Spectral Indicator Method for a Non-selfadjoint Steklov Eigenvalue Problem

Abstract

We propose an efficient numerical method for a non-selfadjoint Steklov eigenvalue problem. The Lagrange finite element is used for discretization and the convergence is proved using the spectral perturbation theory for compact operators. The non-selfadjointness of the problem leads to non-Hermitian matrix eigenvalue problem. Due to the existence of complex eigenvalues and lack of a priori spectral information, we employ the recently developed spectral indicator method to compute eigenvalues in a given region on the complex plane. Numerical examples are presented to validate the effectiveness of the proposed method.

1 Introduction

Steklov eigenvalue problems arise in mathematical physics with spectral parameters in the boundary conditions [24]. Applications of Steklov eigenvalues include surface waves, mechanical oscillators immersed in a viscous fluid, the vibration modes of a structure in contact with an incompressible fluid, etc. [13, 14, 24]. Recently, Steklov eigenvalues have been used in the inverse scattering theory to reconstruct the index of refraction of an inhomogeneous medium [12]. Note that most Steklov eigenvalue problems considered in the literature are related to partial differential equations of second order. However, Steklov eigenvalue problems of higher order were also studied, e.g., the fourth order Steklov eigenvalue problem [2].

In contrast to the theoretical study of the Steklov eigenvalue problems, numerical methods, in particular, finite element methods have attracted some researchers rather recently [1, 3, 7, 8, 14, 16, 23, 26, 38]. Various methods have been proposed, including the isoparametric finite element method [3], the virtual element method [26], non-conforming finite element methods [16, 25, 38], the spectral-Galerkin method [2], adaptive methods [7], multilevel methods [18, 35], etc. All of the above works consider the selfadjoint cases. In this paper, we consider a non-selfadjoint Steklov eigenvalue problem arising in the study of inhomogeneous absorbing media in inverse scattering theory [12]. There seems to exist only one paper by Bramble and Osborn [10], which considered the non-selfadjoint Steklov eigenvalue problem. However, the second order non-selfadjoint operator is assumed to be uniformly elliptic and no numerical results were reported in [10]. The current paper contains both finite element theory and numerical examples for a non-selfadjoint Steklov eigenvalue problem. Note that there exist works on other non-selfadjoint eigenvalue problems, e.g., [28, 36, 37]. For the general theory and examples of finite element methods for eigenvalue problems, we refer the readers to the book chapter by Babuška and Osborn [4], the review paper by Boffi [9], and the monograph by Sun and Zhou [34].

Finite element discretizations for non-selfadjoint eigenvalue problems usually lead to non-Hermitian generalized eigenvalues problems. Most classical eigensolvers for large sparse matrices are designed for Hermitian (or symmetric) matrices. For non-Hermitian problems, there exist less methods including the Arnoldi method, Lanczos method and Jacobi–Davidson method [5]. Unfortunately, these methods are still far from satisfactory [30], in particular, when the number and distribution of eigenvalues are not known.

In a recent paper [20], a novel spectral indicator eigenvalue solver RIM (recursive integral method) is developed for non-Hermitian eigenvalue problems. RIM computes all eigenvalues in a region on the complex plane ${\mathbb {C}}$ without any a priori spectral information. Roughly speaking, given a region $S \subset {\mathbb {C}}$ whose boundary $\Gamma :=\partial S$ is a simple closed curve, RIM computes an indicator $\delta _S$ for S using spectral projection defined by a Cauchy contour integral on $\Gamma$. The indicator is used to decide if S contains eigenvalue(s). In case of positive answers, S is divided into sub-regions and indicators for these sub-regions are computed. The procedure continues until the size of the region is smaller than a specified precision $d_0$ (e.g., $d_0 = 10^{-9}$). The centers of the regions are the approximations of eigenvalues.

Spectral projection (contour integral) is a classical tool in operator theory [22] and has been used extensively in the finite element theory for eigenvalues problem [4, 27]. It became popular recently for matrix eigenvalue problems using invariant subspaces [6, 29, 31]. RIM differs from these integral based methods [29, 31] in the sense that the basic idea is to define an indicator of a region and use bisection-like technique to compute the eigenvalues accurately. Contour integrals are only used in RIM to compute an indicator of the region. The approximation of the integral need not be accurate. Furthermore, no estimation of the number of eigenvalues in the region is needed as a prior.

In this paper, we propose a simple finite element method and extend the spectral indicator method RIM to compute complex Steklov eigenvalues in the region of interest. The contributions of the paper are (1) it provides a finite element analysis for a non-selfadjoint Steklov eigenvalue problem; and (2) it extends a new eigensolver for the resulting non-Hermitian matrix eigenvalue problem. The rest of the paper is organized as follows. In Sect. 2, we introduce the Steklov eigenvalue problem, its adjoint problems, variational formulations, and prove the well-posedness. In Sect. 3, we propose a linear finite element method and prove the convergence. In Sect. 4, we extend the new spectral indicator method for the resulting non-Hermitian matrix eigenvalue problems. Numerical examples are presented in Sect. 5. Finally, some conclusions and future work are discussed in Sect. 6.

2 A Non-selfadjoint Steklov Eigenvalue Problem

Let $\Omega \subset {\mathbb {R}}^2$ be a bounded polygon with Lipshitz boundary $\partial \Omega$. Let $\nu$ be the unit outward normal to $\partial \Omega$. Let k be the wavenumber and n(x) be the index of refraction. We consider the Steklov eigenvalue problem to find $\lambda \in {\mathbb {C}}$ and a nontrivial function $u \in H^1(\Omega )$ such that

$$\begin{aligned} \triangle u + k^2n(x) u&=0 \qquad \text {in } \Omega , \end{aligned}$$

(2.1a)

$$\begin{aligned} \frac{\partial u}{ \partial \nu }+\lambda u&=0\qquad \text {on } \partial \Omega . \end{aligned}$$

(2.1b)

Define

$$\begin{aligned} (u, v) = \int _\Omega u {\overline{v}} \,dx, \qquad \langle f, g \rangle = \int _{\partial \Omega } f {{\overline{g}}} \,ds, \end{aligned}$$

and the continuous sesquilinear form

$$\begin{aligned} a(u, v):= (\nabla u, \nabla v) - k^2(nu, v) \quad \text {for all } u, v \in H^1(\Omega ). \end{aligned}$$

The weak formulation for (2.1) is to find $(\lambda , u) \in {\mathbb {C}} \times H^1(\Omega )$ such that

$$\begin{aligned} a(u, v) = -\lambda \langle u, v \rangle \quad \text {for all } v \in H^1(\Omega ). \end{aligned}$$

(2.2)

The associated source problem is, given $g \in L^2(\partial \Omega )$, to find $u \in H^1(\Omega )$ such that

$$\begin{aligned} a(u, v) = \langle g, v \rangle \quad \text {for all } v \in H^1(\Omega ). \end{aligned}$$

(2.3)

In this paper, we assume that n(x) is a bounded complex valued function given by

$$\begin{aligned} n(x) = n_1(x) + i \frac{n_2(x)}{k}, \end{aligned}$$

where $i = \sqrt{-1}$, $n_1(x) > 0$ and $n_2(x) \ge 0$ are bounded smooth functions.

It is easy to verify that $a(\cdot , \cdot )$ satisfies the Gårding’s inequality [11], i.e., there exist constants $K < \infty$ and $\alpha _0 > 0$ such that

$$\begin{aligned} \text {Re}\left\{ a(v, v)\right\} + K\Vert v\Vert ^2_{L^2(\Omega )} \ge \alpha _0 \Vert v\Vert ^2_{H^1(\Omega )} \quad \text {for all } u \in H^1(\Omega ). \end{aligned}$$

(2.4)

Let K be a positive constant, which is large enough. Define the sesquilinear form $A: H^1 \times H^1 \rightarrow {\mathbb {C}}$ such that

$$\begin{aligned} A(u,v) := a(u, v) + K(u, v) = (\nabla u, \nabla v) - k^2(nu, v) + K(u, v), \quad u, v \in H^1(\Omega ). \end{aligned}$$

(2.5)

The following lemma shows that A is $H^1(\Omega )$-elliptic [19].

Lemma 2.1

For K large enough, the sesquilinear form A is $H^1(\Omega )$-elliptic, i.e., there exists $\alpha _0 > 0$ such that

$$\begin{aligned} |A(v,v)| \ge \alpha _0 \Vert v\Vert ^2_{H^1(\Omega )} \quad \text {for all } v \in H^1(\Omega ). \end{aligned}$$

Proof

Since $n_1(x)$ is bounded, there exist a constant B such that $n_1(x) < B$ for all $x \in \Omega$. Using the Gårding’s inequality (2.4), we have that

$$\begin{aligned} |A(v,v)|= & {} |(\nabla v, \nabla v) - k^2(nv, v) + K(v, v)| \\\ge & {} \text {Re} \left\{ (\nabla v, \nabla v) - k^2(nv, v) + K(v, v) \right\} \\= & {} (\nabla v, \nabla v) - k^2(n_1(x)v, v) + K(v,v) \\\ge & {} (\nabla v, \nabla v) - k^2B(v, v) + K(v,v)\\\ge & {} \alpha _0 \Vert v\Vert ^2_{H^1(\Omega )}, \end{aligned}$$

where $\alpha _0=\min \{1, K-k^2B\}$ for K large enough. $\square$

As a consequence, the Fredholm alternative can be used to show the existence of a unique solution for (2.3). To this end, we need to define the (generalized) Neumann eigenvalues.

Definition 2.1

The Neumann eigenvalue problem associated with n(x) is to find $k^2 \in {\mathbb {C}}$ and a nontrivial $u \in H^1(\Omega )$ such that

$$\begin{aligned} \triangle u +k^2 n(x) u&=0\qquad \text {in } \Omega , \end{aligned}$$

(2.6a)

$$\begin{aligned} \frac{\partial u}{ \partial \nu }&=0 \qquad \text {on } \partial \Omega . \end{aligned}$$

(2.6b)

Theorem 2.1

Let $g \in L^2(\partial \Omega )$. Assuming that $k^2$ is not a Neumann eigenvalue associated with n(x) on $\Omega$, there exists a unique solution $u \in H^1(\Omega )$ to (2.3) such that

$$\begin{aligned} \Vert u\Vert _{H^1(\Omega )} \le C \Vert g\Vert _{L^2(\partial \Omega )}. \end{aligned}$$

(2.7)

Proof

Since $k^2$ is not a Neumann eigenvalue, then the uniqueness holds for (2.3). By Fredholm Alternative (see e.g., Section 5.3 of [19]), there exists a unique solution u to (2.3) and the regularity follows readily. $\square$

Consequently, one can define an operator, which is in fact the Neumann-to-Dirichlet mapping, $T: L^2(\partial \Omega ) \rightarrow L^2(\partial \Omega )$ [12]

$$\begin{aligned} Tg = u|_{\partial \Omega }. \end{aligned}$$

(2.8)

The mapping T is compact since $Tg \in H^{1/2}(\partial \Omega )$ and $H^{1/2}(\partial \Omega )$ is compactly embedded in $L^2(\partial \Omega )$. Denote an eigenpair of T by $(\mu , g)$ such that

$$\begin{aligned} Tg = \mu g. \end{aligned}$$

It is clear that $\mu$ and $\lambda$ are related

$$\begin{aligned} \lambda = -1/\mu . \end{aligned}$$

We shall also need the adjoint operator $T^*$ of T for the proof of convergence of the finite element method later. Consider the adjoint problem for (2.3). Given $g \in L^2(\partial \Omega )$, find $v \in H^1(\Omega )$ such that

$$\begin{aligned} a(u, v) = \langle u, g\rangle \quad \text {for all } u \in H^1(\Omega ). \end{aligned}$$

(2.9)

Then (2.9) has a unique solution v. The solution operator for (2.9) is the adjoint operator $T^*g: L^2(\partial \Omega ) \rightarrow L^2(\partial \Omega )$ such that $T^*g = v|_{\partial \Omega }$.

3 Finite Element Approximation

In this section, we present a finite element approximation $T_h$ for T. Let ${\mathcal {T}}_h$ be a regular triangular mesh for $\Omega$ with mesh size h. Let $V_h \subset H^1(\Omega )$ be the Lagrange finite element space associated with ${\mathcal {T}}_h$ and $V^B_h := V_h|_{\partial \Omega }$ be the restriction of $V_h$ on $\partial \Omega$. It is clear that $V^B_h \subset L^2(\partial \Omega )$. The finite element formulation for the Steklov eigenvalue problem is to find $(\lambda _h, u_h) \in {\mathbb {C}} \times V_h$ such that

$$\begin{aligned} (\nabla u_h, \nabla v_h) - k^2(nu_h, v_h) = -\lambda _h \langle u_h, v_h \rangle \quad \text {for all } v_h \in V_h. \end{aligned}$$

(3.1)

For the convergence of eigenvalues, we first study the source problem. Given $g \in L^2(\partial \Omega )$, let $g_h$ be the projection of g onto $V^B_h$. The discrete problem is to find $u_h \in V_h$ such that

$$\begin{aligned} (\nabla u_h, \nabla v_h) - k^2(nu_h, v_h) = \langle g_h, v_h \rangle \quad \text {for all } v_h \in V_h. \end{aligned}$$

(3.2)

In the rest of this section, we assume that $u \in H^2(\Omega )$ and the same regularity holds for the solution of the adjoint problem. This is the case when $\Omega$ is convex. In general, when $\Omega$ is non-convex, u does not belong to $H^2(\Omega )$. We refer the readers to [17] for further discussions on the regularity of u, which is out of the scope of the current paper.

We have the following estimate for (3.2). For completeness, we provide the proof as well, which is based on Theorem 5.7.6 of [11].

Theorem 3.1

Let u be the solution to (2.3). Assume that k is not a Neumann eigenvalue. There exists a unique solution $u_h$ to (3.2) such that, for h small enough,

$$\begin{aligned} \Vert u-u_h\Vert _{H^1(\Omega )} \le C \inf _{v \in V_h} \Vert u-v\Vert _{H^1(\Omega )} \end{aligned}$$

(3.3)

and

$$\begin{aligned} \Vert u-u_h\Vert _{L^2(\Omega )} \le C h \Vert u-u_h\Vert _{H^1(\Omega )}. \end{aligned}$$

(3.4)

Proof

For the finite dimensional problem (3.2), existence of a solution can be established using uniqueness. Assuming that there exists a nontrivial solution $u_h$ to (3.2) for $g_h=0$. For the continuous problem, $g=0$ implies that the solution $u=0$. Then (3.6) asserts that $u_h=0$. Thus the uniqueness holds, which implies the existence of the solution $u_h$ as well.

Using (2.3) and (3.2), one has the Galerkin orthogonality

$$\begin{aligned} a(u-u_h, v_h) = 0 \quad \text {for all } v_h \in V_h. \end{aligned}$$

The Gårding’s inequality (2.4) implies that

$$\begin{aligned} \alpha _0 \Vert u-u_h\Vert ^2_{H^1(\Omega )}\le & {} |a(u-u_h, u-u_h) + K(u-u_h, u-u_h)| \\= & {} | a(u-u_h, u-v_h) + K\Vert u-u_h\Vert ^2_{L^2(\Omega )} | \\\le & {} C \Vert u-u_h\Vert _{H^1(\Omega )} \Vert u-v_h\Vert _{H^1(\Omega )} + K\Vert u-u_h\Vert ^2_{L^2(\Omega )}. \end{aligned}$$

Assume that the estimate (3.4) holds, i.e.,

$$\begin{aligned} \Vert u-u_h\Vert _{L^2(\Omega )} \le C_1h \Vert u-u_h\Vert _{H^1(\Omega )} \end{aligned}$$

(3.5)

for some constant $C_1 > 0$. One has that

$$\begin{aligned} \alpha _0 \Vert u-u_h\Vert ^2_{H^1(\Omega )} \le C \Vert u-u_h\Vert _{H^1(\Omega )} \Vert u-v_h\Vert _{H^1(\Omega )} + KC_1h^2 \Vert u-u_h\Vert ^2_{H^1(\Omega )}. \end{aligned}$$

(3.6)

Then for h small enough, we obtain

$$\begin{aligned} \Vert u-u_h\Vert _{H^1(\Omega )} \le C \inf _{v \in V_h} \Vert u-v_h\Vert _{H^1(\Omega )} \quad \text {for all } v_h \in V_h. \end{aligned}$$

The rest of the proof is devoted to verifying (3.4). Let w be the solution to the adjoint problem

$$\begin{aligned} a(v, w) = (u-u_h, v) \quad \text {for all } v \in V. \end{aligned}$$

Then, for any $w_h \in V_h$,

$$\begin{aligned} (u-u_h, u-u_h)= & {} a(u-u_h, w) \\= & {} a(u-u_h, w-w_h) \\\le & {} C\Vert u-u_h\Vert _{H^1(\Omega )} \Vert w-w_h\Vert _{H^1(\Omega )}\\\le & {} Ch \Vert u-u_h\Vert _{H^1(\Omega )} |w|_{H^2(\Omega )} \\\le & {} Ch \Vert u-u_h\Vert _{H^1(\Omega )} \Vert u-u_h\Vert _{L^2(\Omega )}, \end{aligned}$$

where we have used the regularity of the solution for the adjoint problem. Consequently,

$$\begin{aligned} \Vert u-u_h\Vert _{L^2(\Omega )} \le C h \Vert u-u_h\Vert _{H^1(\Omega )}. \end{aligned}$$

$\square$

As a result, problem (3.2) defines a discrete operator $T_h: L^2(\partial \Omega ) \rightarrow V^B_h$ such that

$$\begin{aligned} T_h g = u_h|_{V^B_h}. \end{aligned}$$

(3.7)

The following theorem shows that $T_h$ converges to T in norm in $L^2(\partial \Omega )$.

Theorem 3.2

Assume that $g \in H^{1/2}(\partial \Omega ) \subset L^2(\partial \Omega )$. Let T and $T_h$ be defined as in (2.8) and (3.7) using linear Lagrange element, respectively. Then

$$\begin{aligned} \Vert T-T_h\Vert _{L^2(\partial \Omega ), L^2(\partial \Omega )} \le C h^{3/2}. \end{aligned}$$

(3.8)

Proof

Using the approximation property of the linear Lagrange finite element (see Eq. (3.9) of [34]), for $u \in H^2(\Omega )$, one has that

$$\begin{aligned} \inf _{v \in V_h} \Vert u-v\Vert _{H^1(\Omega )} \le Ch\Vert u\Vert _{H^2(\Omega )} \le C h\Vert g\Vert _{L^2(\partial \Omega )}. \end{aligned}$$

(3.9)

Therefore, by Theorem 3.1,

$$\begin{aligned} \Vert u-u_h\Vert _{L^2(\Omega )} \le C h^2 \Vert g\Vert _{L^2(\partial \Omega )}. \end{aligned}$$

One has that

$$\begin{aligned} \Vert (T-T_h)g\Vert _{L^2(\partial \Omega )}= & {} \Vert (u-u_h)|_{\partial \Omega }\Vert _{L^2(\partial \Omega )} \\\le & {} C \Vert u-u_h\Vert _{L^2(\Omega )}^{1/2} \Vert u-u_h\Vert _{H^1(\Omega )}^{1/2} \\\le & {} Ch^{3/2}\Vert g\Vert _{L^2(\partial \Omega )}, \end{aligned}$$

where we have applied Theorem 3.1 and the trace theorem (Theorem 1.6.6 in [11]). Hence (3.8) follows immediately and the proof is complete. $\square$

Similarly, the discrete problem for the adjoint problem (2.9) can be defined as follows. Given $g \in L^2(\partial \Omega )$, find $v_h \in H^1(\Omega )$ such that

$$\begin{aligned} (\nabla u_h, \nabla v_h) - k^2(nu_h, v_h) = \langle u_h, g \rangle \quad \text {for all } u_h \in V_h. \end{aligned}$$

(3.10)

Then all results in this section also hold for the adjoint problem (2.9). In particular, one has that discrete adjoint operator $T^*_h: L^2(\partial \Omega ) \rightarrow V^B_h$ such that $T_h g_h = v_h|_{V^B_h}$ where $v_h$ the solution of (3.10) and

$$\begin{aligned} \Vert T^*-T^*_h\Vert _{L^2(\partial \Omega ), L^2(\partial \Omega )} \le C h^{3/2}. \end{aligned}$$

(3.11)

Now we turn to the finite element spectral approximation of Steklov eigenvalues based on the theory in [27]. We shall need some preliminaries on the spectral theory of compact operators (see, e.g., [22]). Let $T: X \rightarrow X$ be a compact operator on a complex Hilbert space X. Let $z \in {\mathbb {C}}$. The resolvent operator of T is defined as

$$\begin{aligned} R_z(T) = (z-T)^{-1}. \end{aligned}$$

(3.12)

The resolvent set of T is

$$\begin{aligned} \rho (T)=\{ z \in {\mathbb {C}}: (z-T)^{-1} \text { exists and is bounded} \}. \end{aligned}$$

(3.13)

The spectrum of T is $\sigma (T)={\mathbb {C}} \setminus \rho (T)$.

Since T is compact, each $\mu \in \sigma (T)$ is an isolated eigenvalue of T and the generalized eigenspace associated with $\mu$ is finite dimensional. Furthermore, there exists a smallest positive integer $\alpha$ such that

$$\begin{aligned} {\mathcal {N}}\left( (\mu -T)^\alpha \right) = {\mathcal {N}}\left( (\mu -T)^{\alpha +1}\right) , \end{aligned}$$

where ${\mathcal {N}}$ denotes the null space. The integer $m=\dim {\mathcal {N}}\left( (\mu -T)^\alpha \right)$ is called the algebraic multiplicity of $\mu$. The functions in ${\mathcal {N}}\left( (\mu -T)^\alpha \right)$ are called the generalized eigenfunctions of T corresponding to $\mu$. Note that the geometric multiplicity of $\mu$ is defined as $\dim {\mathcal {N}}(\mu - T)$.

Let $\Gamma$ be a simple closed curve on the complex plane $\mathbb C$ lying in $\rho (T)$, which contains an eigenvalue $\mu$ and no other eigenvalues. Let the algebraic multiplicity of $\mu$ be m. The spectral projection is defined by

$$\begin{aligned} E(\mu ):= \frac{1}{2\pi i} \int _\Gamma R_z(T) dz. \end{aligned}$$

It is well-known that E is a projection onto the space spanned by the generalized eigenfunctions ${\phi }_j, j=1,\ldots , m$ associated with $\mu$, i.e., the range of E, ${\mathcal {R}}(E)$, coincides wth ${\mathcal {N}}\left( (\mu -T)^\alpha \right)$.

Since $T_h$ converges to T in norm as $h \rightarrow 0$, $\Gamma \subset \rho (T_h)$ for h small enough. In addition, there exists m eigenvalues $\mu _h^1, \ldots , \mu _h^m$ of $T_h$ inside $\Gamma$ such that

$$\begin{aligned} \lim _{h \rightarrow 0} \mu ^j_h = \mu _j \quad \text {for } j = 1, \ldots , m. \end{aligned}$$

The spectral projection

$$\begin{aligned} E_h(\mu ):= \frac{1}{2\pi i} \int _\Gamma R_z(T_h) dz \end{aligned}$$

converges to E pointwise and $\text {dim}{\mathcal {R}}(E_h) = \text {dim}{\mathcal {R}}(E)$.

If $\mu$ is an eigenvalue of T, then ${\overline{\mu }}$ is an eigenvalue of $T^*$. Let $\phi _1, \ldots , \phi _m$ be a basis for ${{\mathcal {R}}}(E)$ and $\phi _1^*, \ldots , \phi _m^*$ be the dual basis to $\phi _1, \ldots , \phi _m$ (see Section 1.1 of [34]). The following lemma (Theorem 3 of [27]) will be used to prove the convergence of Steklov eigenvalues.

Lemma 3.1

Let $\mu$ be an eigenvalue of T with algebraic multiplicity m. Let $\mu _h^1, \ldots , \mu _h^m$ be the m eigenvalues of $T_h$ converge to $\mu$ and define

$$\begin{aligned} {\hat{\mu }}_h = \frac{1}{m} \sum _{j=1}^m \mu _h^j. \end{aligned}$$

Then there exists a constant C such that

$$\begin{aligned} |\mu - {\hat{\mu }}_h| \le \frac{1}{m} \sum _{i, j=1}^m |\langle (T-T_h) \phi _i, \phi _j^* \rangle | + C \Vert (T-T_h)|_{{{\mathcal {R}}}(E)}\Vert \, \Vert (T^*-T^*_h)|_{{{\mathcal {R}}}(E^*)}\Vert , \end{aligned}$$

(3.14)

where ${{\mathcal {R}}}(E^*) = \text {span}\{ \phi _1^*, \ldots , \phi _m^*\}$.

Using the convergence results of the finite element method for the source problem and the above lemma, we have the following theorem.

Theorem 3.3

Let $\mu$ be an eigenvalue of T with multiplicity m and $\mu _h^j, j=1,\ldots ,m$ be the m eigenvalues of $T_h$ approximating $\mu$. Then there exists a constant C, independent of h, such that

$$\begin{aligned} |\mu - {\hat{\mu }}_{h}| \le C h^2, \qquad \text {where } {\hat{\mu }}_{h} = \frac{1}{m} \sum _{j=1}^m \mu _h^j. \end{aligned}$$

Proof

Let $u_j$ and $u_{h,j}$ be the solutions of (2.3) and (3.2) with the right hand side being $\phi _j$, respectively. Let $u_j^*$ and $u_{h,j}^*$ be the solution of the adjoint problem (2.9) and the corresponding finite element solution with the right hand side being $\phi _j^*$, respectively. In view of Theorem 3.2 and (3.11), we only need to estimate the first term of (3.14).

$$\begin{aligned} |\langle (T-T_h) \phi _j, \phi _j^*\rangle |= & {} |\langle \phi _j, (T^*-T^*_h) \phi _j^* \rangle | \\= & {} | a(u_j, u_j^*-u_{h,j}^*)| \\= & {} | a( u_j -u_{h,j}, u_j^*-u_{h,j}^*)| \\\le & {} C \Vert u_j -u_{h,j}\Vert _{H^1(\Omega )} \Vert u_j^*-u_{h,j}^*\Vert _{H^1(\Omega )}\\\le & {} Ch^2. \end{aligned}$$

$\square$

As a consequence, we have the following convergence result on the Steklov eigenvalues.

Theorem 3.4

Let $\lambda$ be a Steklov eigenvalue with multiplicity m and $\lambda _h^j, j=1,\ldots ,m$ be the m discrete eigenvalues of (3.1) approximating $\lambda$. Then there exists a constant C, independent of h, such that

$$\begin{aligned} |\lambda - {\hat{\lambda }}_{h}| \le C h^2, \qquad \text {where } {\hat{\lambda }}_{h} = \frac{1}{m} \sum _{j=1}^m \lambda _h^j. \end{aligned}$$

4 Spectral Indicator Method

When n(x) is complex, the Steklov eigenvalue problem is non-selfadjoint. The above finite element method leads to a non-Hermitian matrix eigenvalue problem. Due to the lack of a priori spectral information, classical methods might not work effectively. To this end, we extend the spectral indicator method RIM, which was proposed recently in [20] (see also [21]) for the non-selfadjoint transmission eigenvalue problem [15, 32], to compute (complex) Steklov eigenvalues in a given region on the complex plane $\mathbb C$.

From the finite element approximation, $T_h: V^B_h \rightarrow V^B_h$ has the following matrix form, which is also denoted by $T_h$,

$$\begin{aligned} T_h : = I_h (G-k^2 M_n)^{-1} M_{\partial \Omega }, \end{aligned}$$

where $I_h$ corresponds the restriction of a function in $V_h$ onto $V^B_h$. $T_h$ is an $M\times M$ matrix where M is the number of vertices on $\partial \Omega$. Clearly, $M \ll N$ and the following problem is much smaller than (4.2)

$$\begin{aligned} T_h {{\varvec{u}}} = \mu {{\varvec{u}}}. \end{aligned}$$

(4.1)

Unfortunately, one needs to invert the $N \times N$ matrix $(G-k^2 M_n)$ and the cost is prohibitive.

In the following, we use the matrix form for (3.1) given by

$$\begin{aligned} (G-k^2 M_n) {{\varvec{u}}} = -\lambda M_{\partial \Omega } {{\varvec{u}}}, \end{aligned}$$

(4.2)

where G is the stiffness matrix, $M_n$ is the mass matrix, $M_{\partial \Omega }$ is the mass matrix on $\partial \Omega$.

We first introduce RIM proposed in [20] for the generalized eigenvalue problem

$$\begin{aligned} A{{\varvec{x}}} = \lambda B{{\varvec{x}}}, \end{aligned}$$

(4.3)

where $A=(G-k^2 M_n)$ and $B = -M_{\partial \Omega }$.

Let $S \subset {\mathbb {C}}$ be a simply connect domain and $\Gamma =\partial S$. The problem of interest is to compute all eigenvalues of (4.3) in S. Let ${{\varvec{g}}}$ be a random vector. From the previous section, the spectral projection of ${{\varvec{g}}}$ is defined as

$$\begin{aligned} E {{\varvec{g}}} =\frac{1}{2\pi i} \int _\Gamma R_z(A,B) {{\varvec{g}}} dz= \frac{1}{2\pi i} \int _\Gamma (A-zB)^{-1}{{\varvec{g}}} dz. \end{aligned}$$

The idea behind RIM is very simple. The spectral projection $E {{\varvec{g}}}$ can be used to define an indicator $\delta _S$, which decides if there exist eigenvalues in S or not. If there are no eigenvalues inside $\Gamma$, $|E {{\varvec{g}}}| = 0$. Otherwise, if there exist m eigenvalues $\lambda _j, j = 1, \ldots , m$, $|E {{\varvec{g}}}| \ne 0$.

Without loss of generality, let S be a square. $E{{\varvec{g}}}$ can be approximated using a quadrature

$$\begin{aligned} E{{\varvec{g}}} \approx \dfrac{1}{2 \pi i} \sum _{j=1}^W \omega _j {{\varvec{x}}}_j, \end{aligned}$$

(4.4)

where $\omega _j$’s are quadrature weights and ${{\varvec{x}}}_j$’s are the solutions of the linear systems

$$\begin{aligned} (A- z_jB){{\varvec{x}}}_j = {{\varvec{g}}}, \quad j = 1, \ldots , W. \end{aligned}$$

(4.5)

Recall that if there is no eigenvalue inside $\Gamma$, then $E{{\varvec{g}}} = \mathbf{0}$ for all ${{\varvec{g}}} \in \mathbb C^n$. Hence $|E{{\varvec{g}}}|$ can be used as an indicator of S. However, in practice, it is difficult to distinguish between $|E{{\varvec{g}}}| \ne 0$ and $|E{{\varvec{g}}}|= 0$. The solution in [20] is to normalize $E{{\varvec{g}}}$ and project it again. The indicator $\delta _S$ is set to be

$$\begin{aligned} \delta _S := \left| E \left( \frac{E {{\varvec{g}}}}{|E {{\varvec{g}}}|}\right) \right| . \end{aligned}$$

(4.6)

Since we use numerical quadratures, $\delta _S \approx 1$ if there exist eigenvalues in S. In this case, S is divided into sub-regions and indicators for these sub-regions are computed. The procedure continues until the size of the region is smaller than a specified precision $d_0$ (e.g., $d_0 = 10^{-9}$). Then the centers of the regions are the approximations of eigenvalues.

According to (4.6), $\delta _S = 1$ if there exists at least one eigenvalue in S and $\delta _S = 0$ if there is no eigenvalue in S. Since only “yes” ($\delta _S = 1$) or “no” ($\delta _S = 0$) is needed in the algorithm and quadrature is used to evaluate $E{{\varvec{g}}}$, it is natural to use a threshold $\delta _0$ to distiguish “yes” and “no”. Let ${{\varvec{g}}}$ be a random vector and $\delta _0, 0< \delta _0 < 1$, be a threshold value. The following is the basic algorithm for RIM.

• RIM$(A, B, S, d_0, \delta _0, {{\varvec{g}}})$

• Input: matrices A, B, region S, precision $d_0$, threshold $\delta _0$, random vector ${{\varvec{g}}}$.

• Output: generalized eigenvalue(s) $\lambda$ inside S

• 1. Compute ${\delta _S}$ using (4.6), (4.4) and (4.5).

• 2. Decide if S contains eigenvalue(s).

• If $\delta _S < \delta _0$, then exit.

• Otherwise, compute the size h(S) of S.

  • If $h(S) > d_0$,

    • partition S into subregions $S_j, j=1, \ldots J$.

    • for $j=1: J$

    • $\qquad RIM(A, B, S_j, d_0, \delta _0, {{\varvec{g}}})$.

    • end

  • If $h(S) \le d_0$,

    • set $\lambda$ to be the center of S.

    • output $\lambda$ and exit.

The computational cost of RIM mainly comes from solving the linear system (4.5) at each quadrature point. Note that these matrices are $N \times N$, where N is the number of vertices of the triangular mesh if the linear Lagrange element is used. Furthermore, for robustness, the strategy of RIM in [20] selects a small threshold $\delta _0=0.1$, i.e., S contains eigenvalue(s) whenever $\delta _S > \delta _0$. This choice of threshold for selecting a region systematically leans towards further investigation of regions that may potentially contain eigenvalues. Such a strategy leads to more linear systems.

Since both A and B are large and sparse, to employ RIM in a more efficient way, we resort to a new version of RIM using the Krylov subspace and Cayley transformation [21]. Assume that $\sigma$ is not an eigenvalue. Multiplying both sides of (4.5) with $(A- \sigma B)^{-1}$, one gets that

$$\begin{aligned} (A- \sigma B)^{-1}{{\varvec{g}}}= & {} (A- \sigma B)^{-1}(A-z_j B){{\varvec{x}}}_j \\= & {} (A- \sigma B)^{-1}(A-\sigma B+(\sigma -z_j)B) {{\varvec{x}}}_j \\= & {} (I+(\sigma - z_j)(A- \sigma B)^{-1}B) {\varvec{x}}_j. \end{aligned}$$

Let $M=(A- \sigma B)^{-1}B$ and ${{\varvec{b}}}=(A- \sigma B)^{-1}{{\varvec{g}}}$. Then (4.5) becomes

$$\begin{aligned} (I+(\sigma -z_j)M) {{\varvec{x}}}_j = {{\varvec{b}}}. \end{aligned}$$

(4.7)

Given a matrix M, a vector ${{\varvec{b}}}$, and a non-negative integer m, the Krylov subspace is defined as

$$\begin{aligned} K_{m}(M; {{\varvec{b}}}):=\text {span} \{{{\varvec{b}}}, M {{\varvec{b}}}, \ldots , M^{m-1}{{\varvec{b}}} \}, \end{aligned}$$

(4.8)

which processes the shift-invariant property

$$\begin{aligned} K_{m}(a M+ b I; {{\varvec{b}}})=K_{m}(M; {{\varvec{b}}}), \end{aligned}$$

(4.9)

where a and b are two scalars. Consequently, $K_{m}(I+(\sigma -z_j)M; {{\varvec{b}}})=K_{m}(M; {{\varvec{b}}})$.

The idea is to built one Krylov subspace $K_{m}(M; {{\varvec{b}}})$ for a shift $\sigma$ to solve (4.5) for as many $z_j$’s as possible. As a result, one expects that Krylov subspaces for only a few shifts are needed to solve (4.5) for all quadrature points $z_j$’s. We refer the readers to [21] for more details. We simply denote this version of spectral indicator method by SIM in the rest of the paper.

5 Numerical Examples

We present some numerical results in this section. For all examples, we choose $k=1$. Consider three domains: $\Omega _1$ is the unit disk, $\Omega _2$ is the square whose vertices are

$$\begin{aligned} (0, -1), \quad (1, 0), \quad (0, 1), \quad (-1, 0), \end{aligned}$$

and $\Omega _3$ is an L-shaped domain given by

$$\begin{aligned} (-0.9, 1.1) \times (-1.1, 0.9) \setminus [0.1, 1.1] \times [-1.1, -0.1]. \end{aligned}$$

For the disk with radius R and constant index of refraction n, separation of variables in polar coordinates can be used to obtain exact Steklov eigenvalues. Since u is the solution of the Helmholtz equation (2.1a), it has the expansion

$$\begin{aligned} u(r,\theta )=\sum _{m=-\infty }^{+\infty }a_m J_{|m|}(k\sqrt{n}r)e^{im\theta },\quad r<R, \quad \theta \in (0,2\pi ], \end{aligned}$$

(5.1)

where m’s are integers and $J_{|m|}$ denotes the Bessel function of order |m|. By the boundary condition (2.1b), the coefficients $a_m$ satisfy

$$\begin{aligned} \sum _{m=-\infty }^{+\infty } a_m\bigg (k\sqrt{n} J_{|m|}'(k\sqrt{n}R)+\lambda J_{|m|}(k\sqrt{n}R)\bigg )e^{im\theta }=0, \end{aligned}$$

i.e.,

$$\begin{aligned} a_m\bigg (k\sqrt{n} J_{|m|}'(k\sqrt{n}R)+\lambda J_{|m|}(k\sqrt{n}R)\bigg )=0, \ \ \ m=-\infty , \ldots ,\infty . \end{aligned}$$

(5.2)

If $\lambda$ is a Steklov eigenvalue, there exists at least one m such that $a_m \ne 0$. Then from (5.2), $\lambda$ must satisfy

$$\begin{aligned} k\sqrt{n} J_{m}'(k\sqrt{n}R)+\lambda J_{m}(k\sqrt{n}R)=0, \ \ \text {for some}\ m \ge 0. \end{aligned}$$

Therefore, the Steklov eigenvalues are given by

$$\begin{aligned} \lambda =-\frac{k\sqrt{n} J_{m}'(k\sqrt{n}R)}{J_{m}(k\sqrt{n}R)}, \ \ \text {for some}\ m \ge 0. \end{aligned}$$

On the other hand, it is clear that all $\lambda _m=-\frac{k\sqrt{n} J_{m}'(k\sqrt{n}R)}{J_{m}(k\sqrt{n}R)},\ m=0, 1, 2, \ldots ,$ are Steklov eigenvalues since $J_{m}(k\sqrt{n}r)e^{im\theta }$ are non-trivial solutions of the Steklov eigenvalue problem.

From the above discussion, for the unit disk, Steklov eigenvalues are given by

$$\begin{aligned} \lambda _m = - k \sqrt{n} \frac{J'_m(k\sqrt{n})}{J_m(k\sqrt{n})}, \quad m = 0, 1, 2, \ldots . \end{aligned}$$

(5.3)

In Fig. 1, we show $\lambda _m$ against the index of refraction n for $m=0, 1, 2$.

Using (5.3), when $n=4$ the 6 largest Steklov eigenvalues are

$$\begin{aligned} \lambda _1=5.151841, \quad \lambda _{2,3}=0.223578, \quad \lambda _{4,5}=-1.269100, \quad \lambda _6=-2.472703 \end{aligned}$$

(5.4)

and when $n=4+4i$ the 4 complex Steklov eigenvalues with largest imaginary parts are

$$\begin{aligned} \lambda _1=\begin{array}{l} -0.320506 \\ \quad + 3.124689i \end{array}, \quad \lambda _{2,3}=\begin{array}{l} -0.136861 \\ \quad + 1.396737i\end{array}, \quad \lambda _4=\begin{array}{l} -1.353076 \\ \quad + 0.791723i\end{array}. \end{aligned}$$

(5.5)

5.1 Selfadjoint Cases

When the index of refraction n(x) is real, the problem is selfadjoint and all Steklov eigenvalues are real. We compute the six largest Steklov eigenvalues for $n(x) = 4$ on a series of uniformly refined meshes for each domain. The results are shown in Tables 1, 2, and 3. The mesh sizes are denoted by h. The values are consistent with those in [12].

Table 1 The largest six Steklov eigenvalues for the circle $n(x)=4$

Note that for the unit disk, the first 3 eigenvalues are given by (5.3) for $m=0, 1, 2$. The values of the columns 2, 3, 4 in Table 1 approximate the intersections of $\lambda _m, m=0, 1, 2$ and $n=4$ in Fig. 1.

Table 2 The largest six Steklov eigenvalues for the square $n(x)=4$

Table 3 The largest six Steklov eigenvalues for the L-shaped domain $n(x)=4$

Fig. 1

$\lambda _m$ versus n for $m=0,1,2$

Fig. 2

Convergence rates of Steklov eigenvalues $n(x)=4$. Top left: the unit circle. Top right: the square. Bottom left: the L-shaped domain

In Fig. 2, we show the convergence rates of Steklov eigenvalues of three domains. Since we use the linear Lagrange finite element, the second order convergence is achieved for the unit circle and the square. For the L-shaped domain, which is non-convex, the convergence rate of the second Steklov eigenvalue is lower than 2, while the other 5 eigenvalues have second order convergence.

Note that the convergence rate of the eigenvalues relates to the regularity of the associated eigenfunctions. The result in Fig. 2 implies that the eigenfunction associated with the second eigenvalues does not belong to $H^2(\Omega )$.

5.2 Non-selfadjoint Cases

When n(x) is complex, the solution operator is non-selfadjoint. Consequently, we end up with non-Hermitian matrix eigenvalue problem. Computation of complex eigenvalues of non-Hermitian matrices are challenging, in particular, when there is no a priori spectral information on the number and distribution of eigenvalues. To this end, we use the new spectral indicator method introduced in the previous section to compute Steklov eigenvalues. For all examples, we take $n(x) = 4+4i$.

The left picture of Fig. 3 shows the distribution of Steklov eigenvalues for the unit disk on the complex plane. These are eigenvalues computed using Matlab ‘eig’ for the non-Hermitian matrix ($2097 \times 2097$) resulting from the finite element method. The mesh size is $h \approx 0.0613$. Note that ‘eig’ is a direct eigensolver and not suitable for larger matrices.

Fig. 3

Left: Steklov eigenvalues for the unit disk on the complex plane $n(x)=4+4i$. Right: the region explored by SIM

Since we are interested in eigenvalues close to the origin, we choose the search region S on the complex plane to be the square $[-3, 0.5] \times [0, 3.5]$. The right picture of Fig. 3 shows how SIM explores S and finds the eigenvalues inside S.

The computed complex eigenvalues for the three domains are shown in Tables 4,  5, and 6, respectively. We arrange the eigenvalues according to the decreasing order of their real parts. Again, these values are consistent with the values given in [12], which are reconstructed by some inverse algorithm using scattering data.

Table 4 Eigenvalues for the circle $n(x)=4+4i$

Table 5 Eigenvalues for the square $n(x)=4+4i$

Table 6 Eigenvalues for the L-shaped domain $n(x)=4+4i$

In Fig. 4, we show the convergence rates of complex Steklov eigenvalues. The second order convergence is achieved for the unit circle and square. Similar to real n(x), the second eigenvalue of the L-shaped domain has lower convergence rate indicating that the associated eigenfunction has lower regularity.

Fig. 4

Convergence rates of Steklov eigenvalues ($n(x)=4+4i$). Top left: the unit circle. Top right: the square. Bottom left: the L-shaped domain

For the last example, the index of refraction is a function $n(x)=|x|+4+|x|^2i$. The eigenvalues are shown in Table 7, which are computed with the mesh sizes $h \approx 0.2341$ for the circle, $h \approx 0.2441$ for the square, and $h \approx 0.2383$ for the L-shaped domain, respectively. Since n(x) is a smooth function, the convergence rates are similar to the above constant cases and are thus omitted.

Table 7 Eigenvalues for the three domains $n(x)=|x|+4+|x|^2i$

6 Conclusions and Future Work

In this paper, we propose a finite element method of a non-selfadjoint Steklov eigenvalue problem arising from the inverse scattering theory. The current paper contains both convergence theory and numerical examples. An earlier paper by Bramble and Osborn considered a similar but different non-selfadjoint Steklov eigenvalue problem [10]. The second order non-selfadjoint operator is assumed to be uniformly elliptic and no numerical results were reported therein.

The convergence of Lagrange finite elements is proved using the spectral perturbation theory for compact operators. Due to the fact that the problem is non-selfadjoint and no a priori spectral information is available, the recently developed spectral indicator method RIM-C is considered for the resulting non-Hermitian matrix eigenvalue problems.

Non-selfadjoint Steklov eigenvalue problems have many important applications. Numerical computation of these problems is challenging. The problem considered in this paper is related to the Helmholtz equation. Similar problems exist for the Maxwell equation and elasticity equation. In future, we plan to extend the theory and algorithm to these problems.

The current spectral indicator method cannot compute the multiplicity and the eigenspaces. However, it can be done by adding some functions. For example, one can choose a small circle centered at an computed eigenvalue with no other eigenvalues inside. Accurate computation of the spectral projection of several random vectors would lead to the associated eigenspace, which also provides the multiplicity of the eigenvalue [29, 31]. This will be considered in the future version of the method. Note that for the inverse problems considered in [12, 33], the multiplicity is not requried.