1) such that, if h ≤ h 0 , then

(2.15) | λ ⁎ − λ h ⁎ | ≤ C h 2 r .

Let u h ⁎ ∈ M h ⁎ ( λ ⁎ ) , ∥ u h ⁎ ∥ L 2 ( Γ ) = 1 , then there exists u* ∈ M*(λ*) such that

(2.16) ∥ u ⁎ − u h ⁎ ∥ H 1 ( Ω ) ≤ C h r ,

(2.17) ∥ u ⁎ − u h ⁎ ∥ L 2 ( Ω ) ≤ C h 3 r / 2 ,

(2.18) ∥ u ⁎ − u h ⁎ ∥ L 2 ( Γ ) ≤ C h 3 r / 2 .

3 A posteriori error estimates

In this section, we introduce error indicator for the non-self-adjoint Steklov eigenvalue problem in inverse scattering and prove that, up to higher order terms, it is equivalent to the energy norm of the error, i.e., it provides upper and lower bounds for the error of finite element solution. First of all, the local a posteriori error indicator η E for u h will be given. We use the symbol ε E to represent the set of edges of E, E ∈ π h . Define ε = ∪ E ∈ π h ε E and ε Γ = {l ∈ ε; l ⊂ Γ}. We decompose ε = ε Ω ∪ ε Γ , where ε Ω = ε\ε Γ . The following value of 〚 ∂ v h ∂ ν l 〛 l is independent of the choice of ν l :

〚 ∂ v h ∂ ν l 〛 l = ( ∇ ( v h | E 1 ) − ∇ ( v h | E 2 ) ) ⋅ ν l , l ∈ ε Ω ,

where E 1 and E 2 are the elements sharing edge l, the unit normal ν l points outwards E 2. Define

J l ( λ h , u h ) = { 1 2 〚 ∂ u h ∂ ν l 〛 l , l ∈ ε Ω , − λ h u h − ∂ u h ∂ ν l , l ∈ ε Γ ,

J l ⁎ ( λ h ⁎ , u h ⁎ ) = { 1 2 〚 ∂ u h ⁎ ∂ ν l 〛 l , l ∈ ε Ω , − λ h ⁎ u h ⁎ − ∂ u h ⁎ ∂ ν l , l ∈ ε Γ .

The local error indicator is given by

(3.1) η E ( λ h , u h , n ( x ) , k ) = { k 4 h E 2 ∥ n ( x ) u h ∥ L 2 ( E ) 2 + ∑ l ∈ ε E | l | ∥ J l ∥ L 2 ( l ) 2 } 1 2 ,

and the global error indicator is given by

(3.2) η Ω ( λ h , u h , n ( x ) , k ) = ( ∑ E ∈ π h η E 2 ( λ h , u h , n ( x ) , k ) ) 1 2 .

Similarly, we give the local error indicator and the global error indicator of the dual eigenfunction as follows:

(3.3) η E ⁎ ( λ h ⁎ , u h ⁎ , n ( x ) , k ) = { k 4 h E 2 ∥ n ( x ) u h ⁎ ∥ L 2 ( E ) 2 + ∑ l ∈ ε E | l | ∥ J l ⁎ ( λ h ⁎ u h ⁎ ) ∥ L 2 ( l ) 2 } 1 2 ,

(3.4) η Ω ⁎ ( λ h ⁎ , u h ⁎ , n ( x ) , k ) = ( ∑ E ∈ π h ( η E ⁎ ( λ h ⁎ , u h ⁎ , n ( x ) ¯ , k ) ) 2 ) 1 2 .

For convenience of reading, in the next discussion, we simplify J l (λ h ,u h ), η E (λ h ,u h ,n(x),k), and η Ω (λ h ,u h ,n(x),k) as J l , η E , and η Ω if without specification, respectively. Similarly, we simplify J l ⁎ ( λ h ⁎ , u h ⁎ ) , η E ⁎ ( λ h ⁎ , u h ⁎ , n ( x ) , k ) , and η Ω ⁎ ( λ h ⁎ , u h ⁎ , n ( x ) , k ) as J l ⁎ , η E ⁎ , and η Ω ⁎ , respectively.

Second, for analyzing the reliability and efficiency of the error indicators, we make some preparations. Let (λ,u) and (λ h ,u h ) be the eigenpair of (2.3)–(2.4) and (2.5)–(2.6), respectively. On one hand, it is easy to know that e = u−u h satisfies

(3.5) ∫ Ω ( ∇ e ⋅ ∇ v ¯ − k 2 n ( x ) e v ¯ ) = − ∫ Γ ( λ u v ¯ − λ h u h v ¯ ) , ∀   v ∈ V h .

On the other hand, for any v ∈ V h , it follows from Green formula and Δu h = 0 on E that ∫ E ∇ u h ⋅ ∇ v ¯ = ∫ ∂ E ∂ u h ∂ ν ⋅ v ¯ . Thus,

a ( u h , v ) = ∑ E { ∫ E ∇ u h ⋅ ∇ v ¯ − k 2 ∫ E n ( x ) u h v ¯ } = ∑ E { ∫ ∂ E ∂ u h ∂ ν ⋅ v ¯ − k 2 ∫ E n ( x ) u h v ¯ } .

Combining the aforementioned equality with (2.3), we conclude

(3.6) ∫ Ω ( ∇ e ⋅ ∇ v ¯ − k 2 n ( x ) e v ¯ ) = a ( u , v ) − a ( u h , v ) = − ∫ Γ λ u v ¯ − ∑ E ( ∫ ∂ E ∂ u h ∂ ν ⋅ v ¯ − k 2 ∫ E n ( x ) u h v ¯ ) = ∑ E { k 2 ∫ E n ( x ) u h v ¯ + ∑ l ∈ ε E ∩ ε Γ ∫ l ( − λ h u h − ∂ u h ∂ ν l ) ⋅ v ¯   + 1 2 ∑ l ∈ ε E ∩ ε Ω ∫ l 〚 ∂ u h ∂ ν l 〛 l v ¯ } − ∫ Γ ( λ u − λ h u h ) v ¯ .

In addition, we will use the following estimates for the interpolation operator of Clement I h : H 1(Ω) → V h :

(3.7) ∥ v − I h v ∥ L 2 ( E ) ≤ C h E ∥ v ∥ H 1 ( E ˆ ) ,

(3.8)   ∥ v − I h v ∥ L 2 ( l ) ≤ C | l | 1 2 ∥ v ∥ H 1 ( l ˆ ) ,

where E ˆ and l ˆ is the union of all elements sharing at least a vertex with E and l, respectively.

It is easy to verify that a(·,·) satisfies Gårding’s inequality [31], i.e., there exist constants 0 < M < ∞ (M is large enough) and α 0 > 0 such that

(3.9) Re { a ( v , v ) } + M ∥ v ∥ L 2 ( Ω ) 2 ≥ α 0 ∥ v ∥ H 1 ( Ω ) 2 , ∀ v ∈ H 1 ( Ω ) ,

where both M and α 0 are dependent on k and ∥ n ( x ) ∥ L ∞ ( Ω ) .

Third, we will use inequality (3.9) and the arguments in [11] to prove the following theorem. The theorem shows that the global error indicator provides upper bound of the error in the energy norm up to two higher order terms, which guarantees the reliability of indicator.

Theorem 3.1

The following conclusions are valid:

  1. there exists a constant C such that

    (3.10) ∥ u − u h ∥ H 1 ( Ω ) ≤ C { η Ω + ( M + | λ | ) ∥ u − u h ∥ L 2 ( Ω ) + | λ − λ h | } ,

  2. there exists a constant C such that

(3.11) ∥ u ⁎ − u h ⁎ ∥ H 1 ( Ω ) ≤ C { η Ω ⁎ + ( M + | λ ⁎ | ) ∥ u ⁎ − u h ⁎ ∥ L 2 ( Ω ) + | λ ⁎ − λ h ⁎ | } .

Proof

Take v = e I in (3.5), where e I denotes the Clement interpolation of e, then

∫ Ω ( ∇ e ⋅ ∇ e I ¯ − k 2 n ( x ) e e I ¯ ) = − ∫ Γ ( λ u − λ h u h ) e I ¯ ,

which together with (3.6) yields

(3.12) ∫ Ω ( ∇ e ⋅ ∇ e ¯ − k 2 n ( x ) e e ¯ ) = ∫ Ω ( ∇ e ⋅ ∇ ( e ¯ − e I ¯ ) − k 2 n ( x ) e ( e ¯ − e I ¯ ) ) + ∫ Γ ( ∇ e ⋅ ∇ e I ¯ − k 2 n ( x ) e e I ¯ ) = ∑ E { k 2 ∫ E n ( x ) u h ( e ¯ − e I ¯ ) + ∑ l ∈ ε E ∫ l J l ( e ¯ − e I ¯ ) } − ∫ Γ ( λ u − λ h u h ) e ¯ .

Noting that e = u − u h , and using the triangle inequality and Schwarz’s inequality, we have

(3.13) | ∫ Γ ( λ u − λ h u h ) e ¯ | = | ∫ Γ λ e e ¯ + ∫ Γ ( λ − λ h ) u h e ¯ | ≤ | λ | ∥ e ∥ L 2 ( Γ ) 2 + | λ − λ h | ∥ e ∥ L 2 ( Γ ) .

From (3.9), (3.12), and (3.13), we derive that

(3.14)   α 0 ∥ e ∥ H 1 ( Ω ) 2 ≤ | a ( e , e ) | + M ∥ e ∥ L 2 ( Ω ) 2 ≤ | ∑ E { k 2 ∫ E n ( x ) u h ( e ¯ − e I ¯ ) + ∑ l ∈ ε E ∫ l J l ( e ¯ − e I ¯ } |   + | λ | ∥ e ∥ L 2 ( Γ ) 2 + | λ − λ h | ∥ e ∥ L 2 ( Γ ) + M ∥ e ∥ L 2 ( Ω ) 2 .

Using Schwarz’s inequality, (3.7) and (3.8), and that the triangulation satisfies the minimum angle condition we obtain

(3.15) | ∑ E { k 2 ∫ E n ( x ) u h ( e ¯ − e I ¯ ) + ∑ l ∈ ε E ∫ l J l ( e ¯ − e I ¯ ) } | ≤ ∑ E { k 2 ∥ n ( x ) u h ∥ L 2 ( E ) ∥ e − e I ∥ L 2 ( E ) + ∑ l ∈ ε E ∥ J l ∥ L 2 ( l ) ∥ e − e I ∥ L 2 ( l ) } ≤ C ∑ E { k 2 h E ∥ e ∥ H 1 ( E ˆ ) ∥ n ( x ) u h ∥ L 2 ( E ) + ∑ l ∈ ε E | l | 1 2 ∥ e ∥ H 1 ( l ˆ ) ∥ J l ∥ L 2 ( l ) } ≤ C { ∑ E ( k 4 h E 2 ∥ n ( x ) u h ∥ L 2 ( E ) 2 + ∑ l ∈ ε E | l | ∥ J l ∥ L 2 ( l ) 2 ) } 1 2 ∥ e ∥ H 1 ( Ω ) .

Using the trace theorems, we get

(3.16) ∥ e ∥ L 2 ( Γ ) 2 ≤ C ∥ e ∥ L 2 ( Ω ) ∥ e ∥ H 1 ( Ω ) ,

(3.17) ∥ e ∥ L 2 ( Γ ) ≤ C ∥ e ∥ H 1 ( Ω ) .

Substituting (3.15), (3.16), and (3.17) into (3.14) and noting ∥ e ∥ L 2 ( Ω ) ≤ C ∥ e ∥ H 1 ( Ω ) , we deduce (3.10).

Denoting e ⁎ = u ⁎ − u h ⁎ and using the similar arguments to (3.10), we can derive (3.11) easily. Hence proved.□

Finally, we use the arguments in [11] to prove the efficiency of our indicator for practical adaptive refinement. To achieve our goal, we must prove that the local indicator is bounded by the error in the energy norm and higher order terms.

Now, we will prove that the following Lemmas 3.1 and 3.2 are true, which will be used to estimate the first term of η E .

Lemma 3.1

There exists a constant C such that

(3.18) k 2 h E ∥ n ( x ) u h ∥ L 2 ( E ) ≤ C ( ∥ ∇ e ∥ L 2 ( E ) + k 2 h E ∥ n ( x ) e ∥ L 2 ( E ) + h E 2 ∥ n ( x ) ∥ W 1 , ∞ ( E ) ∥ u h ∥ L 2 ( E ) ) .

Proof

Let δ 1,E , δ 2,E , and δ 3,E denote the barycentric coordinates of E ∈ π h . Define the cubic bubble b E by

b E = { δ 1 , E δ 2 , E δ 3 , E i n E , 0 i n Ω \ E .

Let v E ∈ P 4 ( E ) ∩ H 0 1 ( E ) . Then, there exists a unique v E such that

(3.19) ∫ E ∫ E n ( x ) ¯ | E | v E w ¯ = k 2 h E 2 ∫ E | n ( x ) | 2 u h w ¯ , ∀ w ∈ P 1 ( E ) .

We select v E = ∑ i = 1 3 α i φ i , where φ i = δ i,E b E (i = 1,2,3). According to ∫ E δ 1 , E k 1 δ 2 , E k 2 δ 3 , E k 3 = k 1 ! k 2 ! k 3 ! ( k 1 + k 2 + k 3 + 2 ) ! 2 ! | E | and noting that

∫ E δ 1 , E k 1 δ 2 , E k 2 δ 3 , E k 3 = { 12 7 ! | E | , ( k 1 , k 2 , k 3 ) is any possible permutation of ( 3 , 1 , 1 ) ; 8 7 ! | E | , ( k 1 , k 2 , k 3 ) is any possible permutation of ( 2 , 2 , 1 ) ,

then equality (3.19) can be written as:

∫ E n ( x ) ¯ | E | ⋅ 4 7 ! | E | ( 3 2 2 2 3 2 2 2 3 ) ( α 1 α 2 α 3 ) = k 2 h E 2 ( ∫ E | n ( x ) | 2 u h δ 1 , E ∫ E | n ( x ) | 2 u h δ 2 , E ∫ E | n ( x ) | 2 u h δ 3 , E ) .

Obviously, the determinant of coefficient matrix is not equal to zero. Therefore, v E exists uniquely. The solution of this linear system satisfies that

| α i | ≤ C | E | max 1 ≤ i ≤ 3 | k 2 h E 2 ∫ E | n ( x ) | 2 u h δ i , E | ,

then

(3.20) max 1 ≤ i ≤ 3 | α i | ≤ C | E | max 1 ≤ i ≤ 3 | k 2 h E 2 ∫ E | n ( x ) | 2 u h δ i , E | ≤ C | E | max 1 ≤ i ≤ 3 k 2 h E 2 ∥ n ( x ) u h ∥ L 2 ( E ) ( ∫ E δ i , E 2 ) 1 2 ≤ C | E | | E | 1 2 k 2 h E 2 ∥ n ( x ) u h ∥ L 2 ( E ) ,

where C is dependent on ∥ n ( x ) ∥ L ∞ ( Ω ) .

Since v E = ∑ i = 1 3 α i φ i , we get

(3.21) ∥ v E ∥ L 2 ( E ) 2 = ∫ E ( ∑ i = 1 3 α i φ i ) ( ∑ i = 1 3 α i φ i ¯ ) = ∑ i = 1 3 | α i | 2 ∫ E φ i 2 + ∑ 1 ≤ i < j ≤ 3 2 Re ( α i α j ) ∫ E φ i φ j .

Combining (3.21), ∫ E φ i φ j ≤ C | E | ( i , j = 1 , 2 , 3 ) , and (3.20), we deduce

(3.22) ∥ v E ∥ L 2 ( E ) ≤ C | E | 1 2 max 1 ≤ i ≤ 3 | α i | ≤ C k 2 h E 2 ∥ n ( x ) u h ∥ L 2 ( E ) .

Thus,

(3.23) ∥ v E ∥ L 2 ( E ) + h E ∥ ∇ v E ∥ L 2 ( E ) ≤ C ∥ v E ∥ L 2 ( E ) ≤ C k 2 h E 2 ∥ n ( x ) u h ∥ L 2 ( E ) .

Taking w = u h in (3.19), we deduce

k 2 h E 2 ∥ n ( x ) u h ∥ L 2 ( E ) 2 = ∫ E ∫ E n ( x ) ¯ | E | u h ¯ v E = ∫ E ∫ E n ( x ) | E | u h v E ¯ .

Taking v = v E in (3.6) and combining the aforementioned equality, we conclude

(3.24)   k 4 h E 2 ∥ n ( x ) u h ∥ L 2 ( E ) 2 = ∫ E ( ∇ e ⋅ ∇ v E ¯ − k 2 n ( x ) e v E ¯ ) − k 2 ( ∫ E n ( x ) u h v E ¯ − ∫ E ∫ E n ( x ) | E | u h v E ¯ ) ≤ | ∫ E ( ∇ e ⋅ ∇ v E ¯ − k 2 n ( x ) e v E ¯ ) | + k 2 | ∫ E ( n ( x ) − ∫ E n ( x ) | E | ) u h v E ¯ | .

We estimate the first term on the right-hand side of (3.24) by using Schwarz’s inequality and (3.23).

(3.25) | ∫ E ( ∇ e ⋅ ∇ v E ¯ − k 2 n ( x ) e v E ¯ ) | ≤ C ( ∥ ∇ e ∥ L 2 ( E ) ∥ ∇ v E ∥ L 2 ( E ) + k 2 h E ∥ n ( x ) e ∥ L 2 ( E ) 1 h E ∥ v E ∥ L 2 ( E ) ) ≤ C ( ∥ ∇ e ∥ L 2 ( E ) + k 2 h E ∥ n ( x ) e ∥ L 2 ( E ) ) ( ∥ ∇ v E ∥ L 2 ( E ) + 1 h E ∥ v E ∥ L 2 ( E ) ) ≤ C k 2 h E ( ∥ ∇ e ∥ L 2 ( E ) + k 2 h E ∥ n ( x ) e ∥ L 2 ( E ) ) ∥ n ( x ) u h ∥ L 2 ( E ) .

We estimate the second term on the right-hand side of (3.24) by using Schwarz’s inequality and (3.22).

(3.26)   | ∫ E ( n ( x ) − ∫ E n ( x ) | E | ) u h v E ¯ | ≤ ∥ ( n ( x ) − ∫ E n ( x ) | E | ) u h ∥ L 2 ( E ) ∥ v E ∥ L 2 ( E ) ≤ ∥ n ( x ) − ∫ E n ( x ) | E | ∥ L ∞ ( E ) ∥ u h ∥ L 2 ( E ) ∥ v E ∥ L 2 ( E ) ≤ C k 2 h E 3 ∥ n ( x ) ∥ W 1 , ∞ ( E ) ∥ u h ∥ L 2 ( E ) ∥ n ( x ) u h ∥ L 2 ( E ).

Substituting (3.25) and (3.26) into (3.24), we derive (3.18).□

The following Lemma 3.2 will be used to estimate the second term of η E .

Lemma 3.2

The following conclusions are valid:

  1. for l ∈ ε E ∩ ε Γ , there exists a constant C such that

    (3.27) | l | 1 2 ∥ J l ∥ L 2 ( l ) ≤ C ( ∥ ∇ e ∥ L 2 ( E ) + k 2 h E ∥ n ( x ) e ∥ L 2 ( E ) + h E 2 ∥ n ( x ) ∥ W 1 , ∞ ( E ) ∥ u h ∥ L 2 ( E ) + | l | 1 2 ∥ λ u − λ h u h ∥ L 2 ( l ) ) .

  2. for l ∈ ε E ∩ ε Ω , let E 1, E 2 ∈ π h be the two triangles sharing l. Then, there exists a constant C such that

(3.28) | l | 1 2 ∥ J l ∥ L 2 ( l ) ≤ C ( ∥ ∇ e ∥ L 2 ( E 1 ∪ E 2 ) + k 2 h E ∥ n ( x ) e ∥ L 2 ( E 1 ∪ E 2 ) + h E 2 ∥ n ( x ) ∥ W 1 , ∞ ( E 1 ∪ E 2 ) ∥ u h ∥ L 2 ( E 1 ∪ E 2 ) ) .

Proof

Using similar arguments to Lemma 3.2 in [11], we complete the proof. For l ∈ ε E ∩ ε Ω , we denote by E1 and E2 the two triangles sharing l and we enumerate the vertices of E1 and E2 such that the vertices of l are numbered first. Then, we introduce the edge-bubble function b l

b l = { δ 1 , E i δ 2 , E i in E i , 0 in Ω \ E 1 ∪ E 2 .

  1. For l ∈ ε E ∩ ε Γ , let v l ∈ P 3(E) be the function such that v l|l′ = 0 for l′ ∈ ε E , l ≠ l′ and

    { ∫ l v l w ¯ = | l | ∫ l J l w ¯ , ∀ w ∈ P 1 ( E ) , ∥ v l ∥ L 2 ( l ) ≤ C | l | ∥ J l ∥ L 2 ( l ) .

    Then, such v l exists and is unique. Actually, let v l = ∑ i = 1 2 β i ψ i where ψ i = δ i,E b l and δ 1,E , δ 2,E are the barycentric coordinates associated with the vertices of l. Then, using that ∫ l δ 1 , E k 1 δ 2 , E k 2 = k 1 ! k 2 ! ( k 1 + k 2 + 1 ) ! | l | to solve β 1, β 2, we know

    | β i | ≤ C | l | max | ∫ l | l | J l δ i , E | ,

    and

    (3.29) max 1 ≤ i ≤ 2 | β i | ≤ C ∥ J l ∥ L 2 ( l ) max 1 ≤ i ≤ 2 ∥ δ i , E ∥ L 2 ( l ) ≤ C | l | 1 2 ∥ J l ∥ L 2 ( l ) .

    Using the similar arguments to (3.22), we get

    (3.30) ∥ v l ∥ L 2 ( l ) ≤ C | l | 1 2 max 1 ≤ i ≤ 2 | β i | ≤ C | l | ∥ J l ∥ L 2 ( l ) .

    Analogously, combining (3.29) we obtain

    (3.31) ∥ v l ∥ L 2 ( E ) ≤ C h E max 1 ≤ i ≤ 2 | β i | ≤ C | l | 3 2 ∥ J l ∥ L 2 ( l ) ,

    furthermore,

    (3.32) ∥ v l ∥ L 2 ( E ) + h E ∥ ∇ v l ∥ L 2 ( E ) ≤ C | l | 3 2 ∥ J l ∥ L 2 ( l ) .

    Since | l | ∥ J l ∥ L 2 ( l ) 2 = ∫ l v l J l ¯ = ∫ l J l v l ¯ , it follows from (3.6) that

    | l | ∥ J l ∥ L 2 ( l ) 2 = ∫ E ( ∇ e ⋅ ∇ v l ¯ − k 2 n ( x ) e v l ¯ ) + ∫ l ( λ u − λ h u h ) v l ¯ − k 2 ∫ E n ( x ) u h v l ¯ .

    Using Schwarz’s inequality and combining (3.30) and (3.31) with (3.32), we conclude

    (3.33) | l | ∥ J l ∥ L 2 ( l ) 2 ≤ C ( ∥ ∇ e ∥ L 2 ( E ) + k 2 h E ∥ n ( x ) e ∥ L 2 ( E ) + | l | 1 2 ∥ λ u − λ h u h ∥ L 2 ( l ) + k 2 h E ∥ n ( x ) u h ∥ L 2 ( E ) ) | l | 1 2 ∥ J l ∥ L 2 ( l ) .

    From (3.33) and Lemma 3.1 we conclude (3.27).

  2. For l ∈ ε E ∩ ε Ω , let E 1, E 2 ∈ π h be the two triangles sharing l. Let v l ∈ H 0 1 ( E 1 ∪ E 2 ) be an edge-bubble function such that v l | Ei ∈ P 2, i = 1,2, and

{ ∫ l v l w ¯ = | l | ∫ l J l w ¯ , ∀ w ∈ P 0 ( E ) ; ∥ v l ∥ L 2 ( l ) ≤ C | l | ∥ J l ∥ L 2 ( l ) .

Using similar arguments to (a), we deduce

∥ v l ∥ L 2 ( E ) + h E ∥ ∇ v l ∥ L 2 ( E ) ≤ C | l | 3 2 ∥ J l ∥ L 2 ( l ) .

| l | ∥ J l ∥ L 2 ( l ) 2 = ∫ l J l ¯ v l = ∫ l J l v l ¯ and (3.6) imply that

| l | ∥ J l ∥ L 2 ( l ) 2 = ∫ E 1 ∪ E 2 ( ∇ e ⋅ ∇ v l ¯ − k 2 n ( x ) e v l ¯ ) − k 2 ∫ E 1 ∪ E 2 n ( x ) u h v l ¯ ,

thus,

(3.34) | l | ∥ J l ∥ L 2 ( l ) 2 ≤ C ( ∥ ∇ e ∥ L 2 ( E 1 ∪ E 2 ) + k 2 h E ∥ n ( x ) e ∥ L 2 ( E 1 ∪ E 2 ) +   h E ∥ n ( x ) u h ∥ L 2 ( E 1 ∪ E 2 ) ) | l | 1 2 ∥ J l ∥ L 2 ( l ) .

From (3.34) and Lemma 3.1, we conclude (3.28).□

The following theorem provides the lower bound for the error and high-order terms.

Theorem 3.2

There exists a constant C such that

  1. If ∂ E ∩ Γ = ∅ for E ∈ π h , then

    (3.35) η E ≤ C { ∥ e ∥ H 1 ( E u ) + k 2 h E ∥ n ( x ) e ∥ L 2 ( E u ) + h E 2 ∥ n ( x ) ∥ W 1 , ∞ ( E u ) ∥ u h ∥ L 2 ( E u ) } ,

    where E u denotes the union of E and the triangles sharing an edge with E.

  2. If ∂ E ∩ Γ ≠ ∅ for E ∈ π h , then

(3.36) η E ≤ C { ∥ e ∥ H 1 ( E ) + k 2 h E ∥ n ( x ) e ∥ L 2 ( E ) + h E 2 ∥ n ( x ) ∥ W 1 , ∞ ( E ) ∥ u h ∥ L 2 ( E ) + ∑ l ∈ ε E ∩ ε Γ | l | 1 2 ∥ λ u − λ h u h ∥ L 2 ( l ) } .

Proof

It follows from Lemmas 3.1 and 3.2.□

There are similar conclusions for the dual problem.

Lemma 3.3

There exists a constant C such that

(3.37) k 2 h E ∥ n ( x ) u h ⁎ ∥ L 2 ( E ) ≤ C ( ∥ ∇ e ⁎ ∥ L 2 ( E ) + k 2 h E ∥ n ( x ) e ⁎ ∥ L 2 ( E ) + h E 2 ∥ n ( x ) ∥ W 1 , ∞ ( E ) ∥ u h ⁎ ∥ L 2 ( E ) ) .

Lemma 3.4

  1. For l ∈ ε E ∩ ε Γ , there exists a constant C such that

    (3.38) | l | 1 2 ∥ J l ⁎ ∥ L 2 ( l ) ≤ C ( ∥ ∇ e ⁎ ∥ L 2 ( E ) + k 2 h E ∥ n ( x ) e ⁎ ∥ L 2 ( E ) + h E 2 ∥ n ( x ) ∥ W 1 , ∞ ( E ) ∥ u h ⁎ ∥ L 2 ( E ) + | l | 1 2 ∥ λ ⁎ u ⁎ − λ h ⁎ u h ⁎ ∥ L 2 ( l ) ) .

  2. For l ∈ ε E ∩ ε Ω , let E 1, E 2 ∈ π h be the two triangles sharing l. Then, there exists a constant C such that

(3.39) | l | 1 2 ∥ J l ⁎ ∥ L 2 ( l ) ≤ C ( ∥ ∇ e ⁎ ∥ L 2 ( E 1 ∪ E 2 ) + k 2 h E ∥ n ( x ) e ⁎ ∥ L 2 ( E 1 ∪ E 2 ) + h E 2 ∥ n ( x ) ∥ W 1 , ∞ ( E 1 ∪ E 2 ) ∥ u h ⁎ ∥ L 2 ( E 1 ∪ E 2 ) ) .

Theorem 3.3

There exists a constant C such that

  1. If ∂ E ∩ Γ = ∅ for E ∈ π h , then

    (3.40) η E ⁎ ≤ C { ∥ e ⁎ ∥ H 1 ( E u ) + k 2 h E ∥ n ( x ) e ⁎ ∥ L 2 ( E u ) + h E 2 ∥ n ( x ) ∥ W 1 , ∞ ( E u ) ∥ u h ⁎ ∥ L 2 ( E u ) } .

  2. If ∂ E ∩ Γ ≠ ∅ for E ∈ π h , then

(3.41) η E ⁎ ≤ C { ∥ e ⁎ ∥ H 1 ( E ) + k 2 h E ∥ n ( x ) e ⁎ ∥ L 2 ( E ) + h E 2 ∥ n ( x ) ∥ W 1 , ∞ ( E ) ∥ u h ⁎ ∥ L 2 ( E ) + ∑ l ∈ ε E ∩ ε Γ | l | 1 2 ∥ λ ⁎ u ⁎ − λ h ⁎ u h ⁎ ∥ L 2 ( l ) } .

Referring to Lemma 9.1 in [29], we deduce the following lemma.

Lemma 3.5

Let (λ,u) and (λ*,u*) be the eigenpair of (2.3)–(2.4) and (2.5)–(2.6), respectively. Then, for any v, v* ∈ H 1(Ω), b(v,v*) ≠ 0, the generalized Rayleigh quotient satisfies

(3.42) a ( v , v ⁎ ) − b ( v , v ⁎ ) − λ = a ( v − u , v ⁎ − u ⁎ ) − b ( v , v ⁎ ) + λ b ( v − u , v ⁎ − u ⁎ ) − b ( v , v ⁎ ) .

Proof

From (2.3) and (2.7), we get

a ( v − u , v ⁎ − u ⁎ ) + λ b ( v − u , v ⁎ − u ⁎ ) = a ( v , v ⁎ ) − a ( v , u ⁎ ) − a ( u , v ⁎ ) + a ( u , u ⁎ ) + λ ( b ( v , v ⁎ ) − b ( v , u ⁎ ) − b ( u , v ⁎ ) + b ( u , u ⁎ ) ) = a ( v , v ⁎ ) + λ b ( v , v ⁎ ) ,

dividing both sides by −b(v,v*), we obtain the desired conclusion.□

From Lemma 4.1 in [34] and Lemma 4.2 in [35], we have the following hypothesis.

Hypothesis H0 b ( u h , u h ⁎ ) has a positive lower bound uniformly with respect to h.

Theorem 3.4

Let (λ h ,u h ) be an approximate eigenpair of (2.5)–(2.6), which converges to the eigenpair (λ,u). Under hypothesis H0, we have

(3.43) | λ h − λ | ≤ C ( η Ω 2 + ( η Ω ⁎ ) 2 ) .

Proof

From Lemma 3.5, we get

λ h − λ = a ( u h − u , u h ⁎ − u ⁎ ) − b ( u h , u h ⁎ ) + λ b ( u h − u , u h ⁎ − u ⁎ ) − b ( u h , u h ⁎ ) .

According to hypothesis H0 and combining Lemmas 2.1 and 2.2, we derive

| λ h − λ | ≤ C | a ( u h − u , u h ⁎ − u ⁎ ) + b ( u h − u , u h ⁎ − u ⁎ ) | ≤ C { | u h − u | H 1 ( Ω ) ∥ u h ⁎ − u ⁎ ∥ H 1 ( Ω ) + ∥ u h − u ∥ L 2 ( Γ ) ∥ u h ⁎ − u ⁎ ∥ L 2 ( Γ ) } ≤ C ∥ u h − u ∥ H 1 ( Ω ) ∥ u h ⁎ − u ⁎ ∥ H 1 ( Ω ) ≤ C { ∥ u h − u ∥ H 1 ( Ω ) 2 + ∥ u h ⁎ − u ⁎ ∥ H 1 ( Ω ) 2 } .

Substituting (3.10) and (3.11) into the aforementioned inequality and ignoring higher order terms, we conclude (3.43).□

4 Edge residual error estimator

In this section, for primal eigenfunction and dual eigenfunction, we propose simpler global error indicators which are equivalent to the errors up to higher order terms, respectively.

η ˆ Ω = { ∑ E η ˆ E 2 } 1 / 2 , η ˆ Ω ⁎ = { ∑ E ( η ˆ E ⁎ ) 2 } 1 / 2 ,

where η ˆ E = ( ∑ l ∈ ε E | l | ∥ J l ∥ L 2 ( l ) 2 ) 1 / 2 and η ˆ E ⁎ = ( ∑ l ∈ ε E | l | ∥ J l ⁎ ∥ L 2 ( l ) 2 ) 1 / 2 . Actually, they are obtained by omitting the first term in the residual error indicator given in (3.1) and (3.3), respectively. Next, we prove that they are reliable and efficient up to higher order terms. For any P ∈ N Ω , we define Ω P = ∪{E ∈ π h : P ∈ E}.

Lemma 4.1

For any P∈N Ω , we have that

(4.1) ∑ E ⊂ Ω P k 4 h E 2 ∥ n ( x ) u h ∥ L 2 ( E ) 2 ≤ C ( ∑ l ∈ ε Ω P | l | ∥ J l ∥ L 2 ( l ) 2 + | Ω P | 2 ∥ ∇ u h ∥ L 2 ( Ω P ) 2 ) ,  

(4.2) ∑ E ⊂ Ω P k 4 h E 2 ∥ n ( x ) u h ⁎ ∥ L 2 ( E ) 2 ≤ C ( ∑ l ∈ ε Ω P | l | ∥ J l ⁎ ∥ L 2 ( l ) 2 + | Ω P | 2 ∥ ∇ u h ⁎ ∥ L 2 ( Ω P ) 2 ) .  

Proof

Using similar arguments to Lemma 4.1 in [11], we complete the proof. Let I 0(u h ) be the L 2(Ω P ) projection of u h onto the constants, i.e.,

∫ Ω P u h w ¯ = ∫ Ω P I 0 ( u h ) w ¯ , ∀ w is constant i n Ω P .

Obviously,

(4.3) ∑ E ⊂ Ω P k 4 h E 2 ∥ n ( x ) u h ∥ L 2 ( E ) 2 ≤ C | Ω P | ∥ k 2 n ( x ) u h ∥ L 2 ( Ω P ) 2   ≤ C | Ω P | { ∥ I 0 ( k 2 n ( x ) u h ) ∥ L 2 ( Ω P ) 2 + ∥ k 2 n ( x ) u h − I 0 ( k 2 n ( x ) u h ) ∥ L 2 ( Ω P ) 2 } .

Next, we estimate the first term on the right-hand side of (4.3). Let Φ P be the corresponding Lagrange basis function with supp Φ P = Ω P , then

(4.4) ∥ I 0 ( k 2 n ( x ) u h ) ∥ L 2 ( Ω P ) 2 = | I 0 ( k 2 n ( x ) u h ) | 2 | Ω P | = 9 | Ω P | ( ∫ Ω P I 0 ( k 2 n ( x ) u h ) Φ P ¯ ) 2   ≤ 18 | Ω P | { ( ∫ Ω P ( I 0 ( k 2 n ( x ) u h ) − k 2 n ( x ) u h ) Φ P ¯ ) 2 + ( ∫ Ω P k 2 n ( x ) u h Φ P ¯ ) 2 } .

From (2.5)–(2.6), we have

(4.5) ∫ Ω P k 2 n ( x ) u h Φ P ¯ = ∫ Ω P ∇ u h ⋅ ∇ Φ P ¯ = 1 2 ∑ l ∈ ε Ω ∩ ε Ω P | l | 〚 ∂ u h ∂ ν l 〛 l = 1 2 ∑ l ∈ ε Ω ∩ ε Ω P ∫ l 〚 ∂ u h ∂ ν l 〛 l .

The relationships (4.4) and (4.5) imply that

(4.6) | Ω P | ∥ I 0 ( k 2 n ( x ) u h ) ∥ L 2 ( Ω P ) 2 ≤ C { | Ω P | ∥ I 0 ( k 2 n ( x ) u h ) − k 2 n ( x ) u h ∥ L 2 ( Ω P ) 2 + 1 2 ∑ l ∈ ε Ω ∩ ε Ω P | l | ∥ J l ∥ L 2 ( Ω P ) 2 } .

Substituting (4.6) into (4.5), we derive

∑ E ⊂ Ω P k 4 h E 2 ∥ n ( x ) u h ∥ L 2 ( E ) 2 ≤ C { | Ω P | ∥ I 0 ( k 2 n ( x ) u h ) − k 2 n ( x ) u h ∥ L 2 ( Ω P ) 2 + 1 2 ∑ l ∈ ε Ω ∩ ε Ω P | l | ∥ J l ∥ L 2 ( Ω P ) 2 } .

Inequality (4.1) can be concluded by using the standard estimate for the L 2(Ω P ) projection.□

The following theorem indicates that the indicator is globally reliable and locally efficient up to higher order terms.

Theorem 4.1

There exists a constant C such that

(4.7) ∥ u − u h ∥ H 1 ( Ω ) ≤ C { η Ω + ( M + | λ | ) ∥ u − u h ∥ L 2 ( Ω ) + | λ − λ h | + h 2 ∥ ∇ u h ∥ L 2 ( Ω ) } ,

and if ∂ E ∩ Γ = ∅ for E ∈ π h , then

(4.8) η ˆ E ≤ C { ∥ e ∥ H 1 ( E u ) + k 2 h E ∥ n ( x ) e ∥ L 2 ( E u ) + h E 2 ∥ n ( x ) ∥ W 1 , ∞ ( E u ) ∥ u h ∥ L 2 ( E u ) } ,

if ∂ E ∩ Γ ≠ ∅ for E ∈ π h , then

(4.9) η ˆ E ≤ C { ∥ e ∥ H 1 ( E ) + k 2 h E ∥ n ( x ) e ∥ L 2 ( E ) + h E 2 ∥ n ( x ) ∥ W 1 , ∞ ( E ) ∥ u h ∥ L 2 ( E ) + ∑ l ∈ ε E ∩ ε Γ ∥ λ u − λ h u h ∥ L 2 ( l ) } .

Proof

It follows immediately from Theorem 3.1, Theorem 3.2, and Lemma 4.1.□

Theorem 4.2

There exists a constant C such that

(4.10) ∥ u ⁎ − u h ⁎ ∥ H 1 ( Ω ) ≤ C { η ˆ Ω ⁎ + ( M + | λ ⁎ | ) ∥ u ⁎ − u h ⁎ ∥ L 2 ( Ω ) + | λ ⁎ − λ h ⁎ | + h 2 ∥ ∇ u h ⁎ ∥ L 2 ( Ω ) } ,

and if ∂ E ∩ Γ = ∅ for E ∈ π h , then

(4.11) η ˆ E ⁎ ≤ C { ∥ e ⁎ ∥ H 1 ( E u ) + k 2 h E ∥ n ( x ) e ⁎ ∥ L 2 ( E u ) + h E 2 ∥ n ( x ) ∥ W 1 , ∞ ( E u ) ∥ u h ⁎ ∥ L 2 ( E u ) } ,

if ∂ E ∩ Γ ≠ ∅ for E ∈ π h , then

(4.12) η ˆ E ⁎ ≤ C { ∥ e ⁎ ∥ H 1 ( E ) + k 2 h E ∥ n ( x ) e ⁎ ∥ L 2 ( E ) + h E 2 ∥ n ( x ) ∥ W 1 , ∞ ( E ) ∥ u h ∥ L 2 ( E ) + ∑ l ∈ ε E ∩ ε Γ ∥ λ ⁎ u ⁎ − λ h ⁎ u h ⁎ ∥ L 2 ( l ) } .

Proof

It follows immediately from Theorem 3.1, Theorem 3.3, and Lemma 4.1.□

On adaptive mesh with local refinement, the convergence rate of eigenvalue is measured by the total number of degrees of freedom. There are some literature studies that have discussed the relationship between the mesh diameter and the degrees of freedom (e.g., see Remark 2.1 in [35]).

5 Adaptive algorithm and numerical experiments

First of all, we give the following Algorithm 5.1 which is fundamental and important and can be found in [36].

Algorithm 5.1

Choose parameter 0 < θ < 1.

  1. Pick any initial mesh π h 0 with mesh size h 0.

  2. Solve (2.5)–(2.6) and (2.9)–(2.10) on π h 0 for discrete solutions ( λ h 0 , u h 0 ) and ( λ h 0 ⁎ , u h 0 ⁎ ) , respectively.

  3. Let i = 0.

  4. Compute the local indicator η E 2 + ( η E ⁎ ) 2 corresponding to λ h i .

  5. Construct π ˆ h i ⊂ π h i by Marking Strategy M and parameter θ.

  6. Refine π h i to get a new mesh π h i + 1 by Procedure REFINE.

  7. Solve (2.5)–(2.6) and (2.9)–(2.10) on π h i + 1 for discrete solution ( λ h i + 1 , u h i + 1 ) and ( λ h i + 1 ⁎ , u h i + 1 ⁎ ) , respectively.

  8. Let i = i + 1 and go to Step 4.

Marking Strategy M

Given parameter 0 < θ < 1.

Step 1. Construct a minimal subset π ˆ h i of π h i by selecting some elements in π h i such that

∑ E ∈ π ˆ h i ( η E 2 + ( η E ⁎ ) 2 ) ≥ θ ( η Ω 2 + ( η Ω ⁎ ) 2 ) .

Step 2. Mark all the elements in π ˆ h i .

Marking strategy M was first proposed by [37].

The procedure REFINE is some iterative or recursive bisection (see, e.g., [38,39]) of elements with the minimal refinement condition that marked elements are bisected at least once.

Next, we will report some numerical examples for Algorithm 5.1 to validate our theoretical results. For comparison, using linear finite element, we also solve the problem on uniform meshes. The discrete eigenvalue problems are solved in MATLAB 2016b on an Lenovo IdeaPad 500-14ACZ PC with 1.8 GHZ CPU and 8 GB RAM. Our program is compiled under the package of iFEM [40] and we take θ = 0.25 in our program. In step 2 and step 7, we use the sparse solver “eigs” in MATLAB 2016b to solve the discrete eigenvalue problems. Numerical experiments are considered on the L-shaped domain (−1,1)2\([0,1) × (−1,0]) and the square with a slit ( − 2 2 , 2 2 ) 2 \ { 0 ≤ x ≤ 2 2 , y = 0 } . The initial mesh is made up of congruent triangles with the mesh size h 0 = 2 32 . We select the index of refraction n(x) = 4 or n(x) = 4 + 4i when k = 1 (refer to [7]). When n(x) is the real number, the first 30 eigenvalues are sorted in the descending order. When n(x) is the complex number, the first 30 eigenvalues are sorted in the descending order of the absolute value of imaginary part. We select n(x) = 4 + 2i, 4 + i when k = 2 and k = 4, respectively. For convenience and simplicity, the following notations are introduced in tables and figures.

  • i: the ith iteration of Algorithm 5.1.

  • λ j , h i A : the jth eigenvalue from the ith iteration of the Adaptive Algorithm 5.1.

  • λ j,h : the jth eigenvalue in uniform meshes.

  • dof: number of degrees of freedom.

  • η 2: the a posteriori error estimator of approximate eigenvalue obtained by Algorithm 5.1.

In the case of k = 1, when n(x) = 4, we use the approximations λ 1 ≈ 2.53321363, λ 2 ≈ 0.85778759, and λ 3 ≈ 0.12452443 as the reference values for the L-shaped domain. For the square with a slit, the first three reference eigenvalues are 1.48471191, 0.46173362, and −0.18417592, respectively. When n(x) = 4 + 4i, for the L-shaped domain, we take reference values as λ 1 ≈ 0.514287041 + 2.88232331i, λ 2 ≈ 0.39703537 + 1.45898539i, and λ 3 ≈ −0.07717876 + 1.04267799i. For the square with a slit, the reference values are λ 1 ≈ 0.91930585 + 1.77078802i, λ 2 ≈ 0.29263004 + 0.99987320i, and λ 3 ≈ −0.26261456 + 0.75745031i. In the case of k ≠ 1, for the L-shaped domain, we take λ 9 ≈ −1.40118553 + 1.54057954i and λ 19 ≈ −4.33350590 + 1.54440353i as the reference values when k = 2, n(x) = 4 + 2i and k = 4, n(x) = 4 + i, respectively. For the square with a slit, we select λ 2 ≈ 2.19728089 + 5.85519972i and λ 2 ≈ 6.67501844 + 9.59727038i as the reference values, respectively. These reference values are computed by extrapolation.

For the L-shaped domain, using the linear finite element, we solve the first three eigenvalues on uniform meshes, the error curves are depicted in Figure 1. The error curves of eigenvalues obtained by Algorithm 5.1 are depicted in Figures 2 and 3. The numerical results are listed in Tables 1 and 2.

Figure 1 Error curves of the first three eigenvalues on uniform meshes for (2.1)–(2.2) on the L-shaped domain (the left: k = 1, n(x) = 4, the right: k = 1, n(x) = 4 + 4i).
Figure 1

Error curves of the first three eigenvalues on uniform meshes for (2.1)–(2.2) on the L-shaped domain (the left: k = 1, n(x) = 4, the right: k = 1, n(x) = 4 + 4i).

Figure 2 Error comparison of the second eigenvalue for (2.1)–(2.2) on the L-shaped domain (the left: k = 1, n(x) = 4, the right: k = 1, n(x) = 4 + 4i).
Figure 2

Error comparison of the second eigenvalue for (2.1)–(2.2) on the L-shaped domain (the left: k = 1, n(x) = 4, the right: k = 1, n(x) = 4 + 4i).

Figure 3 Error comparison of the eigenvalues for (2.1)–(2.2) on the L-shaped domain (the left: k = 2, n(x) = 4 + 2i, the right: k = 4, n(x) = 4 + i).
Figure 3

Error comparison of the eigenvalues for (2.1)–(2.2) on the L-shaped domain (the left: k = 2, n(x) = 4 + 2i, the right: k = 4, n(x) = 4 + i).

Table 2

The second eigenvalue for (2.1)–(2.2): L-shaped domain (k = 1, n(x) = 4 + 4i)

i j dof λ j , h i A dof λ j,h
20 2 11,365 0.3970403 + 1.4587041i 12,545 0.3966861 + 1.4584678i
32 2 48,797 0.3970372 + 1.4589067i 49,665 0.3968836 + 1.4587931i
43 2 1,81,128 0.3970354 + 1.4589609i 1,97,633 0.3969716 + 1.4589108i
Table 3

The second eigenvalue for (2.1)–(2.2): the square with a slit (k = 1, n(x) = 4)

i j dof λ j , h i A dof λ j,h
22 2 14,505 0.461483 16,705 0.457585
32 2 60,809 0.461670 66,177 0.459664
42 2 2,42,511 0.461715 2,63,425 0.460699

We know that the optimal convergence order of numerical eigenvalues on adaptive meshes can achieve O(dof−1), and error curve of numerical eigenvalue should be parallel to the line with slope −1. In the case of k = 1, in Figure 1, we see that the error curves of the second eigenvalues are not parallel to the line with slope −1, which implies that the eigenfunction corresponding to the eigenvalue is of singularity and the convergence order of numerical eigenvalue computed on uniform meshes cannot achieve the optimal order. In Figure 2, the error curve of eigenvalue computed on adaptive meshes is parallel to the line with slope −1. Therefore, the numerical eigenvalue obtained by Algorithm 5.1 can achieve the optimal order. Using adaptive meshes is more effective than using uniform meshes when the eigenfunction corresponding to eigenvalue is of singularity. In addition, we know that the a posteriori error indicator of numerical eigenvalue is reliable and efficient as the error curve of numerical eigenvalue is basically parallel to the curve of error indicator. In the case of k ≠ 1, the eigenfunctions corresponding to these eigenvalues are of singularity. In Figure 3, we also see that the numerical eigenvalues computed in adaptive meshes can achieve the optimal convergence order O(dof−1).

For the square with a slit, in the case of k = 1, the error curves of the first three eigenvalues are depicted in Figure 4. The error curves of eigenvalues obtained by Algorithm 5.1 are depicted in Figures 5 and 6. The numerical results are listed in Tables 3 and 4. From the figures and tables, we obtain similar conclusion to the L-shaped domain, which coincides in theory results. We also see that the numerical eigenvalues computed in adaptive meshes can achieve the optimal convergence order O(dof−1).

Figure 4 Error curves of the first three eigenvalues on uniform meshes for (2.1)–(2.2) on the square with a slit (the left: k = 1, n(x) = 4, the right: k = 1, n(x) = 4 + 4i).
Figure 4

Error curves of the first three eigenvalues on uniform meshes for (2.1)–(2.2) on the square with a slit (the left: k = 1, n(x) = 4, the right: k = 1, n(x) = 4 + 4i).

Figure 5 Error comparison of the second eigenvalue for (2.1)–(2.2) on the square with a slit (the left: k = 1, n(x) = 4, the right: k = 1, n(x) = 4 + 4i).
Figure 5

Error comparison of the second eigenvalue for (2.1)–(2.2) on the square with a slit (the left: k = 1, n(x) = 4, the right: k = 1, n(x) = 4 + 4i).

Figure 6 Error comparison of the eigenvalues for (2.1)–(2.2) on the square with a slit (the left: k = 2, n(x) = 4 + 2i, the right: k = 4, n(x) = 4 + i).
Figure 6

Error comparison of the eigenvalues for (2.1)–(2.2) on the square with a slit (the left: k = 2, n(x) = 4 + 2i, the right: k = 4, n(x) = 4 + i).

Table 4

The second eigenvalue for (2.1)–(2.2): the square with a slit (k = 1, n(x) = 4 + 4i)

i j dof λ j , h i A dof λ j,h
22 2 15,370 0.292428 + 0.999679i 16,705 0.289067 + 0.997926i
33 2 64,492 0.292578 + 0.999827i 66,177 0.290846 + 0.998908i
44 2 2,59,623 0.292616 + 0.999865i 2,63,425 0.291737 + 0.999395i
Table 1

The second eigenvalue for (2.1)–(2.2): L-shaped domain (k = 1, n(x) = 4)

i j dof λ j , h i A dof λ j,h
16 2 11,536 0.857500 12,545 0.857083
27 2 47,688 0.857715 49,665 0.857523
38 2 1,90,616 0.857770 1,97,633 0.857687

Finally, adaptively refined meshes obtained by Algorithm 5.1 are displayed in Figure 7 as k = 1 and n(x) = 4 + 4i. In theory, the errors of eigenfunctions on the elements around concave vertices should be greater than other elements. In Figure 7, we see that, on each domain, these elements around concave vertices are more refined than other elements, which coincides in theoretical results.

Figure 7 The adaptive meshes obtained by Algorithm 5.1 as i = 25, k = 1, and n(x) = 4 + 4i.
Figure 7

The adaptive meshes obtained by Algorithm 5.1 as i = 25, k = 1, and n(x) = 4 + 4i.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (grant no. 11561014), the Young Scientific and Technical Talents Development of Education Department of Guizhou Province (grant no. KY[2018]153), and the National Natural Science Foundation of China (grant no. 11761022). We cordially thank the editor and the referees for their valuable comments and suggestions which led to the improvement of this paper.

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Received: 2019-04-30
Revised: 2020-01-08
Accepted: 2020-01-18
Published Online: 2020-05-11

© 2020 Yu Zhang et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

  1. Regular Articles
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  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
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  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
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  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
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  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
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  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
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https://doi.org/10.1515/math-2020-0140
Schlagwörter für diesen Artikel
Steklov eigenvalues in inverse scattering; a posteriori error estimates; reliability and efficiency; adaptive algorithm
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Artikel in diesem Heft

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
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  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
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  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
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  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
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Heruntergeladen am 30.10.2025 von https://www.degruyterbrill.com/document/doi/10.1515/math-2020-0140/html
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