# Error estimates and a two grid scheme for approximating transmission eigenvalues^{†}^{†}thanks: This
work was supported by the National Science Foundation of China (Grant Nos.11561014, 11201093).

###### Abstract

In this paper, using the linearization technique we write the Helmholtz transmission eigenvalue problem as an equivalent nonselfadjoint linear eigenvalue problem whose left-hand side term is a selfadjoint, continuous and coercive sesquilinear form. To solve the resulting nonselfadjoint eigenvalue problem, we give an conforming finite element discretization and establish a two grid discretization scheme. We present a complete error analysis for both discretization schemes, and theoretical analysis and numerical experiments show that the methods presented in this paper can efficiently compute real and complex transmission eigenvalues.

Key words. transmission eigenvalues, finite element method, two grid discretization scheme, error estimates

AMS subject classifications. 65N25, 65N30, 65N15

## 1 Introduction

Transmission eigenvalues not only have wide physical
applications, for example, they can be used to obtain estimates for
the material properties of the scattering object
[6, 7], but also have theoretical importance in the
uniqueness and reconstruction in inverse scattering theory
[13]. Many papers such as
[7, 13, 23, 28, 29] study the existence
of transmission eigenvalues, and [7, 8, 15]
explore upper and lower bounds for the index of refraction
from knowledge of the transmission eigenvalues.
The attention from the computational mathematics community is also increasing (see, e.g., [1, 9, 14, 19, 21, 22, 26, 31, 32]).

In this paper, we consider the Helmholtz transmission
eigenvalue problem: Find , ,
such that

\hb@xt@.01(1.1) | |||

\hb@xt@.01(1.2) | |||

\hb@xt@.01(1.3) | |||

\hb@xt@.01(1.4) |

where is a bounded simply
connected set containing an inhomogeneous medium, and is the unit outward normal
to .

From [8, 29] we know that for , the weak formulation for the transmission eigenvalue
problem (LABEL:s1.1)-(LABEL:s1.4) can be stated as follows: Find
, , such that

\hb@xt@.01(1.5) |

where denotes
the inner product. We denote as usual,
then (LABEL:s1.5) is a quadratic
eigenvalue problem.

In recent years, based on this weak formulation, various
numerical methods have been presented to solve the transmission
eigenvalue problem. The first numerical treatment appeared in
[14] where three finite element methods were proposed for
the Helmholtz transmission eigenvalues which have been further
developed in [1, 9, 19, 21, 22, 31]. Among them
[14, 22, 31] studied the conforming finite
element method, [9, 14, 21] mixed finite element
methods, [1] spectral-element method and [19] the
Galerkin-type numerical method.
Inspired by these
works, this paper further studies the conforming finite
element method and has three features as follows:

(1) A complete error analysis is presented. Due to the
fact that the problem is neither elliptic nor self-adjoint, once its
error analysis was viewed as a difficult task.
Sun [31]
first proved an error estimate for the conforming finite element approximation
of (LABEL:s1.5). Following him Ji et al. [22] presented an
accurate error estimate, but Theorems 1-2 therein are only
valid for real eigenvalues and rely on the conditions of Lemma 3.2
in [31] which are not easy to verify in general, especially
for multiple eigenvalues. In this paper, we use the linearization
technique in [33] to write the weak formulation
(LABEL:s1.5) as an equivalent nonselfadjoint linear eigenvalue
problem (LABEL:s2.6). Then the conforming finite element
approximation eigenpair of (LABEL:s2.6) is
exactly the one of (LABEL:s1.5). Fortunately, the left-hand side
term of (LABEL:s2.6) is a selfadjoint, continuous and coercive
sesquilinear form, thus we can use Babuska-Osborn’s spectral
approximation theory [2] to give a complete error
analysis for the conforming finite element approximation of
(LABEL:s2.6), namely the error analysis of (LABEL:s1.5).
The difficulty of the error estimates in low norms lies in the
nonsymmetry of the right-hand side term of (LABEL:s2.6) that
involves derivatives. We introduce two auxiliary problems
and overcome this difficulty by the
Nitche technique in a subtle way. Our theoretical results are proved under general conditions
and are valid for arbitrary real and complex eigenvalues.

(2) A finite element discretization with good algebraic
properties is presented. We give an conforming finite
element discretization (see (LABEL:s3.1) or (LABEL:s5.1)) that will
lead to a positive definite Hermitian and block diagonal stiff
matrix. Then we use this discretization to solve the transmission
eigenvalue problem numerically and obtain both real and complex
transmission eigenvalues of high accuracy as expected. Similar
discretizations already exist in the literatures. Colton et al.
[14] established the conforming finite element
discretization for the formulation (LABEL:s1.5) (see (4.3) in
[14]),
by starting with discretizing (LABEL:s1.5) into a
quadratic algebra eigenvalue problem and then linearizing it, which
is in the opposite order of our treatment.
Though our discrete form (LABEL:s5.1) is formally different from
the one in [14] for the resulting mass matrix in
[14] is positive definite Hermitian and block diagonal,
the two discrete forms are equivalent from the linear algebra point
of view. Gintides et al. [19] has used square root of
matrices, which is a standard tool, and a Hilbert basis in
as test functions to discretizite (LABEL:s1.5) into a
linear eigenvalue problem (see (2.14) in [19]) and proved
the convergence for the approximate eigenvalues.

(3) A
two grid discretization scheme is proposed. As we know, it is
difficult to solve nonselfadjoint eigenvalue problems in general.
So, Sun [31] adopted iterative methods to work on a series of
generalized Hermitian problems and finally to solve for the real
transmission eigenvalues. Ji et al. [22] combined the
iterative methods in [31] with the extended finite element
method
to establish a multigrid algorithm for solving the real transmission eigenvalues.
This paper establishes a two grid
discretization scheme directly for the nonselfadjoint eigenvalue
problem (LABEL:s2.6) which is suitable for arbitrary real and
complex eigenvalues.
The two grid discretization scheme is an
efficient approach which was first introduced by Xu [34] for
nonsymmetric or indefinite problems, and successfully applied to
eigenvalue problems later (see, e.g.,
[17, 24, 35, 36, 37] and the references therein). With
the two grid discretization scheme, the solution of the transmission
eigenvalue problem on a fine grid is reduced to the
solution of the primal and dual eigenvalue problem on a much coarser
grid and the solutions of two linear algebraic systems
with the same positive definite Hermitian and block diagonal
coefficient matrix on the fine grid , and the resulting
solution still maintains an asymptotically optimal accuracy. The key
to our theoretical analysis is to find a
and prove that (see Step 1 of Scheme 4.1) has a
positive lower bound uniformly with respect to the mesh size ,
which is also critical for further establishing multigrid method
and adaptive algorithm. We successfully apply spectral approximation
theory to solve this
problem (see Lemma 4.1 and Remark 4.1).

The conforming finite element discretization is easy
to implement under the package of iFEM [11] with MATLAB, and
the numerical results indicate that this discretization is
efficient for computing transmission eigenvalues. Moreover, we can
further improve the computational efficiency
by using the two grid discretization scheme.

Regarding the basic theory of finite
element methods, we refer to [2, 4, 12, 27].

Throughout this paper, denotes a positive constant
independent of the mesh size , which may not be the same constant
in different places. For simplicity, we use the symbol
to mean that .

## 2 A linear formulation of (LABEL:s1.5)

In this paper, we suppose that the index of refraction satisfies the following assumption

for some constant , although, with obvious changes, the
theoretical analysis in this paper
also holds for strictly less than .

Inspired by the works in [9, 14, 21], we now
apply the linearization technique in [33] to write the
weak
formulation (LABEL:s1.5) as an equivalent linear formulation.

From (LABEL:s1.5) we derive that

\hb@xt@.01(2.1) |

Introduce an auxiliary variable

\hb@xt@.01(2.2) |

then

\hb@xt@.01(2.3) |

Thus, combining (LABEL:s2.1) and (LABEL:s2.3), we arrive at a linear formulation: Find and nontrivial such that

\hb@xt@.01(2.4) | |||

\hb@xt@.01(2.5) |

This is a nonselfadjoint linear eigenvalue problem.

Suppose that is an eigenpair of (LABEL:s1.5),
then it is obvious that is an eigenpair of
(LABEL:s2.4)-(LABEL:s2.5). On the other hand, suppose that satisfies (LABEL:s2.4)-(LABEL:s2.5), from (LABEL:s2.5)
we get , and substituting
it into (LABEL:s2.4) we get (LABEL:s2.1)(or (LABEL:s1.5)). The above argument indicates that
(LABEL:s2.4)-(LABEL:s2.5) and (LABEL:s1.5) are equivalent.

Let be the “negative space”, with norm given by

Define the Hilbert space with norm
, and define
with norm
,
, .
It’s obvious that compactly (see[5, 35] ).

Denote . Let

then (LABEL:s2.4)-(LABEL:s2.5) can be rewritten as: Find , such that

\hb@xt@.01(2.6) |

Thus we get the following theorem.

###### Theorem 2.1

If is an eigenpair of (LABEL:s1.5) and , then satisfies (LABEL:s2.6), and if satisfies (LABEL:s2.6), then is an eigenpair of (LABEL:s1.5) and .

By calculation we derive for any ,

i.e.,
is a selfadjoint, continuous and coercive
sesquilinear form on .

We use and
as an inner product and
norm on , respectively. Then is
equivalent to the norm .

When , , , we deduce

\hb@xt@.01(2.7) |

When , , we have

\hb@xt@.01(2.8) |

And when , , we have

\hb@xt@.01(2.9) |

We can see from (LABEL:s2.7)-(LABEL:s2.9) that
for any given (), is a continuous linear form on .

The
source problem associated with (LABEL:s2.6) is given by: Find
such that

\hb@xt@.01(2.10) |

From the Lax-Milgram theorem we know that (LABEL:s2.10) admits a unique solution. Therefore, we define the corresponding solution operators (s=0,1,2) by

\hb@xt@.01(2.11) |

Then (LABEL:s2.6) has the equivalent operator form:

\hb@xt@.01(2.12) |

Namely, if is an eigenpair of (LABEL:s2.6),
then is an eigenpair of (LABEL:s2.12).
Conversely, if is an eigenpair of
(LABEL:s2.12), then is an eigenpair of (LABEL:s2.6).

###### Theorem 2.2

Suppose , then is compact; suppose , then is compact.

Proof. When , let in (LABEL:s2.11), then from (LABEL:s2.7) we have

thus

\hb@xt@.01(2.13) |

Let be any bounded set in
, because of the compact embedding
and , then is relatively compact in , is relatively compact in
and is relatively
compact in . And from (LABEL:s2.13) we conclude that is
compact.

When , in (LABEL:s2.11), let
, then from (LABEL:s2.8) and (LABEL:s2.9) we have

thus

\hb@xt@.01(2.14) |

which implies that is continuous.
Due to the compact embedding , is compact.

Consider the dual problem of (LABEL:s2.6): Find
, such that

\hb@xt@.01(2.15) |

Define the corresponding solution operators by

\hb@xt@.01(2.16) |

Then (LABEL:s2.15) has the equivalent operator form:

\hb@xt@.01(2.17) |

It can be proved that is the adjoint operator of in the sense of inner product . In fact, from (LABEL:s2.11) and (LABEL:s2.16) we have

\hb@xt@.01(2.18) |

Hence the primal and dual eigenvalues are connected via .

## 3 The conforming finite element approximation and its error estimates

Let be a shape-regular grid of with mesh size and
be a piecewise polynomial space defined
on ; for example, is the finite element space
associated with one of the Argyris element, the Bell element or the
Bogner-Fox-Schmit element (BFS element) (see [12]). Thanks
to (LABEL:s2.2), we choose . Then
be a conforming finite element
space. For the three finite element spaces mentioned above, since
are dense in both and
, it is obvious that the following
condition (C1) holds:

(C1) If , then as ,

From the operator interpolation theory (see e.g. [4]), the following (C2) is also valid:

(C2) If , then

Throughout this paper, we suppose that (C1) is valid.

The conforming finite element approximation of
(LABEL:s2.6) is given by the following: Find , such that

\hb@xt@.01(3.1) |

Consider the approximate source problem: Find such that

\hb@xt@.01(3.2) |

We introduce the corresponding solution operator: (s=0,1):

\hb@xt@.01(3.3) |

Then (LABEL:s3.1) has the operator form:

\hb@xt@.01(3.4) |

Define the projection operators and by

\hb@xt@.01(3.5) | |||

\hb@xt@.01(3.6) |

Let

Then , and

\hb@xt@.01(3.7) |

i.e., is the Ritz projection.

Combining (LABEL:s3.7), (LABEL:s2.11) with (LABEL:s3.3), we
deduce for any that

thus we get

\hb@xt@.01(3.8) |

###### Theorem 3.1

Let , then

\hb@xt@.01(3.9) |

and let , then

\hb@xt@.01(3.10) |

Proof. When , for any , from (C1) we have

Since is compact, is relatively compact. Thus by the definition of operator norm we have

When , for , we have

Since is compact, is relatively compact, thus we have

The proof is completed.

The conforming finite element approximation of (LABEL:s2.15) is given by: Find , such that

\hb@xt@.01(3.11) |

Define the solution operator satisfying

\hb@xt@.01(3.12) |

(LABEL:s3.11) has the following equivalent operator form

\hb@xt@.01(3.13) |

It can be proved that is the adjoint operator of in the sense of inner product . In fact, from (LABEL:s3.3) and (LABEL:s3.12) we have

\hb@xt@.01(3.14) |

Hence, the primal and dual eigenvalues are connected via
.

We need the following
lemma (see Lemma 5 on page 1091 of [18]).

###### Lemma 3.2

Let . Let be an enumeration of the eigenvalues of , each repeated according to its multiplicity. Then there exist enumerations of the eigenvalues of , with repetitions according to multiplicity, such that ().

In this paper we suppose that
and satisfy the above
lemma, and let be the kth eigenvalue with the
algebraic multiplicity and the ascent ,
. Since
, eigenvalues
of (LABEL:s3.1) will
converge to .

Let be the
spectral projection associated with and , then
is the space of generalized
eigenvectors associated with and , where denotes
the range and denotes the null space. Let be the
spectral projection associated with and the eigenvalues
; let
be the space of generalized eigenvectors associated with
and , and let
, then
if is small enough.
In view of the
dual problem (LABEL:s2.15) and (LABEL:s3.11), the definitions of
, , , ,
and are analogous to ,
, , , and
(see
[2]).

Given two closed subspaces and , denote

We define the gaps between and in as

and in as

Define

It follows directly from (C1) that

Suppose that , then from (C2) we get

\hb@xt@.01(3.15) |

Further suppose that , is the Argyris finite element space, then from the interpolation theory we have

\hb@xt@.01(3.16) |

Note that when the functions in and are piecewise
smooth (LABEL:s3.15) and (LABEL:s3.16) are also valid.

Thanks to [2], we get the following Theorem 3.3.

###### Theorem 3.3

Suppose , then

\hb@xt@.01(3.17) | |||

\hb@xt@.01(3.18) | |||