1 Introduction
In this article, we consider the following biharmonic eigenvalue equations:
where \(\Omega\subset \mathbf {R}^{d}\) (\(d=2, 3\)) is an open disk or a ball, \(\frac{\partial u}{\partial \mathbf{n}}\) denotes the outer normal derivative of u on ∂Ω.
$$\begin{aligned}& \Delta^{2}u =\lambda u, \quad \mbox{in } \Omega, \end{aligned}$$
(1)
$$\begin{aligned}& u=\frac{\partial u}{\partial \mathbf{n}}=0, \quad \mbox{on } \partial\Omega, \end{aligned}$$
(2)
The biharmonic eigenvalue equations are used to describe the vibration and buckling of plates in mechanics (see, e.g., [1‐3]) and transmission eigenvalue problem in inverse scattering theory (see, e.g., [4, 5]). Therefore, the biharmonic eigenvalue equations have very wide extensive applications.
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However, most of the existing work were concerned with the second-order elliptic eigenvalue problems and there are relatively few articles treating the biharmonic eigenvalue problems. In recent years, the numerical methods for the biharmonic eigenvalue problems adopted mainly the conforming finite element method (see [6‐9]), the nonconforming finite element method (see [3, 10‐12]), and the mixed finite element method (see [13‐16]). For the conforming finite elements method, it requires globally continuously differentiable finite element spaces, therefore, they are difficult to construct and implement (in particular for three-dimensional problems). For the nonconforming finite element method, a disadvantage is that such elements do not come in a natural hierarchy and existing nonconforming elements only involve low-order polynomials so that they are not efficient for capturing smooth solutions. For the mixed finite element method, it can result in spurious solutions on non-convex domains for the boundary conditions of simply supported plates. In the last decade, the \(C^{0}\) interior penalty Galerkin (\(C^{0}\) IPG) method had been developed for second-order elliptic eigenvalue problems (see [17‐19]). Recently, Brenner et al. in [20] extended the \(C^{0}\) IPG method to biharmonic eigenvalue problems and provided the method converges for all three types of boundary conditions (see [20]).
However, all methods mentioned above are low-order finite element methods so that it is very difficultly and expensively to obtain high accuracy numerical solutions, especially for the three-dimensional spherical domain. To the best of our knowledge, there is not any article on a high accuracy numerical method based on the spectral theory of compact operator for biharmonic eigenvalue equations on the spherical domain. Therefore, the task of this paper is to develop a high precision numerical method based on the spectral theory of compact operator for biharmonic eigenvalue equations in the spherical domain.
The rest of this paper is organized as follows. Section 2 provides some preliminaries. In Section 3, by employing the orthogonal spherical polynomials approximation and the spectral theory of compact operator, we derive the error estimates of approximate eigenvalues and eigenfunctions. In Section 4, by adopting orthogonal spherical base functions, we establish the discrete model with sparse mass and stiff matrices which is very efficient for finding the numerical solutions of biharmonic eigenvalue equations on the spherical domain. In Section 5, we provide some numerical examples to validate that the theoretical results are correct. Finally, we provide some conclusions in Section 6.
2 Some preliminaries
The Sobolev spaces and norms used in this paper are standard ([21]). For example, \(H^{s}(\Omega)\) denotes the usual Sobolev space on Ω with real order s, equipped with the norm \(\Vert \cdot \Vert _{s}\), and \(H^{0}(\Omega)=L^{2}(\Omega)\), \(H_{0}^{2}(\Omega)=\{u\in H^{2}(\Omega):u=\frac{\partial u}{\partial \mathbf{n}}=0 \mbox{ on } \partial\Omega\}\).
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The unit circle and the unit sphere in \(R^{d}\) are denoted, respectively, by
$$\begin{aligned} \mathbf {B}^{d}:=\bigl\{ \mathbf{x}\in R^{d}:\Vert \mathbf{x}\Vert \leq1 \bigr\} \quad \text{and} \quad \mathbb{S}^{d-1}:=\bigl\{ \boldsymbol{\xi}\in R^{d}:\Vert \boldsymbol{\xi} \Vert =1\bigr\} . \end{aligned}$$
Moreover, we denote the sets of all nonnegative integers and all real numbers by N and R, respectively. Further, for \(n\in \mathbf {N}\), we denote the collection of all polynomials in d variables with a total degree ≤n by \(\Pi_{n}^{d}\).
Let \(\mathcal{H}_{n}^{d}\) denote the space of (solid) spherical harmonics of degree n with d variables, i.e., It is easily known that where \(\delta_{n,0}\) is the Kronecker delta.
$$\begin{aligned} \mathcal{H}_{n}^{d} = \bigl\{ q \in\Pi_{n}^{d}: \Delta q = 0 \text{ and } q(\rho \mathbf{x}) = \rho^{n} q( \mathbf{x}) \text{ for any } \rho\in R \text{ and } \mathbf{x} \in R^{d} \bigr\} . \end{aligned}$$
$$a_{n}^{d}: = \dim\mathcal{H}_{n}^{d} = \binom{n+d-1}{n} - \binom{n+d-3}{n-2} = \textstyle\begin{cases} 2-\delta_{n,0}, & d=2,\\ 2n+1, & d=3, \end{cases} $$
Let \(\{Y_{\ell}^{n} :1\le\ell\le a_{n}^{d} \}\) be an orthonormal basis of \(\mathcal{H}_{n}^{d}\) such that where dω is the surface measure and \(\omega_{d} = 2\pi ^{\frac{d}{2}} / \Gamma( \frac{d}{2})\) is the surface area.
$$\begin{aligned} \int_{\mathbb{S}^{d-1}} Y_{\ell}^{n}(\boldsymbol{\xi}) \overline{Y_{\ell '}^{n'}(\boldsymbol{\xi})}\,\mathrm{d}\omega = \omega_{d} \delta_{n,n'} \delta _{\ell,\ell'},\quad 1\le\ell \le a_{n}^{d}, 1\le\ell' \le a_{n'}^{d}, n,n'\ge0, \end{aligned}$$
Let \(P_{j}^{(\mu, \nu)}(\zeta)\) denote the usual Jacobi polynomial of degree j with respect to the weight function \((1-\zeta)^{\mu }(1+\zeta)^{\nu}\) on \([-1,1]\). Let \((a)_{n} :=a(a+1)\cdots(a+n-1) \) be the Pochhammer symbol. Define Then the set \(\{P_{j,\ell}^{\mu,n}( \mathbf{x}) : 0\leq j\leq\frac {n}{2},1 \le\ell\le a_{n-2j}^{d}\} \) is mutually orthogonal with respect to the weight function \(\varpi_{\mu}( \mathbf{x})=(1-\Vert \mathbf{x}\Vert )^{\mu}\) whenever \(\mu>-1\) (see [22]). More precisely, where \(b^{d}_{\mu}\) and \(h_{j,n}^{\mu}\) are given by
$$ P_{j,\ell}^{\mu,n}( \mathbf{x}) := \frac{(n-j+\frac{d}{2})_{j}}{(n-j+\frac{d}{2}+\mu)_{j}} P_{j}^{(\mu, n-2j+\frac{d-2}{2})}\bigl(2 \Vert \mathbf{x}\Vert ^{2} -1 \bigr) Y_{\ell}^{n-2j}( \mathbf{x}). $$
(3)
$$\int_{\mathbf {B}^{d}} P_{j,\ell}^{\mu,n}( \mathbf{x}) \overline{P_{j',\ell'}^{\mu,n'}( \mathbf{x})} \varpi_{\mu}( \mathbf{x})\, \mathrm{d} \mathbf{x} = b^{\mu}_{d} h_{j,n}^{\mu} \delta _{n,n'} \delta_{j,j'} \delta_{\ell,\ell'}, $$
$$ b^{\mu}_{d}:=\frac{\pi^{\frac{d}{2}} \Gamma(\mu+1)}{\Gamma(\mu +\frac{d}{2}+1)},\qquad h_{j,n}^{\mu}: = \frac{ (\mu+1)_{j} (1-n-\frac{d}{2})_{j} (\frac{d}{2})_{n} }{j! (1-n-\frac{d}{2}-\mu)_{j} (\frac{d}{2}+\mu+1)_{n} }. $$
(4)
For \(f\in H^{s}_{0}(\mathbf {B}^{d})\) with \(s\in \mathbf {N}\), let \(S_{N}^{-s}f\) denote its best polynomial approximation of degree at most N by where \(\nabla^{2m}=\Delta^{m}\) and \(\nabla^{2m-1}=\nabla\Delta ^{m-1}\) for any \(m\in \mathbf {N}\). We have the following error estimate for \(S_{N}^{-s}\) (see Theorem 4.3 in [22]).
$$\begin{aligned} \bigl\langle \nabla^{s} \bigl(S_{N}^{-s}f-f \bigr), \nabla^{s} v\bigr\rangle = 0,\quad v\in \Pi_{N}^{d} \cap H^{s}_{0}\bigl(\mathbf {B}^{d}\bigr), \end{aligned}$$
Theorem 2.1
Let
\(r,s\in \mathbf {N}\)
and
\(k\in \mathbf {N}_{0}\). If
\(f\in H_{0}^{s}(\mathbf {B}^{d})\cap H^{r}(\mathbf {B}^{d})\)
with
\(r\geq s\), then, for
\(N\geq s\),
where
C
used next is a positive constant which is possibly different at different occurrences, being independent of the degree
N
of polynomials.
$$\begin{aligned} &\bigl\vert f-S_{N}^{-s}f\bigr\vert _{H^{k}(\mathbf {B}^{d})} \leq C N^{-r+k} \Vert f\Vert _{H^{r}(\mathbf {B}^{d})}, \end{aligned}$$
3 The operator formulations and error estimates
3.1 Operator formulations
Find \(\lambda\in \mathbf {R}\), \(0\neq u \in V\), such that where \(a(u,v)=\int_{\Omega}\triangle u \triangle v \,\mathrm{d} \mathbf{x}\), \(b(u,v)=\int_{\Omega}u v\,\mathrm{d} \mathbf{x}\).
$$\begin{aligned} a(u,v)=\lambda b(u,v), \quad \forall v\in V, \end{aligned}$$
(5)
The source problem associated with (5) is written as follows.
Find \(u \in V\) such that
$$\begin{aligned} a(u,v)=b(f,v), \quad \forall v\in V. \end{aligned}$$
(6)
It is obvious that \(a(\cdot,\cdot)\) is a continuous, symmetric, and V-elliptic bilinear form on \(V\times V\) and \(b(f,\cdot)\) is a continuous and linear functional on V. Thus, we can use \(\Vert \cdot \Vert _{a}=\sqrt{a(\cdot,\cdot)}\) as a norm in V which is equivalent to the norm \(\Vert \cdot \Vert _{2}\) induced by \(H^{2}(\Omega)\) and we know from the Lax-Milgram theorem that (6) has a unique solution. Thus, according to the source problem (6), we define the operator \(T: V\rightarrow V\) by Thanks to [24], we know that (5) has the equivalent operator form
$$\begin{aligned} a(Tf,v)=b(f,v), \quad \forall v\in V. \end{aligned}$$
(7)
$$\begin{aligned} Tu=\frac{1}{\lambda}u. \end{aligned}$$
(8)
For the above operator T, we have the following result.
Lemma 3.1
The operator
\(T:V\mapsto V\)
is a self-adjoint compact one.
Proof
For \(\forall u,v \in V\), we have Thus, \(T:V\mapsto V\) is self-adjoint. By taking \(f=u\), \(v=Tu\) in (7), we can obtain From the Poincaré inequality, we can derive Thus, we have where γ is a positive constant.
$$\begin{aligned} a(Tu,v)=b(u,v)=b(v,u)=a(Tv,u)=a(u,Tv), \end{aligned}$$
$$\begin{aligned} a(Tu,Tu)=b(u,Tu). \end{aligned}$$
$$\begin{aligned} \gamma \Vert Tu\Vert _{2}^{2}&\leq a(Tu,Tu)=b(u,Tu) \leq \Vert u\Vert _{0} \Vert Tu\Vert _{0} \leq \Vert u \Vert _{0} \Vert Tu\Vert _{2}. \end{aligned}$$
$$\begin{aligned} \Vert Tu\Vert _{2}\leq\frac{1}{\gamma} \Vert u\Vert _{0}, \end{aligned}$$
(9)
Let E be the bounded set in V. Since V is compactly embedded in \(L_{2}(\Omega)\), so E is the sequentially compact set in \(L^{2}(\Omega )\). From (9), we know that TE is the sequentially compact set in V. Thus, \(T:V\rightarrow V\) is a compact operator. □
From the classical theory of abstract elliptic eigenvalue problem (see, e.g., [23, 24]), we know that all eigenvalues of T are real and have finite algebraic multiplicity. We arrange the eigenvalues of T by increasing order:
$$0< \lambda_{1}\leq\lambda_{2}\leq\lambda_{3}\leq \cdots\nearrow +\infty. $$
Since the eigenfunctions corresponding to two arbitrary different eigenvalues of T must be orthogonal, there must exist a standard orthogonal basis in eigenspace corresponding to the same eigenvalue. Thus, by using the eigenfunctions of T corresponding to \(\{\lambda_{j}\}\), we can construct a complete orthonormal system of V as follows:
$$\begin{aligned} u_{1}, u_{2}, \ldots, u_{j},\ldots. \end{aligned}$$
Let \(X_{N}=\Pi_{N}^{d} \cap H_{0}^{2}(\Omega)\), then the spectral-Galerkin approximation of (5) is given as follows.
Find \(\lambda_{N} \in \mathbf {R}\), \(0\neq u_{N}\in X_{N}\) such that
$$\begin{aligned} a(u_{N},v_{N})=\lambda_{N} b(u_{N},v_{N}), \quad \forall v_{N} \in X_{N}. \end{aligned}$$
(10)
The source problem associated with (10) is written as follows.
Find \(u_{N}\in X_{N}\), such that Likewise, we know from the Lax-Milgram theorem that (11) has a unique solution. Thus, we can define the operator \(T_{N}: V\rightarrow X_{N}\) by From [24], we know that (10) has the equivalent operator form It is obvious that \(T_{N}: V\rightarrow X_{N}\) is a finite rank operator.
$$\begin{aligned} a(u_{N},v)=b(f,v), \quad \forall v\in X_{N}. \end{aligned}$$
(11)
$$\begin{aligned} a(T_{N}f,v)=b(f,v), \quad \forall v\in X_{N}. \end{aligned}$$
(12)
$$\begin{aligned} T_{N}u_{N}=\frac{1}{\lambda_{N}}u_{N}. \end{aligned}$$
(13)
Define the projection operator \(\Pi_{N}^{2,0}: V\rightarrow X_{N}\) by
$$\begin{aligned} a\bigl(u-\Pi_{N}^{2,0}u,v\bigr)=0, \quad \text{for all } u \in V, v\in X_{N}. \end{aligned}$$
(14)
We have the following result.
Lemma 3.2
Proof
For \(\forall u\in V\), \(v\in X_{N}\), we have By taking \(v=\Pi_{N}^{2,0}Tu-T_{N}u\) in (15), we can obtain Since \(a(\cdot,\cdot)\) is V-elliptic, we can obtain which completes the proof of Lemma 3.2. □
$$\begin{aligned} a\bigl(\Pi_{N}^{2,0}Tu-T_{N}u,v\bigr)=a\bigl( \Pi_{N}^{2,0}Tu-Tu,v\bigr)+a(Tu-T_{N}u,v)=0. \end{aligned}$$
(15)
$$\begin{aligned} a\bigl(\Pi_{N}^{2,0}Tu-T_{N}u, \Pi_{N}^{2,0}Tu-T_{N}u\bigr)=0. \end{aligned}$$
$$\begin{aligned} T_{N}=\Pi_{N}^{2,0}T, \end{aligned}$$
It is obvious that is a self-adjoint finite rank operator and the eigenvalues of (10) can be arranged as
$$\begin{aligned} T_{N}|_{X_{N}}:X_{N}\rightarrow X_{N} \end{aligned}$$
$$\begin{aligned} 0< \lambda_{1N}\leq\lambda_{2N}\leq\lambda_{3N}\leq \cdots\lambda _{KN},\qquad K=\dim(X_{N}). \end{aligned}$$
3.2 Error estimates
In the following, we provide the error estimates. We first provide the following lemma.
Lemma 3.3
Let
\((\lambda,u)\)
and
\((\lambda_{N},u_{N})\)
be the eigenpair of (5) and (10), respectively. Then we have
$$\begin{aligned} \lambda_{N}-\lambda=\frac{\Vert u_{N}-u\Vert _{a}^{2}}{\Vert u_{N}\Vert _{L^{2}(\Omega )}^{2}}-\lambda\frac{\Vert u_{N}-u\Vert _{L^{2}(\Omega)}^{2}}{\Vert u_{N}\Vert _{L^{2}(\Omega )}^{2}}. \end{aligned}$$
(16)
Proof
From (5), we can derive By dividing \(b(u_{N},u_{N})\) and applying (10) to both sides of the above equation, we obtain which completes the proof of Lemma 3.3. □
$$\begin{aligned} &a(u_{N}-u,u_{N}-u)-\lambda b(u_{N}-u,u_{N}-u) \\ &\quad =a(u_{N},u_{N})-2a(u_{N},u)+a(u,u)- \lambda b(u_{N},u_{N})+2\lambda b(u_{N},u)- \lambda b(u,u) \\ &\quad =a(u_{N},u_{N})-2\lambda b(u_{N},u)+ \lambda b(u,u)-\lambda b(u_{N},u_{N})+2\lambda b(u_{N},u)-\lambda b(u,u) \\ &\quad =a(u_{N},u_{N})-\lambda b(u_{N},u_{N}). \end{aligned}$$
$$\begin{aligned} \lambda_{N}-\lambda=\frac{\Vert u_{N}-u\Vert _{a}^{2}}{\Vert u_{N}\Vert _{L^{2}(\Omega )}^{2}}-\lambda\frac{\Vert u_{N}-u\Vert _{L^{2}(\Omega)}^{2}}{\Vert u_{N}\Vert _{L^{2}(\Omega)}^{2}}, \end{aligned}$$
(17)
Put \(\eta_{N}=\sup_{u\in V,\Vert u\Vert _{a}=1}\inf_{v\in X_{N}}\Vert Tu-v\Vert _{a}\). It is clear that we have Thus, we have the following convergence on operators (also see the proof in [25]).
$$\begin{aligned} \eta_{N}=\sup_{u\in V,\Vert u\Vert _{a}=1}\inf_{v\in X_{N}} \Vert Tu-v\Vert _{a}\rightarrow0 \quad (N\rightarrow \infty). \end{aligned}$$
(18)
Theorem 3.1
We have
$$\begin{aligned} \lim_{N\rightarrow\infty} \Vert T-T_{N}\Vert _{a}= 0. \end{aligned}$$
(19)
Proof
By the definition of the operator norm, we have Then, from (18), we obtain the desired result. □
$$\begin{aligned} \Vert T-T_{N}\Vert _{a}&=\sup_{u\in V,\Vert u\Vert _{a}=1} \bigl\Vert (T-T_{N})u\bigr\Vert _{a} =\sup _{u\in V,\Vert u\Vert _{a}=1}\bigl\Vert Tu-\Pi_{N}^{2,0}Tu\bigr\Vert _{a} \\ &=\sup_{u\in V,\Vert u\Vert _{a}=1}\inf_{v\in X_{N}}\Vert Tu-v\Vert _{a}=\eta_{N}. \end{aligned}$$
Let \(M(\lambda)\) denote the eigenfunctions space of (5) corresponding to the eigenvalue λ. We have the following results (also see the proof in [25]).
Theorem 3.2
Let
\((\lambda,u)\)
and
\((\lambda_{N},u_{N})\)
be the eigenpair of (5) and (10), respectively. Then we have
$$\begin{aligned} &\Vert u-u_{N}\Vert _{a} \leq\sup_{u\in M(\lambda),\Vert u\Vert _{a}=1} \frac {C}{\lambda} \bigl\Vert u-\Pi_{N}^{2,0}u\bigr\Vert _{a}, \end{aligned}$$
(20)
$$\begin{aligned} &\lambda_{N}-\lambda\leq\sup_{u\in M(\lambda),\Vert u\Vert _{a}=1} \frac {C}{\lambda^{2}}\frac{\Vert u-\Pi_{N}^{2,0}u\Vert _{a}^{2}}{\Vert u_{N}\Vert _{L^{2}(\Omega )}^{2}}. \end{aligned}$$
(21)
Proof
From Theorem 3.1, we know that \(\Vert T-T_{N}\Vert _{a}\rightarrow0\) (\(N\rightarrow\infty\)). Thus, according to Theorem 7.4 in [24], we have Therefore, for any \(u\in M(\lambda)\) satisfying \(\Vert u\Vert _{a}=1\), we have
By combining (23) and (24) with (22), we get the desired result (20). By Lemma 3.3, we obtain which together with (20) yields (21). □
$$\begin{aligned} \Vert u-u_{N}\Vert _{a}\leq C\bigl\Vert (T-T_{N})|_{M(\lambda)}\bigr\Vert _{a}. \end{aligned}$$
(22)
$$\begin{aligned} &\bigl\Vert (T-T_{N})u\bigr\Vert _{a}= \bigl\Vert Tu-\Pi_{N}^{2,0}Tu\bigr\Vert _{a}= \frac{1}{\lambda}\bigl\Vert u-\Pi _{N}^{2,0}u\bigr\Vert _{a}, \end{aligned}$$
(23)
$$\begin{aligned} &\bigl\Vert (T-T_{N})|_{M(\lambda)} \bigr\Vert _{a}=\sup_{u\in M(\lambda),\Vert u\Vert _{a}=1}\bigl\Vert (T-T_{N})u\bigr\Vert _{a}. \end{aligned}$$
(24)
$$\begin{aligned} &\lambda_{N}-\lambda\leq\frac{\Vert u_{N}-u\Vert _{a}^{2}}{\Vert u_{N}\Vert _{L^{2}(\Omega)}^{2}}, \end{aligned}$$
(25)
Theorem 3.3
Let
\((\lambda,u)\)
and
\((\lambda_{N},u_{N})\)
be the eigenpair of (5) and (10), respectively. If
\(u\in H_{0}^{2}(\Omega)\cap H^{r}(\Omega)\)
with
\(r\geq2\), then, for
\(N\geq2\),
$$\begin{aligned} &\Vert u-u_{N}\Vert _{a} \leq\frac{C}{\lambda}N^{(2-r)} \sup_{u\in M(\lambda ),\Vert u\Vert _{a}=1}\Vert u\Vert _{r}, \\ &\lambda_{N}-\lambda\leq\frac{C}{\lambda^{2}}N^{2(2-r)}\sup _{u\in M(\lambda),\Vert u\Vert _{a}=1}\frac{\Vert u\Vert _{r}^{2}}{\Vert u_{N}\Vert _{L^{2}(\Omega)}^{2}}. \end{aligned}$$
Proof
From (14) and the continuity of \(a(u,v)\), we have Thus, from Theorem 2.1, we can obtain By combing (26) with Theorem 3.2, we can get the desired result. □
$$\begin{aligned} \bigl\Vert u-\Pi_{N}^{2,0}u\bigr\Vert _{a}^{2}&=a\bigl(u-\Pi_{N}^{2,0}u,u- \Pi_{N}^{2,0}u\bigr) \\ &=\inf_{\phi_{N}\in X_{N}}a(u-\phi_{N},u-\phi_{N}) \\ &\leq C\inf_{\phi_{N}\in X_{N}}\vert u-\phi_{N}\vert _{2}^{2} \\ & \leq C\bigl\vert u-S_{N}^{-2}u\bigr\vert _{2}^{2}. \end{aligned}$$
$$\begin{aligned} &\bigl\Vert u-\Pi_{N}^{2,0}u\bigr\Vert _{a}^{2} \leq C\bigl\vert u-S_{N}^{-2}u \bigr\vert _{2}^{2} \leq CN^{2(2-r)}\Vert u\Vert _{r}^{2}. \end{aligned}$$
(26)
4 Matrix formulation of the spectral-Galerkin approximation
For \(n\in \mathbf {N}\), \(2\leq j\leq\frac{n}{2}\), and \(1\le\ell\le a_{n-2j}^{d}\), put
$$\begin{aligned} P^{-2,m}_{j,\ell}( \mathbf{x}):= &\frac{(1-m-2j-\frac{d}{2})_{j}}{ j(j-1) (5-m-2j-\frac{d}{2})_{j-2} } \bigl(\Vert x \Vert ^{2}-1\bigr)^{2} P^{2,m}_{j-2,\ell}(\mathbf{x}). \end{aligned}$$
The following formulas are very useful for the numerical analysis (see [22]):
Thus, \(X_{N}\) can be denoted by Therefore, from (27), (28), and the property of orthogonal polynomials in the unit sphere, we have the following result (see [22, 26]).
$$\begin{aligned} & P^{-2,n}_{j,\ell}( \mathbf{x})=P^{0,n}_{j,\ell}( \mathbf{x})-\frac {2(2n+d-4)}{2n+d-6}P^{0,n-2}_{j-1,\ell}( \mathbf{x}) + \frac {2n+d-2}{2n+d-6}P^{0,n-4}_{j-2,\ell}( \mathbf{x}), \quad j\ge2, \end{aligned}$$
(27)
$$\begin{aligned} &\Delta P^{-2,n}_{j,\ell}( \mathbf{x}) = (2n+d-4) (2n+d-2) P^{0,n-2}_{j-1,\ell}( \mathbf{x}),\quad j\ge2. \end{aligned}$$
(28)
$$\begin{aligned} X_{N}=\biggl\{ P^{-2,n}_{j,\ell}: 4\leq n\leq N, 2\leq j\leq\frac {n}{2}, 1\le\ell\le a_{n-2j}^{d} \biggr\} . \end{aligned}$$
Lemma 4.1
We have the following formulas:
$$\begin{aligned} &\bigl\langle \Delta P^{-2,n}_{j,\ell}, \Delta P^{-2,n'}_{j',\ell'} \bigr\rangle = d (2n+d-2)^{2}(2n+d-4) b_{d}^{0} \delta_{n,n'} \delta_{j,j'} \delta _{\ell,\ell'}, \\ & \bigl\langle P^{-2,n}_{j,\ell}, P^{-2,n'}_{j',\ell'} \bigr\rangle \\ &\quad = b^{0}_{d} \delta_{\ell,\ell'} \times \textstyle\begin{cases} \frac{d}{2n+d}+\frac{4d(2n+d-4)}{(2n+d-6)^{2}} +\frac {d(2n+d-2)^{2}}{(2n+d-6)^{2}(2n+d-8)}, & j=j',n=n',\\ -\frac{2d}{2n+d-2}-\frac{2d(2n+d+2)}{(2n+d-6)(2n+d-2)}, &j=j'-1,n=n'-2, \\ \frac{d(2n+d+6)}{(2n+d+2)(2n+d)}, & j=j'-2,n=n'-4,, \\ -\frac{2d}{2n+d-6}-\frac{2d(2n+d-2)}{(2n+d-6)(2n+d-10)}, &j=j'+1,n=n'+2, \\ \frac{d(2n+d-2)}{(2n+d-6)(2n+d-8)}, & j=j'+2,n=n'+4, \\ 0, & \textit{otherwise}. \end{cases}\displaystyle \end{aligned}$$
Thus, the unknown functions can be rewritten as follows: with the coefficients \(\{\widehat{u}_{j,\ell}^{n}\}\) to be determined. Then the Legendre-Galerkin approximation scheme (10) is simplified into a system of algebraic eigenvalue equations: It is easily known from Lemma 4.1 that the stiff matrix \(A^{n}_{j,\ell }\) is essentially diagonal and the mass matrix \(B^{n}_{j,\ell}\) is hepta-diagonal such that equation (30) can be efficiently solved.
$$\begin{aligned} u_{N}&= \sum_{n=4}^{N}\sum _{j=2}^{ \lfloor(n)/2 \rfloor}\sum_{\ell =1}^{a_{n-2j}^{d}} \widehat{u}_{j,\ell}^{n} P_{j,\ell}^{-2,n} \end{aligned}$$
(29)
$$ A^{n}_{j,\ell} \widehat{u}^{n}_{j,\ell}= \lambda_{N} B^{n}_{j,\ell} \widehat{u}^{n}_{j,\ell}. $$
(30)
5 Numerical experiments
In the following, two numerical tests are provided for computing eigenvalues of (1)-(2) on disk and ball, respectively, which is operated in MATLAB. Their numerical results are shown in Table 1 and Table 2.
Table 1
The approximate eigenvalues on the unit disk
\(\pmb{\mathbf {B}^{2}}\)
in
\(\pmb{\mathbf {R}^{2}}\)
N
|
\(\boldsymbol{\lambda_{1}}\)
|
\(\boldsymbol{\lambda_{2}}\)
|
\(\boldsymbol{\lambda_{4}}\)
|
|---|---|---|---|
10 | 104.3631056 | 452.0074329 | 1216.451916 |
15 | 104.3631056 | 452.0045101 | 1216.407600 |
20 | 104.3631056 | 452.0045101 | 1216.407600 |
Table 2
The approximate eigenvalues on unit ball
\(\pmb{\mathbf {B}^{3}}\)
in
\(\pmb{\mathbf {R}^{3}}\)
N
|
\(\boldsymbol{\lambda_{1}}\)
|
\(\boldsymbol{\lambda_{2}}\)
|
\(\boldsymbol{\lambda_{4}}\)
|
|---|---|---|---|
10 | 237.7210683 | 769.9765197 | 1818.29231 |
15 | 237.7210675 | 769.9634832 | 1818.167926 |
20 | 237.7210675 | 769.9634832 | 1818.167924 |
25 | 237.7210675 | 769.9634832 | 1818.167924 |
It is easily seen from Table 1 that the results have at least ten-digit accuracy with \(N\geq 15\). From numerical results, it is shown that \(\lambda_{1}\) is a simple eigenvalue, but \(\lambda_{2}\) and \(\lambda _{4}\) are all eigenvalues with multiplicity 2.
It is easily seen from Table 2 that the results have at least ten-digit accuracy with \(N\geq20\). From numerical results, we know that \(\lambda_{1}\) also is a simple eigenvalue, but \(\lambda_{2}\) and \(\lambda_{5}\) are eigenvalues with multiplicity 3 and 5, respectively.
By comparing the relative errors of numerical solutions, we found that the convergence rates of numerical solutions approximate \(O(10^{-12})\) when \(N\ge20\), which is consistent with those obtained for the theoretical case. It is also shown that finding the approximate solutions for the biharmonic eigenvalue equations with spectral-Galerkin method is computationally very effective.
6 Conclusions
In this study, we have developed a high accuracy numerical method by means of the spectral theory of compact operator for biharmonic eigenvalue equations on a spherical domain. By employing the orthogonal spherical polynomials approximation and the spectral theory of compact operator, we have derived the error estimates of approximate eigenvalues and eigenfunctions. We also, respectively, provided two numerical experiments in the two-dimensional case and three-dimensional cases to verify that our method is very effective. While we have restricted our attention in this study to the cases of biharmonic eigenvalue problems with constant coefficient. Whereas the approach presented in this article can be extended to biharmonic eigenvalue problems with variable coefficients.
Acknowledgements
This research was supported by National Science Foundation of China grant 11271127.
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Competing interests
The author declares to have no competing interests.
Author’s contributions
The author wrote, read, and approved the final manuscript.