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Research article

Blow-up of solutions for a time fractional biharmonic equation with exponentional nonlinear memory

  • Received: 30 September 2024 Revised: 29 October 2024 Accepted: 06 November 2024 Published: 08 November 2024
  • In the paper, we focus on the local existence and blow-up of solutions for a time fractional nonlinear equation with biharmonic operator and exponentional nonlinear memory in an Orlicz space. We first establish a LpLq estimate for solution operators of a time fractional nonlinear biharmonic equation, and obtain bilinear estimates for mild solutions. Then, based on the contraction mapping principle, we establish the local existence of mild solutions. Moreover, by using the test function method, we obtain the blow-up result of solutions.

    Citation: Yuchen Zhu. Blow-up of solutions for a time fractional biharmonic equation with exponentional nonlinear memory[J]. Electronic Research Archive, 2024, 32(11): 5988-6007. doi: 10.3934/era.2024278

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  • In the paper, we focus on the local existence and blow-up of solutions for a time fractional nonlinear equation with biharmonic operator and exponentional nonlinear memory in an Orlicz space. We first establish a LpLq estimate for solution operators of a time fractional nonlinear biharmonic equation, and obtain bilinear estimates for mild solutions. Then, based on the contraction mapping principle, we establish the local existence of mild solutions. Moreover, by using the test function method, we obtain the blow-up result of solutions.



    The Steklov spectrum coincides with that of the Dirichlet-to-Neumann map for the Laplacian (see, e.g., [1]), and the Steklov eigenvalue problem for the Laplace operator has been well-studied in the mathematical community. In linear elasticity, the study of the Dirichlet-to-Neumann map is important in elastostatic problems, and has attracted the attention of scholars (see, e.g., [2,3,4]). In 2021, Domˊinguez [5] first introduced the Steklov-Lamé eigenvalue problem in which the spectral parameter appears on a Robin boundary condition. [5] investigated the existence of the countable spectrum of this problem and studied the conforming finite element methods for the Steklov Lamé problem. Later, Li and Bi [6] proposed a discontinuous finite element method for this problem and gave the a priori error estimates.

    As we know, for numerical solutions of the problems in linear planar elasticity, standard conforming finite elements may suffer a deterioration in performance as the Lamˊe constant λ, that is locking phenomenon (see [7,8]). To overcome the locking phenomenon, several numerical approaches have been developed including the p-version method [9], the PEERS method [10], the mixed method [11], the Galerkin least squares method [12], the nonconforming triangular elements [13,14] and the discontinuous finite element method [15,16,17], and so on.

    On the other hand, based on standard finite element methods, people design many efficient discretization schemes/algorithms to get approximations with high accuracy or to reduce the computation costs. The finite element multigrid discretizations is one of such design approaches. This method benefits from the two-grid discretization scheme which was first proposed by Xu and Zhou [18,19]. The basic idea of the two-grid discretizations is to transform solving an eigenvalue problem on a fine grid into solving the eigenvalue problem on a coarse grid and solving a series of algebraic equations on the fine grid. This kind of method can save calculation time while keeping the accuracy of approximations, or improving the accuracy under the same degrees of freedom. So far, two-grid and multigrid finite element discretization schemes have been successfully applied to solving eigenvalue problems, such as elliptic eigenvalue problem [20], Steklov eigenvalue problem [21,22,23,24], biharmonic eigenvalue problem [25], semilinear elliptic eigenvalue problem [26], quantum eigenvalue problem [27], Stokes eigenvalue problem [28,29], Maxwell eigenvalue problem [30], 2m-order elliptic eigenvalue problem [31], etc.

    At present, there is not much numerical research report on the Steklov-Lamé eigenproblem. In view of the characteristics of discontinuous finite element method (DGFEM) and multigrid discretizations and based on the work in [6,32], for the Steklov-Lamé eigenvalue problem we will design and analyze a multigrid discretization scheme of DGFEM based on the shifted-inverse iteration. The rest of this paper is organized as follows. In Section 2, the discontinuous finite element approximation of the Steklov-Lamé eigenvalue problem and its a prior error estimates are given. In Section 3, a multigrid discretization scheme of DGFEM based on the shifted-inverse iteration is established, and the error estimates of the proposed scheme is presented. Finally, in Section 4, an adaptive multigrid algorithm is provided coupled with some numerical experiment results. The numerical results show that our method is efficient and locking-free.

    Before the discussion, let us specify some notations. Scalars are denoted by general letters, vectors are denoted by bold letters and tensors in bold Greek letters. For tensors σ,τRn×n, the double dot product notation σ:τ=tr(τTσ) where tr() denotes the trace of a tensor (sum of the main diagonal). This inner product induces the Frobenius norm for tensors which is denoted as . Let Hs(Ω) and Hs(Ω) be the usual Sobolev space with order s of scalar fields on Ω and Ω, respectively, whereas for tensor fields we use the symbols Hs(Ω) and Hs(Ω) and each element in Hs belongs to Hs. The norm in Hs(Ω) and Hs(Ω) are denoted by s and s,Ω, respectively, and the same symbols are also used for the norms in Hs(Ω) and Hs(Ω) when there is no ambiguity. H0(Ω)=L2(Ω). Throughout this paper, we use the letter C, with or without subscript, to denote a generic positive constant independent of the mesh size h and the Lamé parameters, which may not be the same at each occurrence. We use the symbol ab to mean that aCb.

    Suppose that an isotropic and linearly elastic material occupies the region Ω in Rn (n=2or3) where Ω is a bounded convex polygonal with Lipschitz continuous boundary Ω. Consider the following Steklov-Lamé eigenvalue problem: Find non-zero displacement vector u and the frequencies ωR satisfying

    {divσ(u)=0in Ω,σ(u)n=ωpuon Ω, (2.1)

    where n is the unit outward normal to Ω, σ(u) is the Cauchy stress tensor defined as

    σ(u)=2μϵ(u)+λtr(ϵ(u))I,

    where IRn×n is the identity matrix, ϵ(u) is the strain tensor given by

    ϵ(u)=12(u+(u)T),

    u is the displacement gradient tensor, and λR and μ>0 are the Lamé parameters satisfying 0<μ1<μ<μ2 and 0<λ<.

    Suppose that the density of material pL(Ω) has positive lower bound on Ω.

    Denote

    RM(Ω):={vH1(Ω)|v(x)=a+Bx,aRn,BRn×n,BT=B,xΩ}.

    It is obvious that 0 is an eigenvalue of (2.1) with the associated eigenfunction uRM(Ω) (see [5]). To find non-zero eigenvalues of (2.1), we adopt the following weak formulation: Seek (κ,u)R×H1(Ω) such that

    a(u,v)=κb(u,v), vH1(Ω), (2.2)

    where κ=ω+1,

    a(u,v):=Ωσ(u):ϵ(v)dx+Ωpuvds=2μΩϵ(u):ϵ(v)dx+λΩ(divu)(divv)dx+Ωpuvds, u,vH1(Ω),b(u,v):=Ωpuvds, u,vH1(Ω).

    Reference[5] proved that a(,) is a continuous and H1-coercive bilinear form in H1(Ω), b(,) is bounded.

    Without losing generality, we assume that p1 in the rest of this paper. Denote vb=b(v,v)12, then it is clear that b=0,Ω.

    Let Th={K} be a shape-regular partition of Ω, and h=max{hK:KTh} is the diameter of Th where hK is the diameter of element K. When n=2, K is a triangle and a tetrahedron when n=3. Let eK be an edge/face of element K with diameter he, and let Γh=ΓihΓbh where Γih denotes the interior edges/faces set and Γbh denotes the set of edges/faces lying on the boundary Ω. In the following, when there is no confusion we always use n to represent the unit outward normal on the boundary of Ω or element K.

    Define the broken Sobolev space:

    Hs(Th)={v[L2(Ω)]n:vK[Hs(K)]n,KTh}.

    For any vHs(Th), define the jump [[v]] and the average {v} on e as follows:

    [[v]]={v+v,eΓih,v+ ,eΓbh,{v}={v++v2,eΓih,v+ , eΓbh,

    where v+=v|K+,v=v|K, eK+K.

    Define the DGFEM space:

    Sh={v[L2(Ω)]n:vK[Pk(K)]n,KTh},

    where Pk(K) is the space of polynomials defined on K with degree less than or equal to k1.

    The DGFEM discretization for the problem (2.2) is to find (κh,uh)R×Sh,uh0,κh=ωh+1, such that

    ah(uh,vh)=κhbh(uh,vh), vhSh, (2.3)

    where

    ah(uh,vh)=2μ(KThKϵ(uh):ϵ(vh)dxeΓihe{ϵ(uh)n}[[vh]]dseΓihe{ϵ(vh)n}[[uh]]ds+eΓihγμhee[[uh]][[vh]]ds)+λ(KThK(divuh)(divvh)dxeΓihe{divuh}[[vhn]]dseΓihe{divvh}[[uhn]]ds+eΓihγλhee[[uhn]][[vhn]]ds)+eΓbheuhvhds,bh(uh,vh)=eΓbheuhvhds,

    and the penalty constants γμ,γλ are independent of the shape of K and h. The determination of γμ and γλ is to ensure that (2.4) is valid. It is easy to see that the discretization (2.3) is symmetric which is called symmetric internal penalty method (SIPG) in DGFEM.

    Define the DG norm:

    uh2G=2μKThϵ(uh)20,K+2μeΓihγμh1e[[uh]]20,e+λKThdivuh20,K+λeΓihγλh1e[[uhn]]20,e+eΓbhuh20,e,

    and the energy-like norm:

    uh2h=uh2G+2μeΓihhe{ϵ(uh)n}20,e+λeΓihhe{divuh}20,e.

    From Lemma 4 in [33] we know that there exist constants Cμ and Cλ, independent of h,he,μ and λ, such that

    h1/2eϵ(v)n20,eCμϵ(v)20,K,h1/2edivv20,eCλdivv20,K.

    Then, for 0<β<1, when γμCμ/(1β)2, γλCλ/(1β)2, the bilinear form ah(,) is coercive on Sh (see Lemma 2.2 in [6]):

    βvh2Gah(vh,vh),vhSh. (2.4)

    Using Cauchy-Schwartz inequality, it is easy to prove that the bilinear form ah(,) is continuous:

    |ah(u,v)|Muhvh,u,vH1+s(Th),s>12.

    In order to derive the convergence and the error estimates of DG approximations by using Babu˘ska-Osborn spectral approximation theory, we consider the following source problem associated with the eigenvalue problem (2.2): find wH1(Ω) such that

    a(w,v)=b(f,v), vH1(Ω). (2.5)

    The DG approximation of (2.5) is to find whSh such that

    ah(wh,vh)=bh(f,vh), vhSh. (2.6)

    Since a(,) and ah(,) are continuous and coercive on H1(Ω) and Sh, respectively, b(,) and bh(,) are bounded, from Lax-Milgram Theorem we know that (2.5) and (2.6) admit the unique solution w and wh, respectively.

    The following regularity estimates of the solution of (2.5) has been discussed in Lemma 3.1 of [6].

    (1) Let w be the solution of (2.5). If fHr12(Ω), then wHr+1(Ω) and

    wr+1+λdivwrCRfr12,Ω,

    where r=1 when Ω is a convex polygonal, and r can be large enough when Ω is sufficiently smooth;

    (2) If fH12(Ω), then wH1(Ω) and

    w1+λdivw0CRf12,Ω;

    (3) If fL2(Ω), then wH1+12(Ω) and

    w1+12+λdivw12CRf0,Ω, (2.7)

    where the constant CR is independent of μ and λ.

    For any given fL2(Ω), from (2.7) we have wH1+r(Ω),r<12 and r can be arbitrarily close to 12, and

    w1+r+λdivwrCRf0,Ω. (2.8)

    Let w and wh be the solution of (2.5) and (2.6), respectively, then the SIPG approximation (2.6) of (2.5) is consistent (see Lemma 3.3 in [6]):

    ah(wwh,vh)=0, vhSh. (2.9)

    For the source problem (2.5), let fL2(Ω), define the solution operator  A: L2(Ω)H1(Ω) by

    a(Af,v)=b(f,v), vH1(Ω),

    and define the operator  T: L2(Ω)L2(Ω):

    Tf=(Af),

    where   denotes the restriction on  Ω. Then, (2.2) has the following equivalent operator form:

    Au=1κu.

    Similarly, from (2.6) we can define the discrete solution operator  Ah: L2(Ω)Sh by

    ah(Ahf,v)=bh(f,v), vSh,

    and the operator  Th:L2(Ω)δShL2(Ω) satisfying

    Thf=(Ahf),

    where  δSh is the restriction of Sh on Ω. Then (2.3) has the following equivalent operator form:

    Ahuh=1κhuh.

    Denote ρ=1κ,ρh=1κh. In this paper, κ, κh and ρ, ρh are all called eigenvalues.

    From the definition of Ah and (2.4), noticing that h and G are equivalent on Sh, we can deduce that

    Ahf2hah(Ahf,Ahf)=bh(f,Ahf)f0,ΩAhf0,Ωf0,ΩAhfh,

    thus,

    Ahfh (2.10)

    Reference [6] gave the a priori error estimates of DG approximation of (2.5).

    Theorem 2.1. For any given , let be the solution of , and let be the solution of . Assume that the regularity estimate (2.8) is valid, then there hold

    Further, when , there hold

    Proof. See Theorems 3.6–3.8 in [6].

    Suppose that is the th eigenvalue of with algebraic multiplicity , i.e., . [5] proved that when , therefore, eigenvalues of (2.3) will converge to . Let be the space of eigenfunctions of (2.2) associated with eigenvalue , and be the direct sum of the generalized eigenspace of associated with that converge to , and . From [34] we have the following error estimates.

    Theorem 2.2. Assume that the regularity estimate (2.8) is valid, and let , then there holds

    (2.11)

    Let be an eigenfunction of , then there exists such that

    (2.12)
    (2.13)

    Let be an eigenfunction of , then there exists such that

    (2.14)

    Proof. See Theorem 3.10 in [6] for the proofs of (2.11)–(2.13). By similar arguments we can get (2.14).

    Let be a family of regular meshes of , , and let be the DG space defined on . Denote . Now, for the eigenvalue problem (2.3) we give the following multigrid discretization scheme of DGFEM based on the shifted inverse iteration.

    Scheme 3.1. Given the iteration times .

    Step 1: Solve on : Find such that and

    Step 2: , .

    Step 3: Solve a linear system on : Find such that

    Set

    Step 4: Compute the Rayleigh quotient

    Step 5: If , then output , stop; else, and return to Step 3.

    Next we will conduct the error analysis on Scheme 3.1.

    From (2.9) we define the projection operator satifying

    (3.1)

    Then, from (2.9) and (3.1) together with , we can prove easily that .

    We first give the following lemmas to prepare for the error analysis.

    Lemma 3.1. Let be an eigenpair of , then for any and , the Rayleigh quotient satisfies

    (3.2)

    Proof. From (2.9) we have

    thus,

    dividing both sides by we obtain (3.2).

    Lemma 3.2. For any non-zero elements in any normed linear space , it is valid that

    Proof. See Lemma 3.1 in [20].

    Denote , Referring to Lemma 4.1 in [20] we prove the following result which plays an important role in our analysis.

    Lemma 3.3. Let be an approximation of the th eigenpair of (2.2) where is not an eigenvalue of , And let . Suppose that

    where is the separate constant of the eigenvalue ;

    and satisfy

    (3.3)

    Then

    Proof. Let be eigenfunctions of satisfying . Then

    Since is not an eigenvalue of , from (3.3) we can get

    (3.4)

    Using triangle inequality and the condition we derive

    where . Hence, we have

    (3.5)

    Because the operator is selfadjoint with respect to , in fact, for , from the symmetry of and and we have and , then, for , there holds

    (3.6)

    Noticing that is a standard orthogonal basis of with respect to the inner product , from , (3.4), (3.6), (2.10) and (3.5) we deduce

    (3.7)

    Taking the norm on both sides of (3.4), and noting that , the condition and (3.6), we get

    (3.8)

    From (3.7) and (3.8) we derive

    The proof is completed.

    Now we can analyze the error of multigrid discretization scheme 3.1 by using Theorem 2.2 and Lemma 3.3. We first consider the case of . Denote .

    Theorem 3.1. Suppose that , and . Let be an approximate eigenpair obtained by Scheme 3.1 () and is sufficiently small, then there exists such that

    (3.9)
    (3.10)
    (3.11)

    Proof. We will use Lemma 3.3 to complete the proof. Take and . From (2.13) we know that there exists such that

    From the triangle inequality and (2.14) we have

    (3.12)

    thus,

    when is small enough, the condition in Lemma 3.3 is valid.

    From (2.11) we get

    i.e., the condition in Lemma 3.3 holds.

    From the definition of we know that Step 3 in Scheme 3.1 is equivalent to the following:

    and , i.e.,

    Note that and differ by only one constant, thus, Step 3 in Scheme 3.1 is equivalent to

    So far, all conditions of Lemma 3.3 are valid.

    Since is a -dimensional space, there must exist such that

    For , according to (2.11) we have

    (3.13)

    Therefore, from Lemma 3.3, (3.12) and (3.13) we get

    (3.14)

    From (2.13) we know that there exists , such that , and

    then

    that is (3.9).

    Next, we will prove (3.10). From (2.12) we have

    which together with (3.14) yields

    Finally, we use Lemma 3.1 to derive (3.11). From Step 4 of Scheme 3.1, Lemma 3.1, (3.9) and (3.10) we deduce that

    The proof is completed.

    Remark 3.1. Using Theorem 3.1 and referring to Theorem 4.2 in [32], we can give the error estimates of Scheme 3.1. To ensure that the error is independent of the number of iterations in the multigrid refinement, we also need the following conditions.

    Condition 3.1. For any given , there exists , such that , and

    Condition 3.1 is easy to be satisfied. For instance, for smooth eigenfunctions, using uniform meshes and linear elements and taking , then , thus, , and when .

    Theorem 3.2. Suppose that Condition 3.1 holds and , and . Let be an approximate eigenpair obtained by Scheme 3.1, then, when is small enough, there exists such that

    (3.15)
    (3.16)
    (3.17)

    In this section, we will report some numerical experiments to show the efficiency of the DG-multigrid method (Scheme 3.1) for solving the Steklov-Lamé eigenproblem. We conduct the numerical experiments on the MATLAB 2022a on a ThinkBook 14p Gen 2 PC with 16G memory, and our program makes use of the package of iFEM [35]. The test domains are set to be the unit square and the L-shaped domain .

    Example 4.1. We use Scheme 3.1 to compute the approximation for the 1st eigenvalue of the problem (2.2). We adopt piecewise polynomial of degree 1 ( element) to compute on uniform isosceles right triangulations. We produce the initial coarse grid and refine the coarse grid in a uniform way (each triangle is divided into four congruent triangles) repeatedly to obtain fine grids . By using the basis functions of , the eigenvalue problem on the initial coarse grid in Step 1 of Scheme 3.1 can be rewritten as a generalized matrix eigenvalue problem

    (4.1)

    where the elements of array are the coordinates of under the basis functions in . Similarly, by using the basis functions of , the algebraic system in Step 3 of Scheme 3.1 can be rewritten as

    (4.2)

    and where is actually the projection of the solution obtained on the previous grid in . For example, if contains elements with the associated solution , denote with , and being the coordinates of the basis function on the element in , and encrypt once (each triangle is divided into four congruent triangles) to get which contains elements, then the projection of in is as follows:

    where is the projection (restriction) operator:

    If bisecting encryption is used, that is, each triangle is divided into two triangles, then just replace with . We use the command "" of MATLAB to solve the discrete algebraic eigenvalue problem (4.1), and use the command "" in MATLAB to solve the linear system (4.2). Further, there has no difficulty with solving the system (4.2) (see Lecture 27.4 in [36]).

    For comparison, we also use the multigrid method of conforming finite elements by adopting element to compute. The error curves are depicted in Figure 1 where the reference value are taken as the most accurate approximations that we can compute. From Figure 1, we can see that as the Lamé parameter increases, the DG-multigrid method is robust compared with the multigrid method of conforming finite elements, which is a major advantage of using DG method to solve elastic problems.

    Figure 1.  The error curves of the approximations for the 1st eigenvalue of (2.2) obtained by multigrid method of conforming finite element and the DG-multigrid method by using element in (left) and (right).

    Example 4.2. Adaptive computation.

    Adaptive algorithm based on the a posterior error estimation is an efficient and important numerical approach for solving partial differential equations. Referring to [37], we combine the multigrid scheme 3.1 and the a posteriori error indicator to establish the adaptive multigrid algorithm. Referring to the a posterior error indicator for the linear elastic source problem in [37], we formally give the following local error indicator for the underlying eigenvalue problem:

    where

    Define the global error indicator:

    Based on the above error indicators and Scheme 3.1, we design the following adaptive multigrid algorithm bases on the shifted inverse iteration.

    Algorithm 4.1. Choose parameter .

    Step 1: Pick any initial mesh .

    Step 2: Solve on for discrete solution .

    Step 3: Let . .

    Step 4: Compute the local indicator .

    Step 5: Construct by Mark Strategy and parameter .

    Step 6: Refine to get a new mesh by procedure REFINE.

    Step 7: Find such that

    Denote and compute the Rayleigh quotient

    Step 8: Let and go to Step 4.

    Mark Strategy

    Given parameter .

    Step 1: Construct a minimal subset of by selecting some elements in such that

    Step 2: Mark all elements in .

    Mark Strategy was first proposed in [38], and the procedure REFINE is some iterative or recursive bisection (see, e.g., [39,40]) of elements with the minimal refinement condition that marked elements are bisected at least once.

    In addition, to investigate the efficiency of Algorithm 4.1, referring to the standard popular adaptive algorithm [41] we give the following Algorithm 4.2 for comparison.

    Algorithm 4.2. Choose parameter .

    Step 1: Pick any initial mesh .

    Step 2: Solve (2.3) on for discrete solution .

    Step 3: Let .

    Step 4: Compute the local indicators .

    Step 5: Construct by Mark Strategy and parameter .

    Step 6: Refine to get a new mesh by procedure REFINE.

    Step 7: Solve (2.3) on for discrete solution .

    Step 8: Let and go to Step 4.

    We use the adaptive DG-multigrid method (Algorithm 4.1) with polynomials of degree 1 ( element) and degree 2 ( element) to compute, and take . For convenience of reading, we specify the following notations in our tables and figures.

    : the degrees of freedom at the th iteration;

    : the th eigenvalue obtained by Algorithm 4.1 at the th iteration;

    : the th eigenvalue obtained by Algorithm 4.2 at the th iteration;

    : the CPU time(s) from the first iteration beginning to the calculate results of the th iteration appearing by using Algorithm 4.1/4.2;

    : the error of the th approximate eigenvalue by Algorithm 4.1;

    : the error indicator of the th approximate eigenvalue by Algorithm 4.1.

    We first give a numerical experiment comparison between using DGFEM to solve directly on fine meshes and using the adaptive DG-multigrid method (Algorithm 4.1) for the 1st nonzero eigenvalue of (2.3). The error curves are shown in Figures 2 and 3. An observation of the left and right subgraphs in Figures 2 and 3 tells us that the regularity of the eigenfunction in is lower than that in , which is consistent with the general conclusion of the regularity of solutions to PDEs. From Figure 2 we can see that the error curves of adaptive DG-multigrid method are all parallel to the line with slope but the error curves of directly computing by DGFEM do not parallel, which indicates that the approximate eigenvalues obtained by the adaptive DG-multigrid method achieve the optimal convergence order. The same conclusion can be seen from Figure 3.

    Figure 2.  The error curves of directly computing by DGFEM and Algorithm 4.1 by using element for the 1st nonzero eigenvalue of (2.3) in (left) and (right).
    Figure 3.  The error curves of directly computing by DGFEM and Algorithm 4.1 by using element for the 1st nonzero eigenvalue of (2.3) in (left) and (right).

    Now we use Algorithms 4.1 and 4.2 with and elements to compute the first 7 non-zero eigenvalues of (2.3) in and , respectively. When using element, the parameters and the diameter of initial mesh is taken as . Limited to space, we list the 1st, the 3rd, the 4th and the 6th approximate eigenvalue in Tables 1 and 2. We also depict the error curves of approximate eigenvalues by Algorithm 4.1 and the curve of error indicators in Figure 4, where the reference values are taken as the most accurate approximations that we can compute. In addition, for the 1st non-zero eigenvalue of (2.3), we investigate the influence of Lamé parameter by taking , and the corresponding error curves are shown in Figure 5. When using element, the parameters and the diameter of initial mesh is taken as . In Tables 3 and 4 we list the 1st, the 3rd, the 4th and the 6th approximate eigenvalue. We also plot the error curves of approximate eigenvalues by Algorithm 4.1 and the curve of error indicators in Figure 6. For the 1st non-zero eigenvalue of (2.3), we investigate the influence of Lamé parameter by taking , and the corresponding error curves are shown in Figure 7.

    Table 1.  The results in by Algorithms 4.1 and 4.2 with element.
    1 1 3600 2.5834151705 0.05433 1 3600 2.5834151539 0.06679
    1 8 26328 2.5365029347 1.62136 8 26328 2.5365029348 2.52953
    1 21 1208232 2.5310939833 249.69061 21 1208232 2.5310939827 391.04103
    1 22 1570164 2.5310583228 485.28739 22 1570164 2.5310583228 651.02095
    1 23 2050932 2.5310346038 817.8231911 23 2050932 2.531034604 1167.502765
    3 1 3324 2.7404121518 0.02745 1 3324 2.7414582353 0.07244
    3 11 30156 2.6778807957 1.75582 16 43908 2.6778901580 7.73674
    3 20 428676 2.6740736381 57.92155 31 605400 2.6740696701 244.04938
    3 23 1026540 2.6739097335 212.85588 34 1025280 2.6739660124 488.60793
    3 24 1349388 2.6738786576 395.09095 35 1219860 2.6739372698 624.67912
    4 1 3888 3.7164345114 0.05795 1 3888 3.7164345114 0.10192
    4 9 44460 3.7115741607 3.84681 9 44460 3.7115741607 5.61523
    4 22 1762944 3.7111432638 194.69801 22 1762944 3.7111432638 341.67732
    4 23 2344044 3.7111398968 269.71127 23 2344044 3.7111398969 470.55583
    4 24 3028620 3.7111378848 385.01813 24 3028620 3.7111378848 654.91996
    6 1 3792 5.2873876626 0.06042 1 3792 5.2873876273 0.09334
    6 5 11784 5.2632333578 0.70311 5 11784 5.2632333578 1.11777
    6 20 942528 5.2537917257 198.46971 20 942528 5.2537917253 306.96848
    6 21 1259328 5.2537597122 334.17056 21 1259328 5.2537597120 452.24058
    6 22 1636512 5.2537348070 605.55218 22 1636512 5.2537348072 796.76924

     | Show Table
    DownLoad: CSV
    Table 2.  The results in by Algorithms 4.1 and 4.2 with element.
    1 1 9252 1.1578406889 0.08684 1 9252 1.1578406889 0.21686
    1 4 9960 1.1568717901 0.51428 4 9960 1.1568717901 1.05144
    1 22 854196 1.1551704849 83.61915 22 854520 1.1551704801 143.97265
    1 23 1125306 1.1551665423 116.92568 23 1125690 1.1551665389 193.87293
    1 24 1469730 1.1551634294 330.33730 24 1470228 1.1551634258 431.50241
    3 1 9852 2.0253499592 0.10770 1 9852 2.0253499590 0.24006
    3 3 13236 2.0200276681 0.62807 3 13236 2.0200276680 1.21075
    3 18 1072320 2.0137654513 221.81309 18 1072320 2.0137654513 344.53795
    3 19 1431672 2.0137487824 351.63554 19 1431672 2.0137487822 590.05031
    3 20 1924536 2.0137351315 711.94221 20 1924536 2.0137351315 1029.63061
    4 1 9540 2.1396949065 0.10559 1 9540 2.1396949061 0.23500
    4 4 13446 2.1322409848 0.70466 4 13446 2.1322409847 1.33492
    4 19 823536 2.1258417612 147.60184 19 823536 2.1258417612 260.40436
    4 20 1085094 2.1258166076 225.83332 20 1085094 2.1258166075 386.16427
    4 21 1437324 2.1257991175 324.24898 21 1437324 2.1257991172 566.48046
    6 1 9828 2.7939491213 0.15553 1 9828 2.7939490858 0.31958
    6 3 12894 2.7668920157 0.78224 3 12894 2.7668920151 1.00058
    6 18 965010 2.7367899041 163.71460 18 965010 2.7367899035 335.07813
    6 19 1292184 2.7367099740 271.08934 19 1292184 2.7367099727 503.74878
    6 20 1729368 2.7366439994 494.77192 20 1729368 2.7366439984 797.84910

     | Show Table
    DownLoad: CSV
    Figure 4.  Convergence study of Algorithm 4.1 by element in (left) and (right).
    Figure 5.  Robustness study of Algorithm 4.1 by element in (left) and (right).
    Table 3.  The results in by Algorithms 4.1 and 4.2 with element.
    1 1 1632 2.5399573563 0.20754 1 1632 2.5399573531 0.33385
    1 5 3600 2.5321126514 0.80285 5 3600 2.5321126514 1.16189
    1 24 760152 2.5309641786 211.46084 24 760152 2.5309641786 239.26807
    1 25 1003368 2.5309641677 278.22507 25 1003368 2.5309641677 320.01464
    1 26 1339404 2.5309641607 378.58015 26 1339404 2.5309641607 439.05795
    3 1 1584 2.6862908794 0.12511 1 1584 2.6862921826 0.40313
    3 18 43920 2.6737990072 8.60574 24 46896 2.6737995054 17.94984
    3 30 813360 2.6737894142 210.84097 42 823224 2.6737894180 409.26749
    3 31 1021272 2.6737894039 275.45868 44 1129272 2.6737894027 587.31383
    3 32 1276776 2.6737893962 363.84821 45 1328904 2.6737893981 708.17810
    4 1 1824 3.7114311740 0.13369 1 1824 3.7114311740 0.21961
    4 5 6360 3.7111583273 0.89324 5 6360 3.7111583273 1.89695
    4 22 754512 3.7111313470 178.28979 22 754512 3.7111313471 250.31766
    4 23 939408 3.7111313465 244.08245 23 939408 3.7111313466 341.16368
    4 24 1218768 3.7111313461 339.61412 24 1218768 3.7111313462 460.76539
    6 2 1920 5.2557235447 0.49529 2 1920 5.2557235444 0.59788
    6 8 8904 5.2537750836 1.87161 8 8904 5.2537750836 2.28091
    6 24 838920 5.2536682063 197.38184 24 838920 5.2536682064 267.40500
    6 25 1119096 5.2536682008 270.08041 25 1119096 5.2536682009 376.06184
    6 26 1456968 5.2536681969 380.87500 26 1456968 5.2536681971 520.85769

     | Show Table
    DownLoad: CSV
    Table 4.  The results in by Algorithms 4.1 and 4.2 with element.
    1 1 4656 1.1562458918 0.38986 1 4656 1.1562458918 0.55317
    1 25 66996 1.1551541513 18.31403 25 66996 1.1551541513 22.07079
    1 38 774480 1.1551532609 260.06329 38 774480 1.1551532609 325.52552
    1 39 923052 1.1551532590 320.61556 39 923052 1.1551532591 400.40419
    1 40 1099044 1.1551532576 398.76151 40 1099044 1.1551532576 496.73607
    3 1 4680 2.0159225406 0.46936 1 4680 2.0159225405 0.57221
    3 9 19224 2.0137381688 4.00481 9 19224 2.0137381688 5.20982
    3 23 976428 2.0136927626 233.96942 23 976428 2.0136927626 313.17266
    3 24 1280052 2.0136927550 324.92148 24 1280052 2.0136927550 421.48383
    3 25 1679592 2.0136927505 458.31975 25 1679592 2.0136927505 578.37647
    4 10 11316 2.1259084959 3.98294 10 11316 2.1259084959 4.01302
    4 21 105648 2.1257416544 27.89323 21 105648 2.1257416544 37.43586
    4 32 822120 2.1257391878 297.07841 32 822120 2.1257391878 368.95873
    4 33 980688 2.1257391755 362.49125 33 980688 2.1257391755 456.94719
    4 34 1181328 2.1257391658 446.73985 34 1181328 2.1257391658 572.04947
    6 1 4680 2.7468959498 0.40303 1 4680 2.7468959503 0.61874
    6 3 5232 2.7405213916 0.92105 3 5232 2.7405213907 1.21365
    6 23 1000704 2.7364519542 234.72412 23 1000704 2.7364519542 334.60146
    6 24 1312716 2.7364519227 327.39089 24 1312716 2.7364519227 448.35877
    6 25 1726080 2.7364519037 468.01995 25 1726080 2.7364519038 608.05296

     | Show Table
    DownLoad: CSV
    Figure 6.  Convergence study of Algorithm 4.1 by element in (left) and (right).
    Figure 7.  Robustness study of Algorithm 4.1 by element in (left) and (right).

    It can be seen from Tables 14 that to get the same accurate approximate eigenvalues, Algorithm 4.1 uses less time or less degrees of freedom than Algorithm 4.2. In Figure 4, the error curves and are all parallel to the line with slope , and in Figure 6 the error curves and are parallel to the line with slope , which indicate that the approximate eigenvalues obtained by Algorithm 4.1 achieve the optimal convergence order. Meanwhile, in Figure 5, the error curves and are almost parallel to the curve of and respectively, and in Figure 7, the curves of and are parallel to and , which indicate that the error indicators are reliable and efficient. Figures 5 and 7 then show that Algorithm 4.1 is robust in both and .

    In this paper, we discussed a multigrid discretization scheme of DGFEM based on the shifted-inverse iteration. Theoretical analysis and numerical results all showed that this method can efficiently solve the Steklov-Lamé eigenproblem as we expected. Generally, the time of solving a linear algebraic system is much less than that of solving an eigenvalue problem. Further, we observe from Tables 14 that although the CPU time of the adaptive DG-multigrid method is less than that of the standard adaptive DGFEM, the advantage is not obvious. We think that this may be because we use "" to solve linear algebraic systems. We notice that in recent research, the multigrid method has been combined with other methods to form many efficient algorithms and applied to many problems, as combined with the DG method in this paper. For example, the multigrid-homotopy method to diffusion equation [42], the multigrid method for the semilinear interface problem based on the modified two-grid method [43], the multigrid method for nonlinear eigenvalue problems based on Newton iteration [44], etc. It is of interest for us to explore more applications of multigrid methods and more efficient solvers for solving linear algebraic equations in multigrid methods.

    This work was supported by the National Natural Science Foundation of China (No. 12261024) and Science and Technology Planning Project of Guizhou Province (Guizhou Kehe fundamental research-ZK[2022] No.324).

    This work does not have any conflicts of interest.



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