Startseite Mathematik An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
Artikel Open Access

An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem

  • Hui Zhang , Zixin Liu EMAIL logo und Jun Zhang
Veröffentlicht/Copyright: 23. August 2022
Artikel downloaden (PDF)
Open Mathematics
Aus der Zeitschrift Open Mathematics Band 20 Heft 1

Abstract

In this article, an effective finite element method based on dimension reduction scheme is proposed for a fourth-order Steklov eigenvalue problem in a circular domain. By using the Fourier basis function expansion and variable separation technique, the original problem is transformed into a series of radial one-dimensional eigenvalue problems with boundary eigenvalue. Then we introduce essential polar conditions and establish the discrete variational form for each radial one-dimensional eigenvalue problem. Based on the minimax principle and the approximation property of the interpolation operator, we prove the error estimates of approximation eigenvalues. Finally, some numerical experiments are provided, and the numerical results show the efficiency of the proposed algorithm.

Keywords: fourth-order Steklov eigenvalue problem; weighted Sobolev space; dimension reduction scheme; finite element approximation; error estimation
MSC 2010: 34L15; 65M60; 97N20

1 Introduction

Fourth-order Steklov eigenvalue problems with eigenvalue parameter in boundary conditions are widely used in mathematics and physics, such as the surface wave research, the stability analysis of mechanical oscillator in a viscous fluid, the study of vibration mode of structure in contact with an incompressible fluid, and so on [1,2,3, 4,5]. The first eigenvalue λ 1 also plays an important role in the positivity-preserving properties for the biharmonic-operator Δ 2 under the boundary conditions ϕ ∣ ∂ Ω = ( Δ ϕ − λ ϕ ν ) ∣ ∂ Ω = 0 in [6,7].

There are many existing results about the fourth-order Steklov eigenvalue problems, but they mainly focus on the qualitative analysis. Kuttler [8] proved that the first eigenvalue is simple and the corresponding eigenfunction does not change the sign. Ferrero et al. [7] and Bucur et al. [9] studied the spectrum on a bounded domain, and the explicit representation of the spectrum is given when the domain is a ball. Recently, the existence of an optimal convex shape among domains of a given measure is proved in [10], and the Weyl-type asymptotic formula for the counting function of the biharmonic Steklov eigenvalues also is established in [11]. For the numerical methods of the fourth-order Steklov eigenvalue problems, a conforming finite element method was first proposed in [12], then some spectral methods are also developed [13].

As we all know, if the conforming finite element method is directly used to solve a fourth-order problem, the boundary of the element requires the continuity of the first derivative, which not only brings the difficulty of constructing the basis function but also costs a lot of calculation time and memory capacity, especially for some special regions, such as circular region, spherical region, and so on. How to efficiently solve a fourth-order Steklov eigenvalue problem in a circular domain? To the best of our knowledge, there are few reports on using some efficient numerical to solve this problem. Thus, the aim of this article is to propose an effective finite element method based on a dimension reduction scheme for a fourth-order Steklov eigenvalue problem in a circular domain. By using the Fourier basis function expansion and variable separation technique, the original problem is transformed into a series of radial one-dimensional eigenvalue problems with boundary eigenvalue. Then we introduce essential polar conditions and establish the discrete variational form for each radial one-dimensional eigenvalue problem. Based on the minimax principle and the approximation property of the interpolation operator, we prove the error estimates of approximation eigenvalues. Finally, some numerical experiments are provided, and the numerical results show the efficiency of the proposed algorithm.

This article is organized as follows. In Section 2, a reduced scheme based on polar coordinate transformation is presented. In Section 3, the weighted space and discrete variational form are derived. In Section 4, the error estimation of approximation solutions is proved. In Section 5, we present the process of effective implementation of the algorithm. We present some numerical experiments in Section 6 to illustrate the accuracy and efficiency of our proposed algorithm. Finally, we give in Section 7 some concluding remarks.

2 Reduced scheme based on polar coordinate transformation

The fourth-order Steklov eigenvalue problems read:

(2.1) Δ 2 ϕ = 0 , in Ω ,

(2.2) ϕ = 0 , on ∂ Ω ,

(2.3) Δ ϕ = λ ∂ ϕ ∂ ν , on ∂ Ω ,

where Ω = { ( x , y ) ∈ R 2 : 0 < x 2 + y 2 < R } , ν is the unit outward normal to the boundary ∂ Ω . Let x = r cos θ , y = r sin θ , ψ ( r , θ ) = ϕ ( x , y ) . Then we derive that

(2.4) Δ ϕ ( x , y ) = L ψ ( r , θ ) = 1 r ∂ ∂ r r ∂ ψ ( r , θ ) ∂ r + 1 r 2 ∂ 2 ψ ( r , θ ) ∂ θ 2 .

Then the equivalent form of (2.1)–(2.3) in polar coordinates is as follows:

(2.5) L 2 ψ ( r , θ ) = 0 , in D = ( 0 , R ) × [ 0 , 2 π ) ,

(2.6) ψ ( R , θ ) = 0 , θ ∈ [ 0 , 2 π ) ,

(2.7) L ψ ( R , θ ) = λ ∂ ψ ∂ r ( R , θ ) , θ ∈ [ 0 , 2 π ) .

Since ψ ( r , θ ) is 2 π periodic in θ , then we have

(2.8) ψ ( r , θ ) = ∑ ∣ m ∣ = 0 ∞ ψ m ( r ) e i m θ .

Substituting (2.8) into (2.4), we derive that

(2.9) L ψ ( r , θ ) = ∑ ∣ m ∣ = 0 ∞ 1 r ∂ ∂ r r ∂ ψ m ( r ) ∂ r − m 2 r ψ m ( r ) e i m θ .

Following the discussion in [14,15], to overcome the pole singularity introduced by polar coordinate transformation, we need to introduce the essential pole conditions, which make (2.9) meaningful, as follows:

(2.10) m 2 ψ m ( 0 ) = 0 , lim r → 0 + ∂ ∂ r r ∂ ψ m ( r ) ∂ r − m 2 r ψ m ( r ) = 0 .

Using the fact that ∂ ∂ r r ∂ ψ m ( r ) ∂ r = ∂ ψ m ( r ) ∂ r + r ∂ 2 ψ m ( r ) ∂ r 2 , (2.10) can be reduced to

(2.11) m 2 ψ m ( 0 ) = 0 , ( 1 − m 2 ) ∂ ψ m ( 0 ) ∂ r = 0 .

From (2.11) we can further obtain that

(2.12) ( 1 ) ∂ ψ m ( 0 ) ∂ r = 0 , ( m = 0 ) ;

(2.13) ( 2 ) ψ m ( 0 ) = 0 , ( ∣ m ∣ = 1 ) ;

(2.14) ( 3 ) ψ m ( 0 ) = 0 , ∂ ψ m ( 0 ) ∂ r = 0 , ( ∣ m ∣ ≥ 2 ) .

Let r = t + 1 2 R , u m ( t ) = ψ m ( r ) , L m u m ( t ) = 1 t + 1 ∂ ∂ t t ∂ u m ( t ) ∂ t − m 2 ( t + 1 ) 2 u m ( t ) . From (2.8), (2.11)–(2.14), and the orthogonal properties of Fourier basis functions, (2.5)–(2.7) are equivalent to a series of one-dimensional eigenvalue problems

(2.15) L m 2 u m ( t ) = 0 , t ∈ ( − 1 , 1 ) ,

(2.16) ( 1 ) ∂ u m ( − 1 ) ∂ t = 0 , u m ( 1 ) = 0 , L m u m ( 1 ) = R 2 λ m ∂ u m ( 1 ) ∂ t , ( m = 0 ) ;

(2.17) ( 2 ) u m ( − 1 ) = 0 , u m ( 1 ) = 0 , L m u m ( 1 ) = R 2 λ m ∂ u m ( 1 ) ∂ t , ( ∣ m ∣ = 1 ) ;

(2.18) ( 3 ) u m ( ± 1 ) = 0 , ∂ u m ( − 1 ) ∂ t = 0 , L m u m ( 1 ) = R 2 λ m ∂ u m ( 1 ) ∂ t , ( ∣ m ∣ ≥ 2 ) .

3 Weighted space and discrete variational form

Without losing generality, we only consider the case of m ≥ 0 . First, we divide the solution interval I = ( − 1 , 1 ) as follows:

− 1 = t 0 < t 1 < ⋯ < t i < ⋯ < t n = 1 .

Define the usual weighted Sobolev space:

L ω 2 ( I ) ≔ ρ : ∫ I ω ρ 2 d t < ∞

equipped with the following inner product and norm:

( ρ , v ) ω = ∫ I ω ρ v d t , ‖ ρ ‖ w = ∫ I ω ρ 2 d t 1 2 ,

where ω = 1 + t , t ∈ ( − 1 , 1 ) . We further introduce the following weighted Sobolev space:

H 0 , ω , m 2 ( I ) ≔ u m : L m u m ∈ L ω 2 ( I ) , m 2 u m ( − 1 ) = ( 1 − m 2 ) ∂ u m ( − 1 ) ∂ t = u m ( 1 ) = 0 ,

equipped with the inner product and norm:

( u m , v m ) 2 , ω , m = ( L m u m , L m v m ) ω , ‖ u m ‖ 2 , ω , m = ( u m , u m ) 2 , ω , m .

Then the variational form of (2.15)–(2.18) is: Find ( λ m , u m ≠ 0 ) ∈ R × H 0 , ω , m 2 ( I ) , such that

(3.1) A m ( u m , v m ) = λ m B m ( u m , v m ) , ∀ v m ∈ H 0 , ω , m 2 ( I ) ,

where

A m ( u m , v m ) = ∫ I ( t + 1 ) L m u m L m v m d t , B m ( u m , v m ) = R ∂ u m ( 1 ) ∂ t ∂ v m ( 1 ) ∂ t .

Let us denote by U h a piecewise cubic Hermite interpolation function space. Define the approximation space S h ( m ) = U h ∩ H 0 , ω , m 2 ( I ) . Then the discrete variational form associated with (3.1) is: Find ( λ m h , u m h ≠ 0 ) ∈ R × S h ( m ) , such that

(3.2) A m ( u m h , v m h ) = λ m h B m ( u m h , v m h ) , ∀ v m h ∈ S h ( m ) .

4 Error estimation of approximation solutions

For the sake of brevity, we shall use the expression a ≲ b which denotes a ≤ c b , where c is a positive constant.

Lemma 1

For any u m , v m ∈ H 0 , ω , m 2 ( I ) , the following equalities hold:

(4.1) ∫ I ( t + 1 ) L m u m L m v m d t = ∫ I ( t + 1 ) u m ″ v m ″ d t + ( 2 m 2 + 1 ) ∫ I 1 t + 1 u m ′ v m ′ d t + m 2 ( m 2 − 4 ) ∫ I 1 ( t + 1 ) 3 u m v m d t + u m ′ ( 1 ) v m ′ ( 1 )

with m ≠ 1 , and

(4.2) ∫ I ( t + 1 ) L m u m L m v m d t = ∫ I ( t + 1 ) u m ′ − m t + 1 u m ′ v m ′ − m t + 1 v m ′ d t + ( 1 + m ) 2 ∫ I 1 t + 1 u m ′ − m t + 1 u m v m ′ − m t + 1 v m d t + ( 1 + m ) u m ′ ( 1 ) v m ′ ( 1 )

with m = 1 .

Proof

Using integration by parts, pole conditions, and boundary conditions, we derive that

(4.3) ∫ I ( u m ″ v m ′ + u m ′ v m ″ ) d x = u m ′ ( 1 ) v m ′ ( 1 ) ,

(4.4) ∫ I 1 ( t + 1 ) 2 ( u m ′ v m + u m v m ′ ) d t = 2 ∫ I 1 ( t + 1 ) 3 u m v m d t ,

(4.5) ∫ I 1 t + 1 ( u m ″ v m + u m v m ″ ) d t = 2 ∫ I 1 ( t + 1 ) 3 u m v m d t − 2 ∫ I 1 ( t + 1 ) u m ′ v m ′ d t .

Then when m ≠ 1 , we derive from (4.3)–(4.5) that

∫ I ( t + 1 ) L m u m L m v m d t = ∫ I ( t + 1 ) u m ″ + u m ′ − m 2 t + 1 u m v m ″ + v m ′ t + 1 − m 2 ( t + 1 ) 2 v m d t = ∫ I ( t + 1 ) u m ″ v m ″ d t + ∫ I u m ″ v m ′ + u m ′ v m ″ d t − m 2 ∫ I 1 ( t + 1 ) 2 ( u m ′ v m + u m v m ′ ) d t − m 2 ∫ I 1 t + 1 ( u m ″ v m + u m v m ″ ) d t + m 4 ∫ I 1 ( t + 1 ) 3 u m v m d t + ∫ I 1 ( t + 1 ) u m ′ v m ′ d t = ∫ I ( t + 1 ) u m ″ v m ″ d t + ( 2 m 2 + 1 ) ∫ I 1 t + 1 u m ′ v m ′ d t + m 2 ( m 2 − 4 ) ∫ I 1 ( t + 1 ) 3 u m v m d t + u m ′ ( 1 ) v m ′ ( 1 ) .

When m = 1 , we have

∫ I ( t + 1 ) L m u m L m v m d t = ∫ I ( t + 1 ) u m ″ + u m ′ t + 1 − m 2 ( t + 1 ) 2 u m v m ″ + v m ′ t + 1 − m 2 ( t + 1 ) 2 v m d t = ∫ I ( t + 1 ) u m ′ − m t + 1 u m ′ v m ′ − m t + 1 v m ′ d t + ( 1 + m ) 2 ∫ I 1 t + 1 u m ′ − m t + 1 u m × v m ′ − m t + 1 v m d t + ( 1 + m ) ∫ I u m ′ − m t + 1 u m v m ′ − m t + 1 v m ′ d t = ∫ I ( t + 1 ) u m ′ − m t + 1 u m ′ v m ′ − m t + 1 v m ′ d t + ( 1 + m ) 2 ∫ I 1 t + 1 u m ′ − m t + 1 u m × v m ′ − m t + 1 v m d t + ( 1 + m ) u m ′ ( 1 ) v m ′ ( 1 ) . □

Theorem 1

A m ( u m , v m ) is a bounded and coercive bilinear functional on H 0 , ω , m 2 ( I ) × H 0 , ω , m 2 ( I ) , i.e.,

∣ A m ( u m , v m ) ∣ ≲ ‖ u m ‖ 2 , ω , m ‖ v m ‖ 2 , ω , m ,

A m ( u m , u m ) ≳ ‖ u m ‖ 2 , ω , m 2 .

Proof

From Cauchy-Schwarz inequality, we derive that

∣ A m ( u m , v m ) ∣ = ∫ I ( t + 1 ) L m u m L m v m d t ≤ ∫ I ( t + 1 ) ∣ L m u m ∣ 2 d t 1 2 ∫ I ( t + 1 ) ∣ L m v m ∣ 2 d t 1 2 ≲ ‖ u m ‖ 2 , ω , m ‖ v ‖ 2 , ω , m , A m ( u m , u m ) = ∫ I ( t + 1 ) ∣ L m u m ∣ 2 d t ≳ ‖ u m ‖ 2 , ω , m 2 . □

Lemma 3.2. Let λ m l be the lth eigenvalue of the variational form (3.1). Use V l to denote any l -dimensional subspace of H 0 , ω , m 2 ( I ) . For λ m 1 ≤ λ m 2 ≤ ⋯ ≤ λ m l ≤ ⋯ , it holds

(4.6) λ m l = min V l ⊂ H 0 , ω , m 2 ( I ) max v m ∈ V l A m ( v m , v m ) B m ( v m , v m ) .

Proof

See Theorem 3.1 in [17].□

Lemma 3.3. Let λ m i be the eigenvalue of the variational form (3.1) and be arranged in the ascending order. Define

W i , j = span { u m i , … , u m j } ,

where u m i is the eigenfunction associated with λ m i . Then there hold

(4.7) λ m l = max v m ∈ W k , l A m ( v m , v m ) B m ( v m , v m ) k ≤ l ,

(4.8) λ m l = min v m ∈ W l , n A m ( v m , v m ) B m ( v m , v m ) l ≤ n .

Proof

See Lemma 3.2 in [17].□

For the discrete form (3.2), the following minimax principle is also effective (see [17]).

Lemma 3.4. Let λ m h l be the eigenvalue of the discrete variational form (3.2). Use V l h to denote any l -dimensional subspace of S h ( m ) . For λ m h 1 ≤ λ m h 2 ≤ ⋯ ≤ λ m h l ≤ ⋯ , it holds

(4.9) λ m h l = min V h l ⊂ S h ( m ) max v m ∈ V h l A m ( v m , v m ) B m ( v m , v m ) .

Define an orthogonal projection Q h 2 , m : H 0 , ω , m 2 ( I ) → S h ( m ) by

A m ( u m − Q h 2 , m u m , v m h ) = 0 , ∀ v m h ∈ S h ( m ) .

Theorem 2

Let λ m h l be the approximation solution of λ m l . Then it holds

(4.10) 0 < λ m l ≤ λ m h l ≤ λ m l max v m ∈ W 1 , l B m ( v m , v m ) B m ( Q h 2 , m v m , Q h 2 , m v m ) .

Proof

According to the positive definite property of A m ( u m , v m ) and B m ( u m , v m ) we derive that λ m l > 0 . Since S h ( m ) ⊂ H 0 , ω , m 2 ( I ) , then from (4.6) and (4.9) we obtain λ m l ≤ λ m h l . Let Q h 2 , m W 1 , l be the space spanned by Q h 2 , m u m 1 , Q h 2 , m u m 2 , … , Q h 2 , m u m l . From the statements of Lemma 4.1 in [17], we know that Q h 2 , m W 1 , l is an l -dimensional subspace of S h ( m ) . We derive from the minimax principle that

λ m h l ≤ max v m ∈ Q h 2 , m W 1 , l A m ( v m , v m ) B m ( v m , v m ) = max v m ∈ W 1 , l A m ( Q h 2 , m v m , Q h 2 , m v m ) B m ( Q h 2 , m v m , Q h 2 , m v m ) .

From the bilinear property of A m ( v m , v m ) , we have A m ( v m , v m ) = A m ( Q h 2 , m v m , Q h 2 , m v m ) + 2 A m ( v m − Q h 2 , m v m , Q h 2 , m v m ) + A m ( v m − Q h 2 , m v m , v m − Q h 2 , m v m ) . Further from A m ( v m − Q h 2 , m v m , Q h 2 , m v m ) = 0 and the non-negativity of A m ( v m − Q h 2 , m v m , v m − Q h 2 , m v m ) , we have

A m ( Q h 2 , m v m , Q h 2 , m v m ) ≤ A m ( v m , v m ) .

Thus, we obtain that

λ m h l ≤ max v m ∈ W 1 , l A m ( v m , v m ) B m ( Q h 2 , m v m , Q h 2 , m v m ) = max v m ∈ W 1 , l A m ( v m , v m ) B m ( v m , v m ) B m ( v m , v m ) B m ( Q h 2 , m v m , Q h 2 , m v m ) ≤ λ m l max v m ∈ W 1 , l B m ( v m , v m ) B m ( Q h 2 , m v m , Q h 2 , m v m ) .

The proof is complete.□

Define the interpolation operator I m h : H 0 , ω , m 2 ( I ) → U h by

I m h u m ( t ) = H 3 , m , i ( t ) , t ∈ I i ,

where I i = [ t i − 1 , t i ] , H 3 , m i ( t ) is a cubic Hermite interpolation polynomial of u m in I i . Let

u m i ( t ) = u m ( t ) , t ∈ I i .

From the remainder theorem of cubic Hermite interpolation, we have

u m i ( t ) − H 3 , m , i ( t ) = ( u m i ) ( 4 ) ( ξ m i ( t ) ) 4 ! ( t − t i − 1 ) 2 ( t − t i ) 2 ,

where ξ m i ( t ) ∈ I i is a function depending on t .

Theorem 3

Let E m i ( t ) = ( u m i ) ( 4 ) ( ξ m i ( t ) ) 4 ! , u m ∈ H 0 , ω , m 2 ( I ) . Assume that u m is sufficiently smooth such that ∣ ∂ t k E m i ( t ) ∣ ≤ M ( k = 0 , 1 , 2 ) , where M is a constant greater than zero. Then the following inequality holds:

(4.11) ‖ ∂ t 2 ( I m h u m − u m ) ‖ ≲ h 2 ,

where h = max 1 ≤ i ≤ n { h i } , h i = t i − t i − 1 , ‖ u m ‖ = ∫ I u m 2 d t 1 2 .

Proof

Since

u m i ( t ) − H 3 , m , i ( t ) = E m i ( t ) ( t − t i − 1 ) 2 ( t − t i ) 2 ,

then we have

∂ t 2 ( u m i ( t ) − H 3 , m , i ( t ) ) = ∂ t 2 E m i ( t ) ( t − t i − 1 ) 2 ( t − t i ) 2 + 2 ∂ t E m i ( t ) ∂ t [ ( t − t i − 1 ) 2 ( t − t i ) 2 ] + E m i ( t ) ∂ t 2 [ ( t − t i − 1 ) 2 ( t − t i ) 2 ] .

Thus, we obtain

∣ ∂ t 2 ( u m i ( t ) − H 3 , m , i ( t ) ) ∣ 2 ≲ [ ( t − t i − 1 ) 2 ( t − t i ) 2 ] 2 + [ ∂ t ( ( t − t i − 1 ) 2 ( t − t i ) 2 ) ] 2 + { ∂ t 2 [ ( t − t i − 1 ) 2 ( t − t i ) 2 ] } 2 = [ ( t − t i − 1 ) ( t − t i ) ] 4 + 4 [ ( t − t i − 1 ) ( t − t i ) ( 2 t − t i − 1 − t i ) ] 2 + 4 [ 4 ( t − t i − 1 ) ( t − t i ) + ( t − t i − 1 ) 2 + ( t − t i ) 2 ] 2 ≤ h i 2 8 + 4 h i h i 2 2 2 + 8 h i 2 2 + h i 2 2 ≲ h i 4 .

Thus,

‖ ∂ t 2 ( I m h u m − u m ) ‖ 2 = ∑ i = 1 n ∫ I i [ ∂ t 2 ( u m i ( t ) − H 3 , m , i ( t ) ) ] 2 d t ≲ ∑ i = 1 n h i 5 ≲ h 4 .

Furthermore, we have

‖ ∂ t 2 ( I m h u m − u m ) ‖ ≲ h 2 .

The proof is complete.□

Theorem 4

Let λ m h l be the approximate eigenvalue of λ m l . Assume that u m ∈ H 0 , ω , m 2 ( I ) and satisfies the condition of Theorem 3, then the following inequality holds:

∣ λ m h l − λ m l ∣ ≲ h 4 ,

where c ( l ) is a constant independent of h.

Proof

For brief, we only give the proof for the case of m ≠ 1 , and it can be similarly proven for the case of m = 1 . For ∀ q ∈ W 1 , l , we have q = ∑ i = 1 l q i u m i . By using the orthogonality of the characteristics function u m i and B m ( u m i , u m i ) = 1 , we have

B m ( q , q ) − B m ( Q h 2 , m q , Q h 2 , m q ) B m ( q , q ) ≤ 2 ∣ B m ( q , q − Q h 2 , m q ) ∣ B m ( q , q ) ≤ 2 ∑ i , j = 1 l ∣ q i q j B m ( u m i − Q h 2 , m u m i , u m j ) ∣ ∑ i = 1 l ∣ q i ∣ 2 ≤ 2 l max i , j = 1 , … , l ∣ B m ( u m i − Q h 2 , m u m i , u m j ) ∣ .

From Cauchy-Schwarz inequality we have

∣ B m ( u m i − Q h 2 , m u m i , u m j ) ∣ = 1 λ m j ∣ λ m j B m ( u m j , u m i − Q h 2 , m u m i ) ∣ = 1 λ m j ∣ A m ( u m j , u m i − Q h 2 , m u m i ) ∣ = 1 λ m j ∣ A m ( u m j − Q h 2 , m u m j , u m i − Q h 2 , m u m i ) ∣ ≤ 1 λ m j [ A m ( u m j − Q h 2 , m u m j , u m j − Q h 2 , m u m j ) ] 1 2 [ A m ( u m i − Q h 2 , m u m i , u m i − Q h 2 , m u m i ) ] 1 2 .

When m ≥ 2 , from Hardy inequality (cf. B8.6 in [16]) we derive

∫ I 1 ( t + 1 ) 3 ( u m i ) 2 d t ≲ ∫ I 1 t + 1 ( ∂ t u m i ) 2 d t ,

∫ I 1 ( t + 1 ) 2 ( ∂ t u m i ) 2 d t ≲ ∫ I ( ∂ t 2 u m i ) 2 d t .

Since

[ ∂ t u m i ( 1 ) ] 2 = 1 4 ∫ I ∂ t ( ( t + 1 ) ∂ t u m i ) d t 2 = 1 4 ∫ I ∂ t u m i + ( t + 1 ) ∂ t 2 u m i d t 2 ≲ ∫ I ( ∂ t u m i ) 2 d t + ∫ I ( t + 1 ) ( ∂ t 2 u m i ) 2 d t ≲ ∫ I 1 t + 1 ( ∂ t u m i ) 2 d t + ∫ I ( t + 1 ) ( ∂ t 2 u m i ) 2 d t ,

from Lemma 1 we have

‖ u m i ‖ 2 , ω , m 2 ≲ ∫ I ( t + 1 ) ( ∂ t 2 u m i ) 2 d t + ∫ I 1 t + 1 ( ∂ t u m i ) 2 d t + ∫ I 1 ( t + 1 ) 3 ( u m i ) 2 d t ≲ ∫ I ( ∂ t 2 u m i ) 2 d t + ∫ I 1 t + 1 ( ∂ t u m i ) 2 d t ≲ ∫ I ( ∂ t 2 u m i ) 2 d t + ∫ I 1 ( t + 1 ) 2 ( ∂ t u m i ) 2 d t ≲ ∫ I ( ∂ t 2 u m i ) 2 d t .

Then we derive that

∣ B m ( u m i − Q h 2 , m u m i , u m j ) ∣ = 1 λ m j ∣ λ m j B m ( u m j , u m i − Q h 2 , m u m i ) ∣ ≤ 1 λ m j [ A m ( u m j − Q h 2 , m u m j , u m j − Q h 2 , m u m j ) ] 1 2 [ A m ( u m i − Q h 2 , m u m i , u m i − Q h 2 , m u m i ) ] 1 2 ≤ 1 λ m j [ A m ( u m j − I m h u m j , u m j − I m h u m j ) ] 1 2 ⋅ [ A m ( u m i − I m h u m i , u m i − I m h u m i ) ] 1 2 ≤ M λ m j ‖ u m j − I h u m j ‖ 2 , ω , m ‖ u m i − I h u m i ‖ 2 , ω , m ≲ ∫ I [ ∂ t 2 ( u m j − I m h u m j ) ] 2 d t 1 2 ∫ I [ ∂ t 2 ( u m i − I m h u m i ) ] 2 d t 1 2 ≲ ‖ ∂ t 2 ( u m j − I m h u m j ) ‖ ⋅ ‖ ∂ t 2 ( u m i − I m h u m i ) ‖ .

Similarly, when m = 0 , we derive that

∣ B m ( u m i − Q h 2 , m u m i , u m j ) ∣ ≲ ‖ ∂ t 2 ( u m j − I m h u m j ) ‖ ⋅ ‖ ∂ t 2 ( u m i − I m h u m i ) ‖ .

Since

B m ( q , q ) B m ( Q h 2 , m q , Q h 2 , m q ) ≤ 1 1 − 2 l max i , j = 1 , … , l ∣ B m ( u m i − Q h 2 , m u m i , u m j ) ∣ ,

we obtain from Theorems 2 and 3 the desired results.□

5 Efficient implementation of the algorithm

In order to efficiently solve the problems (3.2), we start by constructing a set of basis functions which satisfy boundary conditions. Let

φ 0 0 ( t ) = 2 t − t 0 h 1 + 1 t − t 0 h 1 − 1 2 , t i ≤ t ≤ t i + 1 , 0 , others , φ 0 1 ( t ) = h 1 − 2 ( t − t 0 ) ( t − t 1 ) 2 , t 0 ≤ t ≤ t 1 0 , others , φ i 0 ( t ) = 1 + t − t i h i 2 1 − 2 t − t i h i , t i − 1 ≤ t ≤ t i , 2 t − t i h i + 1 + 1 t − t i h i + 1 − 1 2 , t i ≤ t ≤ t i + 1 , 0 , others , φ i 1 ( t ) = h i − 2 ( t − t i ) ( t − t i − 1 ) 2 , t i − 1 ≤ t ≤ t i , h i + 1 − 2 ( t − t i ) ( t − t i + 1 ) 2 , t i ≤ t ≤ t i + 1 , 0 , others , φ n 1 ( t ) = h n − 2 ( t − t n ) ( t − t n − 1 ) 2 , t n − 1 ≤ t ≤ t n , 0 , others ,

where i = 1 , … , n − 1 . It is clear that

S h ( 0 ) = span { φ 0 0 ( t ) , … , φ n − 1 0 ( t ) , φ 1 1 ( t ) , … , φ n 1 ( t ) } ; S h ( 1 ) = span { φ 1 0 ( t ) , … , φ n − 1 0 ( t ) , φ 0 1 ( x ) , … , φ n 1 ( t ) } ; S h ( m ) = span { φ 1 0 ( t ) , … , φ n − 1 0 ( t ) , φ 1 1 ( x ) , … , φ n 1 ( t ) } , ( m ≥ 2 ) .

Denote

a i j p q = ∫ I ( t + 1 ) ( φ j p ) ″ ( φ i q ) ″ d t , b i j p q = ∫ I 1 t + 1 ( φ j p ) ′ ( φ i q ) ′ d t , c i j p q = ∫ I 1 ( t + 1 ) 3 φ j p φ i q d t , d i j p q = φ j p ( 1 ) φ i q ( 1 ) ,

where p , q = 0 , 1 .

Next, we will derive the matrix form of the discrete variational scheme (3.2).

Case 1. When m = 0 , let

(5.1) u 0 h = u 0 0 φ 0 0 + u n 1 φ n 1 + ∑ i = 1 n − 1 ( u i 0 φ i 0 + u i 1 φ i 1 ) .

Plugging the expression (5.1) in (3.2) and taking v 0 h through all the basis functions in S h ( 0 ) , we derive that

(5.2) ( A 0 + B 0 + D 0 ) U 0 = λ 0 h R D 0 U 0 ,

where

A 0 = ( a i j 00 ) ( a i j 10 ) ( a i j 01 ) ( a i j 11 ) , B 0 = ( b i j 00 ) ( b i j 10 ) ( b i j 01 ) ( b i j 11 ) , D 0 = ( d i j 00 ) ( d i j 10 ) ( d i j 01 ) ( d i j 11 ) ,

U 0 = ( u 0 0 , … , u N − 1 0 , u 1 1 , … , u n 1 ) T .

Similarly, when m = 1 , let

(5.3) u 1 h = u 0 1 φ 0 1 + u n 1 φ n 1 + ∑ i = 1 n − 1 ( u i 0 φ i 0 + u i 1 φ i 1 ) .

Plugging the expression (5.3) in (3.2) and taking v 1 h through all the basis functions in S h ( 1 ) , we obtain

(5.4) [ A 1 + ( 1 + m ) D 1 ] U 1 = λ 1 h R D 1 U 1 ,

where

A 1 = ( a ˜ i j 00 ) ( a ˜ i j 10 ) ( a ˜ i j 01 ) ( a ˜ i j 11 ) , D 1 = ( d ˜ i j 00 ) ( d ˜ i j 10 ) ( d ˜ i j 01 ) ( d ˜ i j 11 ) ,

a ˜ i j p q = ∫ I ( t + 1 ) L 1 φ j p L 1 φ i q d t , d ˜ i j p q = φ j p ( 1 ) φ i q ( 1 ) , U 1 = ( u 1 0 … , u n − 1 0 , u 0 1 , … , u n 1 ) T .

When m ≥ 2 , let

(5.5) u 2 h = u n 1 φ n 1 + ∑ i = 1 n − 1 ( u i 0 φ i 0 + u i 1 φ i 1 ) .

Plugging the expression (5.5) in (3.2) and taking v m h through all the basis functions in S h ( m ) , we gain

(5.6) [ A m + ( 2 m 2 + 1 ) B m + m 2 ( m 2 − 4 ) C m + D m ] U m = λ m h R D m U m ,

where

(5.7) A m = ( a i j 00 ) ( a i j 10 ) ( a i j 01 ) ( a i j 11 ) , B m = ( b i j 00 ) ( b i j 10 ) ( b i j 01 ) ( b i j 11 ) , C m = ( c i j 00 ) ( c i j 10 ) ( c i j 01 ) ( c i j 11 ) , D m = ( d i j 00 ) ( d i j 10 ) ( d i j 01 ) ( d i j 11 )

U m = ( u 1 0 , … , u n − 1 0 , u 1 1 , … , u n 1 ) T .

Note that we know from the properties of cubic hermit interpolation basis function that the stiff matrices and mass matrices in (5.4)–(5.6) are all sparse. Thus, they can be efficiently solved.

6 Numerical experiments

In order to show the accuracy and convergence of the proposed algorithm, we will carry out a series of numerical tests. We operate our programs in MATLAB 2016b.

Example 1

We take R = 1 and m = 0 , 1 , 2 , 3 . The eigenvalues for different m and h are listed in Table 1.

Table 1

Eigenvalues for m = 0 , 1 , 2 , 3 and different h

h λ 0 h λ 1 h λ 2 h λ 3 h
1/8 2.000000024461073 4.000000073005518 6.000025208583129 8.000307091454056
1/16 2.000000015899493 4.000000072449404 6.000001539514262 8.000019101762424
1/32 2.000000006492949 4.000000084945128 6.000000095601314 8.000001192525161
1/64 2.000000002660813 4.000000091047035 6.000000005945074 8.000000074519297

We know from Table 1 that the eigenvalues achieve at least six-digit accuracy with h ≤ 1 32 for m = 0 , 1 , 2 , 3 . In order to further show the convergence of the algorithm, we choose the numerical solutions of h = 1 64 as reference solutions, and the error figures of the approximate eigenvalues λ m h ( m = 0 , 1 , 2 , 3 ) with different h are presented in Figure 1. We observe from Figure 1 that the numerical eigenvalues are also convergent.

Figure 1 Errors between numerical solutions and the reference solution for R = 1 R=1 .
Figure 1

Errors between numerical solutions and the reference solution for R = 1 .

Example 2

We take R = 2 and m = 0 , 1 , 2 , 3 . The eigenvalues for different m and h are listed in Table 2.

Table 2

Eigenvalues for m = 0 , 1 , 2 , 3 and different h

h λ 0 h λ 1 h λ 2 h λ 3 h
1/8 1.000000012230537 2.000000036502759 3.000012604291565 4.000153545727028
1/16 1.000000007949747 2.000000036224702 3.000000769757131 4.000009550881212
1/32 1.000000003246474 2.000000042472564 3.000000047800657 4.000000596262581
1/64 1.000000001330406 2.000000045523517 3.000000002972537 4.000000037259649

Similarly, we observe from Table 2 that the eigenvalues have at least six-digit accuracy with h ≤ 1 32 for m = 0 , 1 , 2 , 3 . We still choose the numerical solutions of h = 1 64 as reference solutions, the error figures of the approximate eigenvalues λ m h ( m = 0 , 1 , 2 , 3 ) with different h are listed in Figure 2. We see from Figure 2 that the approximation eigenvalues are also convergent. Besides, in order to show the convergence rate of our algorithm more intuitively, we also plot the error figures in semilog scale in Figures 3 and 4.

Figure 2 Errors between numerical solutions and the reference solution for R = 2 R=2 .
Figure 2

Errors between numerical solutions and the reference solution for R = 2 .

Figure 3 Error curves in semilog scale between the numerical solution and the reference solution for R = 1 R=1 .
Figure 3

Error curves in semilog scale between the numerical solution and the reference solution for R = 1 .

Figure 4 Error curves in semilog scale between the numerical solution and the reference solution for R = 2 R=2 .
Figure 4

Error curves in semilog scale between the numerical solution and the reference solution for R = 2 .

Next, we shall provide a numerical example for some larger Fourier norm m .

Example 3

We take R = 1 and m = 4 , 5 , 6 , 7 . The eigenvalues for different m and h are listed in Table 3.

Table 3

Eigenvalues for m = 4 , 5 , 6 , 7 and different h

h λ 4 h λ 5 h λ 6 h λ 7 h
1/4 10.0238272394670 12.0756338414226 14.1876937235392 16.3931316663901
1/8 10.0014906003953 12.0048786723652 14.0126111745548 16.0277527229022
1/16 10.0000931108253 12.0003074765839 14.0008034680874 16.0017927950984
1/32 10.0000058180331 12.0000192583640 14.0000504622624 16.0001129998648
1/64 10.0000003635851 12.0000012042859 14.0000031577541 16.0000070775063
1/128 10.0000000229184 12.0000000756600 14.0000001974885 16.0000004427145

Likewise, we observe from Table 3 that the eigenvalues have at least five-digit accuracy with h ≤ 1 64 for m = 4 , 5 , 6 , 7 . We choose the numerical solutions of h = 1 128 as reference solutions, and the error figures of the approximate eigenvalues λ m h ( m = 4 , 5 , 6 , 7 ) with different h are listed in Figure 5. We see from Figure 5 that the approximation eigenvalues are also convergent.

Figure 5 Errors between numerical solutions and the reference solution for m = 4 , 5 , 6 , 7 m=4,5,6,7 and R = 1 R=1 .
Figure 5

Errors between numerical solutions and the reference solution for m = 4 , 5 , 6 , 7 and R = 1 .

Introducing the usual definition of convergence order:

(6.1) c ( h ) = log 2 ∣ λ h − λ h 2 ∣ ∣ λ h 2 − λ h 4 ∣ .

For brevity, we shall use formula (6.1) to calculate the convergence order of the approximation eigenvalues of m = 4 , 5 , 6 , 7 in the unit disk and list them in Table 4. We observe from Table 4 that the convergence order is about 4.

Table 4

Convergence order c ( h ) for the eigenvalues with m = 4 , 5 , 6 , 7

h λ 4 h λ 5 h λ 6 h λ 7 h
1/2 3.9924 3.8459 3.6960 3.5770
1/4 3.9985 3.9522 3.8902 3.8150
1/8 4.0008 3.9873 3.9709 3.9499
1/16 4.0003 3.9968 3.9926 3.9872
1/32 4.0010 3.9997 3.9982 3.9968

Finally, we plot the eigenvalue error in log-log scale to describe the algebra convergence rate in Figure 6.

Figure 6 The eigenvalue errors in log-log scale between numerical solutions and the reference solution for m = 4 , 5 , 6 , 7 m=4,5,6,7 and R = 1 R=1 .
Figure 6

The eigenvalue errors in log-log scale between numerical solutions and the reference solution for m = 4 , 5 , 6 , 7 and R = 1 .

7 Conclusion

We present in this article a novel finite element method based on a dimension reduction scheme for a fourth-order Steklov eigenvalue problem in a circular domain. The main advantage of this method is that the original problem is transformed into a series of one-dimensional problems which can be solved in parallel. Then, by introducing the polar conditions and the weighted Sobolev space, we prove the error estimates of approximation eigenvalues by using the minimax principle. The method developed in this article can be applied to more complex problems or more general polar geometric domains which will be the subject of our future endeavors.

  1. Funding information: The work of Zixin Liu is supported by the National Natural Science Foundation of China (No. 62062018), Guizhou Province University Science and Technology top talents project (No. KY[2018]047), and Guizhou Key Laboratory of Big Data Statistics Analysis (No. BDSA20200102). The work of Jun Zhang is supported by the Science and Technology Program of Guizhou Province (No. ZK[2022]006).

  2. Author contributions: H. Zhang analyzed most of the data and wrote the initial draft of the article. J. Zhang contributed to refining the ideas, carrying out additional analyses, and finalizing this article. Z. X. Liu contributed the central idea.

  3. Conflict of interest: The authors declare that we have no conflict of interest.

  4. Data availability statement: Some or all data, models, or code generated or used during the study are available from the corresponding author by request.

References

[1] F. Gazzola and G. Sweers, On positivity for the biharmonic operator under Steklov boundary conditions, Arch. Ration. Mech. Anal. 188 (2008), no. 3, 399–427. 10.1007/s00205-007-0090-4Suche in Google Scholar

[2] E. Sassone, Positivity for polyharmonic problems on domains close to a disk, Ann. Mat. Pura Appl. 186 (2007), no. 3, 419–432. 10.1007/s10231-006-0012-3Suche in Google Scholar

[3] S. Bergman and M. Schiffer, Kernel Functions and Elliptic Differential Equations in Mathematical Physics, Academic Press, New York, NY, USA, 1953. Suche in Google Scholar

[4] C. Conca, J. Planchard, and M. Vanninathan, Fluids and Periodic Structures, John Wiley & Sons, New York, NY, USA, 1995. Suche in Google Scholar

[5] A. Bermudez, R. Rodriguez, and D. Santamarina, A finite element solution of an added mass formulation for coupled fluid-solid vibrations, Numer. Math. 87 (2000), no. 2, 201–227. 10.1007/s002110000175Suche in Google Scholar

[6] E. Berchio, F. Gazzola, and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems, J. Differential Equations 229 (2006), no. 1, 1–23. 10.1016/j.jde.2006.04.003Suche in Google Scholar

[7] A. Ferrero, F. Gazzola, and T. Weth, On a fourth order Steklov eigenvalue problem, Analysis 25 (2005), no. 4, 315–332. 10.1524/anly.2005.25.4.315Suche in Google Scholar

[8] J. R. Kuttler, Remarks on a Stekloff eigenvalue problem, SIAM J. Numer. Anal. 9 (1972), 1–5. 10.1137/0709001Suche in Google Scholar

[9] D. Bucur, A. Ferrero, and F. Gazzola, On the first eigenvalue of a fourth order Steklov eigenvalue problem, Calc. Var. Partial Differ. Equ. 35 (2009), no. 1, 103–131. 10.1007/s00526-008-0199-9Suche in Google Scholar

[10] D. Bucur and F. Gazzola, The first biharmonic Steklov eigenvalue: positivity preserving and shape optimization, Milan J. Math. 79 (2011), no. 1, 247–258. 10.1007/s00032-011-0143-xSuche in Google Scholar

[11] G. Liu, The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues with Dirichlet boundary condition on Riemannian manifolds, Adv. Math. 28 (2009), no. 4, 2162–2217. 10.1016/j.aim.2011.07.001Suche in Google Scholar

[12] H. Bi, S. Ren, and Y. Yang, Conforming finite element approximations for a fourth-order Steklov eigenvalue problem, Math. Probl. Eng. 2011 (2011), 34–35. 10.1155/2011/873152Suche in Google Scholar

[13] J. An, H. Bi, and Z. Luo, A highly efficient spectral-Galerkin method based on tensor product for fourth-order Steklov equation with boundary eigenvalue, J. Inequal. Appl. 2016 (2016), no. 1, 1–2. 10.1186/s13660-016-1158-1Suche in Google Scholar

[14] J. An, H. Li, and Z. Zhang, Spectral-Galerkin approximation and optimal error estimate for biharmonic eigenvalue problems in circular/spherical/elliptical domains, Numer. Algorithms 1 (2019), 1–29. 10.1007/s11075-019-00760-4Suche in Google Scholar

[15] J. An and Z. Zhang, An efficient spectral-Galerkin approximation and error analysis for Maxwell transmission eigenvalue problems in spherical geometries, J. Sci. Comput. 75 (2018), 157–181. 10.1007/s10915-017-0528-2Suche in Google Scholar

[16] J. Shen, T. Tang, and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Science and Business Media, Heidelberg, 2011. 10.1007/978-3-540-71041-7Suche in Google Scholar

[17] M. S. Min and D. Gottlieb, On the convergence of the Fourier approximation for eigenvalues and eigenfunctions of discontinuous problems, SIAM J. Numer. Anal. 40 (2003), no. 6, 2254–2269. 10.1137/S0036142902403012Suche in Google Scholar
Received: 2021-03-20
Revised: 2022-02-16
Accepted: 2022-03-01
Published Online: 2022-08-23

© 2022 Hui 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
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
https://doi.org/10.1515/math-2022-0032
Schlagwörter für diesen Artikel
fourth-order Steklov eigenvalue problem; weighted Sobolev space; dimension reduction scheme; finite element approximation; error estimation
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 8.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/math-2022-0032/html
Button zum nach oben scrollen