Non-conforming Crouzeix-Raviar element approximation for Stekloff eigenvalues in inverse scattering This work is supported by the National Natural Science Foundation of China (Grant Nos.11561014,11761022 ).

Non-conforming Crouzeix-Raviar element approximation for Stekloff eigenvalues in inverse scattering ††thanks: This work is supported by the National Natural Science Foundation of China (Grant Nos.11561014,11761022 ).

Yidu Yang Yidu Yang ()School of Mathematical Sciences, Guizhou Normal University, Guiyang, , China.
22email: ydyang@gznu.edu.cnYu Zhang School of Mathematical Sciences, Guizhou Normal University, Guiyang, , China.
44email: zhang_hello_hi@126.comHai Bi School of Mathematical Sciences, Guizhou Normal University, Guiyang, , China.
66email: bihaimath@gznu.edu.cn
Yu Zhang Yidu Yang ()School of Mathematical Sciences, Guizhou Normal University, Guiyang, , China.
22email: ydyang@gznu.edu.cnYu Zhang School of Mathematical Sciences, Guizhou Normal University, Guiyang, , China.
44email: zhang_hello_hi@126.comHai Bi School of Mathematical Sciences, Guizhou Normal University, Guiyang, , China.
66email: bihaimath@gznu.edu.cn
Hai Bi Yidu Yang ()School of Mathematical Sciences, Guizhou Normal University, Guiyang, , China.
22email: ydyang@gznu.edu.cnYu Zhang School of Mathematical Sciences, Guizhou Normal University, Guiyang, , China.
44email: zhang_hello_hi@126.comHai Bi School of Mathematical Sciences, Guizhou Normal University, Guiyang, , China.
66email: bihaimath@gznu.edu.cn
Abstract

In this paper, we use the non-conforming Crouzeix-Raviart element method to solve a Stekloff eigenvalue problem arising in inverse scattering. The weak formulation corresponding to this problem is non-selfadjoint and does not satisfy -elliptic condition, and its Crouzeix-Raviart element discretization does not meet the Strang lemma condition. We use the standard duality techniques to prove an extension of Strang lemma. And we prove the convergence and error estimate of discrete eigenvalues and eigenfunctions using the spectral perturbation theory for compact operators. Finally, we present some numerical examples not only on uniform meshes but also in an adaptive refined meshes to show that the Crouzeix-Raviart method is efficient for computing real and complex eigenvalues as expected.

Keywords:
Stekloff eigenvalueNonconforming Crouzeix-Raviart elementStrang lemmaError estimates.

1 Introduction

Stekloff eigenvalue problems have important physical background and many applications. For instance, they appear in the analysis of stability of mechanical oscillators immersed in a viscous fluid (see conca () and the references therein), in the study of surface waves bergman (), in the study of the vibration modes of a structure in contact with an incompressible fluid bermudez () and in the analysis of the antiplane shearing on a system of collinear faults under slip-dependent friction law bucur (). Hence, the finite element methods for solving these problems have attracted more and more scholars’ attention. Till now, systematical and profound studies on the finite element approximation mainly focus on Stekloff eigenvalue problems which satisfy -elliptic condition (see, e.g., alonso (); andreev (); armentano1 (); armentano2 (); bermudez (); bi (); bramble (); cao (); garau (); liq2 (); lim (); russo (); xie (); yang1 () and the references therein).
Recently Cakoni et al. cakoni1 () study a new Stekloff eigenvalue problem arising from the inverse scattering theory:

 Δu+k2n(x)u=0  in Ω,    ∂u∂γ=−λu  on ∂Ω, (1.1)

where is a bounded domain in (), is the outward normal derivative, is the wavenumber and is index of refraction that is a bounded complex valued function with and .
Note that the weak formulation of (1.1) (see (2.1)) does not satisfy -elliptic condition. Cakoni et al. cakoni1 () analyze the mathematics properties of (1.1) and use conforming finite element method to solve it. Liu et al. liu () then study error estimates of conforming finite element eigenvalues for (1.1).
In this paper, we will study the non-conforming Crouzeix-Raviart element (C-R element) approximation for the problem. The C-R element was first introduced by Crouzeix and Raviart in crouzeix () in 1973 to solve the stationary Stokes equation. It was also used to solve linear elasticity equations (see falk (); brenner2 ()), the Laplace equation/eigenvalues (see armentano3 (); boffi (); brenner (); carstensen5 (); carstensen6 (); carstensen1 (); duran ()), Darcy’s equation ainsworth (), Stekloff eigenvalue (seealonso (); bi (); liq2 (); russo (); yang1 ()) and so on. The features of our work are as follows:

1. As we know, the convergence and error estimates of the non-conforming finite element method for an eigenvalue problem is based on the convergence and error estimates of the non-conforming finite element method for the corresponding source problem, and Strang lemma (see strang ()) is a fundamental analysis tool. However, the sesquilinear form in the C-R element discretization here does not meet the Strang lemma condition. To overcome this difficulty, referring to §5.7 in brenner (), we use the standard duality techniques to prove an extension of Strang lemma (see Theorem 2). Based on the theorem, we prove the convergence and error estimates of the C-R method for the corresponding source problem. The current paper, to our knowledge, is the first investigation of applying and extending Strang lemma to elliptic boundary value problem that the corresponding sesquilinear form is non-selfadjoint and not coercive.

2. Cakoni et al. cakoni1 () write (1.1) as an equivalent eigenvalue problem of the Neumann-to-Dirichlet operator . In this paper, we write the C-R element approximation of (1.1) as an equivalent eigenvalue problem of the discrete operator , and prove converges in the sense of norm in , thus using Babuska-Osborn spectral approximation theory babuska () we prove first the convergence and error estimates of C-R finite element eigenvalues and eigenfunctions for the problem (1.1).

3. We implement some numerical experiments not only on uniform meshes but also in adaptive refined meshes. It can be seen that the C-R method is efficient for computing real and complex eigenvalues as expected. In addition, we discover, when the index of refraction is real and the corresponding eigenfunctions are singular, the C-R element eigenvalues approximate the exact ones from above and conforming finite element eigenvalues approximate the exact ones from below, thus we get the upper and lower bounds of eigenvalues.

It should be pointed out that the theoretical analysis and conclusions in this paper are also valid for the extension Crouzeix-Raviart element huj ().
In this paper, regarding the basic theory of finite element methods, we refer to babuska (); brenner (); ciarlet (); oden (); shiwang ().
Throughout this paper, denotes a positive constant independent of , which may not be the same constant in different places. For simplicity, we use the symbol to mean that .

2 Preliminary

In this paper, we assume () is a polygonal () or polyhedron () domain. Let denote the Sobolev space with real order on , is the norm on and , and denotes the Sobolev space with real order on with the norm .
Cakoni et al. cakoni1 () give the weak form of (1.1): Find , , such that

 a(u,v)=−λ,   ∀v∈H1(Ω), (2.1)

where

 a(u,v)=(∇u,∇v)−(k2nu,v),   (u,v)=∫Ωu¯vdx,   =∫∂Ωu¯vds.

The source problem associated with (1.1) is as follows: Find such that

 a(φ,v)=,   ∀v∈H1(Ω). (2.2)

Consider the Neumann eigenvalue problem

 Δu+k2n(x)u=0  in Ω,    ∂u∂γ=0  on ∂Ω. (2.3)

In this paper, we always assume is not an interior Neumann eigenvalue of (2.3). Under this assumption, according to cakoni1 () the Neumann-to-Dirichlet map can be defined as follows. Let , define by

 a(Af,v)=,   ∀v∈H1(Ω), (2.4)

and , where denotes the restriction to . And (2.1) can be stated as the operator form:

 Tw=μw. (2.5)

(2.1) and (2.5) are equivalent, namely, if is an eigenpair of (2.5), then is an eigenpair of (2.1), ; conversely, if is an eigenpair of (2.1), then is an eigenpair of (2.5), .
From cakoni1 () we know is compact. If is real, then is also self-adjoint.
Consider the dual problem of (2.1): Find , such that

 a(v,u∗)=−¯¯¯¯¯λ∗,   ∀v∈H1(Ω). (2.6)

The source problem associated with (2.6) is as follows: Find such that

 a(v,φ∗)=,   ∀v∈H1(Ω), (2.7)

Define the corresponding Neumann-to-Dirichlet operator operator by

 a(v,A∗g)=,   ∀v∈H1(Ω), (2.8)

and . Then (2.6) has the equivalent operator form:

 T∗u∗=−λ∗−1u∗. (2.9)

It can be proved that is the adjoint operator of in the sense of inner product . In fact, from (2.4) and (2.8) we have

 =a(Af,A∗g)==,   ∀f,g∈L2(∂Ω).

Note that since is the adjoint operator of , the primal and dual eigenvalues are connected via .
Let be a regular -simplex partition of (see ciarlet (), pp. 131). We denote where is the diameter of element . Let denote the set of all -faces of elements . We split this set as follows: , with and being the sets of inner and boundary edges, respectively. Let be the C-R element space defined on :
, is continuous at the barycenters
of the -faces of element .
The C-R element approximation of (2.1) is: Find , , such that

 ah(uh,v)=−λh,   ∀v∈Sh, (2.10)

where .
Define , . Evidently, is the norm on and it is easy to know that is not uniformly -elliptic.
The C-R element approximation of (2.2) is: Find , such that

 ah(φh,v)=,   ∀v∈Sh. (2.11)

Since is not an interior Neumann eigenvalue of (2.3), from spectral approximation theory we know that when is properly small also is not a C-R element eigenvalue for (2.3). So the discrete source problem (2.11) is uniquely solvable. Thus, we can define the discrete operator , satisfying

 ah(Ahf,v)=,   ∀v∈Sh. (2.12)

Let us denote by the function space defined on , which are restriction of functions in to . Define the discrete operator , satisfying . Then (2.12) has the equivalent operator form:

 Thwh=μhwh, (2.13)

namely, if is an eigenpair of (2.13), then is an eigenpair of (2.10), ; conversely, if is an eigenpair of (2.10), then is an eigenpair of (2.13), .
The non-conforming finite element approximation of (2.6) is given by: Find , such that

 ah(v,u∗h)=−¯¯¯¯¯¯λ∗h,   ∀v∈Sh. (2.14)

The C-R element approximation of (2.7) is: Find , such that

 ah(v,φ∗h)=,   ∀v∈Sh. (2.15)

Define the discrete operator satisfying

 ah(v,A∗hg)=,   ∀ v∈Sh, (2.16)

and denote , then (2.16) has the following equivalent operator form

 T∗hu∗h=−λ∗−1hu∗h. (2.17)

It can be proved that is the adjoint operator of in the sense of inner product . Hence, the primal and dual eigenvalues are connected via .
We need the following regularity estimates which play an important role in our theoretical analysis. Note that for , has a continuous extension, still denoted by , to .

Lemma 1

For any , let be the dual product on in (2.2), then there exists a unique solution to (2.2) such that

 ∥φ∥1≲∥f∥−12,∂Ω. (2.18)
Proof

Since is not an interior Neumann eigenvalue of (2.3), there exists a unique solution to (2.2). Denote

 b(u,v)=∫Ω∇u⋅¯¯¯¯¯¯¯∇v+n(x)u¯¯¯vdx.

Referring to the proof of (14.11) in ciarlet (), it is easy to verify that is a norm on that is equivalent to the norm . (2.3) can be rewritten as: Find , such that

 b(u,v)=~λ(nu,v),   ∀v∈H1(Ω), (2.19)

Since is not an interior Neumann eigenvalue for (2.3), is not an eigenvalue of (2.19). Define the map by

 b(Bg,v)=(ng,v),   ∀v∈H1(Ω).⋮ (2.20)

Then (2.19) has the operator form:

 Bu=~λ−1u.

And is compact, is not an eigenvalue of . So is bounded. Let be solution the following eqution:

 b(ψ,v)=,   ∀v∈H1(Ω), (2.21)

then we have . From (2.20) we obtain

 a(φ,v)=b(φ,v)−(k2+1)(nφ,v) =b(φ,v)−(k2+1)b(Bφ,v)=b((I−(k2+1)B)φ,v),

which, together with (2.2) and (2.21), yields

 b((I−(k2+1)B)φ,v)=b(ψ,v)   ∀v∈H1(Ω),

Thus we have

 ∥φ∥1=∥(1k2+1I−B)−11k2+1ψ∥1 ≤∥(1k2+1I−B)−1∥1|1k2+1|∥ψ∥1≲∥f∥−12,∂Ω,

and the proof is complete.

Lemma 2

Assume that is a polygonal with being the largest interior angle, and is the solution of (2.2). Let , then and

 ∥φ∥1+r2≤CΩ∥f∥0,∂Ω; (2.22)

let , then satisfying

 ∥φ∥1+r≤CΩ∥f∥12,∂Ω, (2.23)

where , when , and when , and is a priori constant.

Proof

Consider the auxiliary boundary value problem:

 Δφ1+φ1=0,   ∂φ1∂γ=f, (2.24) Δφ2+φ2=−k2n(x)(φ1+φ2)+φ1+φ2,   ∂φ2∂γ=0. (2.25)

Let and be the solution of (2.24) and (2.25), respectively, then it is easy to see that . Since , from classical regularity results (seedauge (), or Proposition 4.1 in alonso () and Proposition 4.4 in bermudez ()) we have

 ∥φ1∥1+(12+s)r≲∥f∥s,∂Ω,   s=0,12,

and from classical regularity result for the Laplace problem with homogeneous Neumann boundary condition we have

 ∥φ2∥1+r≲∥−k2n(x)(φ1+φ2)+φ1+φ2∥0,Ω.

Thus we get

 ∥φ∥1+(12+s)r≲∥φ1∥1+(12+s)r+∥φ2∥1+(12+s)r≲∥f∥s,∂Ω+∥φ∥0,   s=0,12.

Substituting (2.18) into the above inequality we get (2.22) and (2.23).

Remark 1 (Regularity in ).  When is a polyhedron domain, regularity of the solution of the Neumann problem (2.24) has been discussed by many scholars. Referring Theorem 4 in savare () and Remark 2.1 in garau (), and using the argument of Lemma 2 in this paper, we think the following regularity assumption is reasonable:
.   Assume that is a polyhedron domain, and is the solution of (2.2). Let , then there is dependent on such that and

 ∥φ∥1+ι≤CΩ∥f∥0,∂Ω. (2.26)

It is easy to know that Lemmas 1-2 and Remark 1 are also valid for the dual problem (2.7).

3 The consistency term and the extension of Strang lemma

Define
Let and be the solution of (2.2) and (2.7), respectively. Define the consistency terms: For any ,

 Dh(φ,v)=ah(φ,v)−, (3.1) D∗h(v,φ∗)=ah(v,φ∗)−. (3.2)

In order to analyze error estimates of the consistency terms, we need the following trace inequalities.

Lemma 3

, , there holds

 ∥w∥0,∂κ≲h−12κ∥w∥0,κ+hϱ−12κ|w|ϱ,κ,   ∀κ∈πh. (3.3)
Proof

The conclusion is followed by using the trace theorem on the reference element and the scaling argument. See, e.g., Lemma 2.2 in yang1 ().

Lemma 4

Let be the solution of (2.2), then there holds

 ∥∇φ⋅γ∥t−12,ℓ≲∥φ∥1+t,κ   ∀κ∈πh, ℓ∈∂κ. (3.4)
Proof

Inequality (3.4) is contained in the proof of Corollary 3.3 on page 1384 of bernardi (), see also Lemma 2.1 in caiz (). For the convenience of readers, we write the proof here.
For any , it is proven by going to a reference element and using the inverse trace theorem that there exists a lifting of such that , , , and

 ∥∇wg∥−t,κ+ht−1κ∥wg∥0,κ≤C∥g∥12−t,ℓ. (3.5)

From Green’s formula, (2.2), Cauchy-Schwarz inequality, the definition of the dual norm and (3.5) we deduce

 ∫ℓ∇φ⋅γ¯gds=∫ℓ∇φ⋅γ¯¯¯¯¯¯wgds=∫κΔφ¯¯¯¯¯¯wgdx+∫κ∇φ⋅¯¯¯¯¯¯¯¯¯¯¯∇wgdx =∫κ−k2nφ¯¯¯¯¯¯wgdx+∫κ∇φ⋅¯¯¯¯¯¯¯¯¯¯¯∇wgdx≲∥−k2nφ∥0,κ∥¯¯¯¯¯¯wg∥0,κ+∥∇φ∥t,κ∥¯¯¯¯¯¯¯¯¯¯¯∇wg∥−t,κ ≲∥−k2nφ∥0,κh1−tκ∥g∥12−t,ℓ+∥∇φ∥t,κ∥g∥12−t,ℓ≲∥φ∥1+t,κ∥g∥12−t,ℓ,

thus by the definition of the dual norm we obtain

 ∥∇φ⋅γ∥t−12,ℓ=supg∈H12−t(ℓ)|∫ℓ∇φ⋅γ¯gds|∥g∥12−t,ℓ≲∥φ∥1+t,κ.

This completes the proof of the lemma.

Based on the standard argument (see, for example alonso (); liq2 (); yang1 ()), the following consistency error estimates will be proved.

Theorem 3.1

Let and be the solution of (2.2) and (2.7), respectively, and suppose that , then

 |Dh(φ,v)|≲ht∥φ∥1+t∥v∥h,   ∀v∈Sh+H1(Ω), (3.6) |D∗h(v,φ∗)|≲ht∥φ∗∥1+t∥v∥h,   ∀v∈Sh+H1(Ω), (3.7)

where .

Proof

Let denote the jump across an inner face . Then by Green’s formula we deduce

 Dh(φ,v)=ah(φ,v)−=∫Ω(−Δφ−k2n(x)φ)¯vdx +∑κ∈πh∫∂κ∂φ∂γ¯vds−∫∂Ω∂φ∂γ¯vds=∑ℓ∈Eih∫ℓ∂φ∂γ¯[[v]]ds. (3.8)

Let be a ()-face of , define

 Pℓf=1|ℓ|∫ℓfds,   Pκf=1|κ|∫κfdx.

For , suppose that such that . Since is a linear function vanishing at the barycenters of , we have

 |∫ℓ∂φ∂γ¯¯¯¯¯¯¯¯¯[[v]]ds|=|∫ℓ(∂φ∂γ−Pℓ(∂φ∂γ))¯¯¯¯¯¯¯¯¯[[v]]ds| =|∫ℓ(∂φ∂γ−Pℓ(∂φ∂γ))¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯([[v]]−Pℓ[[v]])ds| =|∫ℓ∂φ∂γ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯([[v]]−Pℓ[[v]])ds|. (3.9)

Then, when , using Schwarz inequality we deduce

 |∫ℓ∂φ∂γ¯¯¯¯¯¯¯¯¯[[v]]ds|≤∑i=1,2∥∇φ⋅γ−Pℓ(∇φ⋅γ)∥0,ℓ∥v|κi−Pℓ(v|κi)∥0,ℓ ≤∑i=1,2∥∇φ⋅γ−Pκi(∇φ⋅γ)∥0,ℓ∥v|κi−Pκi(v|κi)∥0,ℓ, (3.10)

by (3.3) and the standard error estimates for -projection, we deduce

 ∥∇φ⋅γ−Pκi(∇φ⋅γ)∥0,ℓ≲ht−12||φ||1+t,κi,   ∥v|κi−Pκi(v|κi)∥0,ℓ≲h12||v||1,κi.

Substituting the above two estimates into (3), we obtain

 |∫ℓ∂φ∂γ¯¯¯¯¯¯¯¯¯[[v]]ds|≲∑i=1,2ht∥φ∥1+t,κi∥v∥1,κi, (3.11)

and substituting (3.11) into (3) we conclude that (3.6) holds.
When , from (3) we deduce that

 |∫ℓ∂φ∂γ¯¯¯¯¯¯¯¯¯[[v]]ds|≤∥∇φ⋅γ∥t−12,ℓ∥[[v]]−Pℓ[[v]]∥12−t,ℓ. (3.12)

By using inverse estimate, (3.3) and the error estimate of -projection, we derive

 ∥[[v]]−Pℓ[[v]]∥12−t,ℓ≲ht−12ℓ∥[[v]]−Pℓ[[v]]∥0,ℓ≲∑i=1,2htκi||v||1,κi.

Substituting the above estimate and (3.4) into (3.12), we obtain

 |∫ℓ∂φ∂γ¯¯¯¯¯¯¯¯¯[[v]]ds|≲∑i=1,2∥φ∥1+t,κihtκi||v||1,κi,

plugging the above inequality into (3) we also get (3.6).
Using the same argument as above, we can prove (3.7).

The C-R element approximation (2.11) of (2.2) does not satisfy the condition of Strang lemma, that is is not uniformly -elliptic. To overcome this difficulty, Inspired by the works in §5.7 in brenner (), next we use standard duality techniques to prove an extension version of the well-known Strang lemma.
First, we will use the standard duality argument to prove that is a quantity of higher order than .
Introduce the auxiliary problem: Find , such that

 a(v,ψ)=(v,g),    ∀ v∈H1(Ω). (3.13)

Let be solution of (3.13), then from elliptic regularity estimates for homogeneous Neumann boundary value problem we know that there exists , such that

 ∥ψ∥1+rN≲∥g∥0. (3.14)

Let , then

 |^Eh(v,ψ)|≲hrN∥ψ∥1+rN∥v∥h,   ∀v∈H1(Ω)+Sh. (3.15)
Lemma 5

Let and be the solution of (2.2) and (2.11), respectively, and let and be the solution of (2.7) and (2.15), respectively, then

 ∥φ−φh∥0≲hrN∥φ−φh∥h, (3.16) ∥φ∗−φ∗h∥0≲hrN∥φ∗−φ∗h∥h. (3.17)
Proof

By Riesz representation theorem we have

 ∥φ−φh∥0=supg∈L2(Ω),g≠0|(φ−φh,g)|∥g∥0. (3.18)

Let be C-R non-conforming finite element interpolation function of , then according to the interpolation theory (see ciarlet ()) we have

 ∥ψ−ψI∥h≲hrN∥ψ∥1+rN. (3.19)

By computing, we deduce

 ==a(φ,ψ)−ah(φh,ψI) =a(φ,ψ)−ah(φ,ψI)+ah(φ,ψI)−ah(φh,ψI) =ah(φ,ψ−ψI)+ah(φ−φh,ψI) =ah(φ,ψ−ψI)+ah(φ−φh,ψI−ψ)+ah(φ−φh,ψ),

and

 ah(φ,ψ−ψI)+ah(φ−φh,ψ)=ah(φ,ψ−ψI)− ++ah(φ−φh,ψ)−(φ−φh,g)+(φ−φh,g) =Dh(φ,ψ−ψI)+^Eh(φ−φh,ψ)++(φ−φh,g),

combining the above two inequalities we get

 (φ−φh,g)=ah(φ−φh,ψ−ψI)−Dh(φ,ψ−ψI)−^Eh(φ−φh,ψ).

Substituting the above equality into (3.18) we get

 ∥φ−φh∥0≤supg∈L2(Ω),g≠0|ah(φ−φh,ψ−ψI)|∥g∥0 +supg∈L2(Ω),g≠0|Dh(φ,ψ−ψI)+^Eh(φ−φh,ψ)|∥g∥0. (3.20)

Let be the Lagrange interpolation operator, , according definition of we deduce

 |Dh(φ,ψ−ψI)|=|Dh(φ,IChψ−ψI)|=|ah(φ−φh,IChψ−