Abstract
We consider Helmholtz problems in three-dimensional domains featuring conical points.
We focus on the high-frequency regime and derive novel sharp upper-bounds for the stress intensity factors of the singularities associated with the conical points.
We then employ these new estimates to analyse the stability of finite element discretisations.
Our key result is that lowest-order Lagrange finite elements are stable under the assumption that “
1 Introduction
High-frequency wave propagation problems play a crucial role in several applications ranging from radar imaging [9] to nanophotonics [12], just to cite a few. When considering complex geometries and/or heterogeneous media, finite element methods are a popular approach to discretise such problems and compute approximate numerical solutions. In this work, we focus on scalar Helmholtz problems which is probably the simplest model problem featuring the difficulties linked with high-frequency waves.
When considering smooth domains, finite element discretisations of Helmholtz problems are now widely covered in the literature; see e.g. [4, 5, 18, 20, 21] and the references therein.
In particular, it is now well known that, in the high-frequency regime, the “pollution effect” alters the quality of the finite element solution.
More precisely, the quasi-optimality of the finite element solution (which is always guaranteed when considering coercive problems) is lost in the high-frequency regime, unless the mesh is sufficiently refined.
For the lowest-degree Lagrange elements in smooth (or convex) non-trapping domains (see Assumption 2.1 below), it is well known that the condition “
When dealing with complex geometries and, in particular, with polytopal domains, another concern is the possible presence of singularities in the vicinity of re-entrant corners and edges [6, 16]. These singularities may lower the regularity of the solution and, in turn, diminish the convergence rate of the finite element discretisation. For coercive problem, it is now well established that the effect of singularities is essentially local and may be taken care of by local mesh refinements [1, 3, 4]. Specifically, employing geometrically graded meshes close to edges and corners of the boundary permits to restore the optimal convergence rate of the method without adding a significant number of degrees of freedom.
The goal of this work is to analyse the behaviour of corner singularities in the context of high-frequency problems, and their impact on finite element discretisations.
Actually, the authors already proposed an analysis of the two-dimensional case in [4], and the purpose of the present work is to extend it to three-dimensional domains with conical points.
In fact, we show that the behaviour of corner singularities in three-dimensional domains is fairly similar to the two-dimensional case.
In short, we show that the stress intensity factors associated with each singular function decreases as the frequency increases.
A direct consequence is that the condition “
The remainder of this work is organised as follows. Section 2 makes the model problem we are considering precise and recalls the key assumption that the domain is non-trapping. In Section 3, we consider the case where the domain is actually a cone, thus featuring a single singular point. Section 4 extends this preliminary result to general non-trapping domains via a localisation argument. We build over these abstract results in Section 5, where we establish necessary and sufficient conditions for the stability of finite element discretisations under suitable assumptions on the mesh. In Appendix A, we present technical integration-by-parts results for singular functions.
2 The Setting
In this work, we consider wave propagation problems modelised by the Helmholtz equation in a domain Ω,
where
Classically, assuming that
where
In the remainder of this work, we assume that problem (2.2) is well-posed and “non-trapping”. This is a reasonable assumption often satisfied in applications [2, 11, 17, 22].
We assume that there exists a non-negative constant
for some positive constant
Typically,
When the domain Ω is convex or smooth, one easily obtains a bound for the
3 The Case of a Cone
Given an open subset 𝐺 of the unit sphere of
Its boundary is split into two parts
corresponding respectively to the radial and angular portions of
Since
If Ω has a smooth boundary or if it is convex, it easily follows that
In this section, we require the following key properties linked to Bessel functions.
Before their statements, we recall that, for an arbitrary real number
while the corresponding Hankel function of second kind is defined by
For all
Proof
Except for (3.3), these results are shown in [4, Proposition 3.1].
For (3.3), consider
with
It is also shown in [4, Lemma A.1] that
These properties imply that
as well as
Hence we easily conclude that
from which (3.3) follows. ∎
3.1 Splitting of the Solution
We propose a splitting of the solution 𝑢 into a regular part
that we may suppose to form an orthonormal basis of
Then, for all
and set
where
Now, if
where
Decomposition (3.6) is especially useful when analysing Laplace problems, as
with
where for shortness we have set
and since
In Theorem 3.2, we show that the solution 𝑢 can be decomposed according to (3.6) or (3.7).
Furthermore, we give a relation between the constants
Assume that
Furthermore, (3.7) also holds for some function
Proof
The existence and uniqueness of
where
As 𝑔 belongs to
with the estimate
for some positive constant
Since
As a result, we see that
where
where
Once (3.6) is established, (3.7) and (3.9) directly follow from a careful inspection of the definition of
with
3.2 Estimation of
c
ω
,
j
(
f
)
For each
For all
Proof
First, let us prove that the function
To prove (3.15), consider
for all
To analyse the boundary conditions satisfied by
Since 𝛽 is arbitrary in
In the same manner, we take an arbitrary
As before, using (A.3) to the pair
Since 𝛾 is arbitrary in
Because
We now prove the opposite statement that a function
Then the conclusion follows if we can show that
Indeed, in such a case, we would have
implying that
Using Lemma A.1, we can apply Corollary A.5 to the pair
We then find that
By the above properties of
As we consider a simple geometry, the analytical expression of
Recalling that
with
Proof
We are going to show that the function
so that
We observe that, by construction, both 𝜎 and
and (3.15) is satisfied.
As
hence Theorem A.4 gives a meaning of its trace on
Boundary condition (3.16) is also satisfied by construction.
Indeed, if
so that
Multiplying by
Let us go on with property (3.19). Indeed, as
we have
This automatically leads to (3.19) because
Hence it remains to show that (3.18) holds. We have
We are going to show that
The technique is then to subtract the ball
The beginning of the proof of (3.22) and (3.23) is the same.
Thus, let us set
Since
Obviously, when
and (3.22) follows since
On the other hand, to prove (3.23), since
by (3.24) with
As we have that
and we obtain
Then (3.23) holds by letting
We have
Proof
Estimate (3.25) directly follows from property (3.3) from Proposition 3.1. Then, because of (3.25), we have
In addition, from (3.1) and (3.2), we have
and (3.26) follows. ∎
We are now ready to establish the main result of this section.
The estimate
holds for all
Proof
By definition, for all
But Corollary 3.5 shows that
assuming that
The estimate
holds for all
Proof
Direct consequence of the previous theorem, identity (3.9) and recalling that
3.3 Behaviour of the Regular Part
To complete our analysis, we now need to study the regular part
Assume that
satisfying
Proof
We proceed as in Theorem 3.2 and use the lifting
owing to (2.3) and (3.11). Now, using the splitting (3.13) of 𝑣, we have
where
Since
this last estimate following from the fact that
Then we see that
By applying the Poincaré inequality, we further see that
Using this estimate, we find
This estimate and (3.11) lead to (3.30), recalling that
4 The General Case
We now consider the general case of Helmholtz problems that satisfy Assumptions 2.1 and 2.2.
Recall that ℓ is defined in Assumption 2.2 as the smallest distance between two corners of
where
for
Then, if
where
With this notation, the general case then easily follows from a localisation argument presented in [4, Theorem 4.1], where the “cone case” is applied to the neighbourhood of each corners of
For all
and constants
and it holds that
for all
5 Frequency-Explicit Stability of Finite Element Discretisations
5.1 The Finite Element Space
In this section, we assume that we are given a mesh
We will further assume that there exists an interpolation operator such that
for all
The solution
For the sake of simplicity, we assume in the remainder of this work that
This assumption is natural and means that the number of elements per wavelength is bounded from below. We will actually see that more restrictive conditions on ℎ must be met anyway to ensure the well-posedness of the finite element problem.
In order to simplify notation, we introduce the 𝜔-dependent norm
that is equivalent to the standard
We start our analysis with an interpolation result for the singular functions.
For all
for all
Proof
It is clear that
Then the core of our theoretical findings is the following frequency-explicit interpolation estimate for the solution to the Helmholtz problem.
For
Then we have
where
Proof
We recall that we have the decomposition
where
we have
Recalling that
On the other hand, recalling (4.1) and (5.4), we have
Since
Using Lemma 5.2, we can easily derive sufficient condition for the quasi-optimality of the finite element solution following the so-called Schatz argument [10, 20, 25].
Interestingly, the resulting quasi-optimality, “
For all
and the error estimate
holds.
Proof
The proof uses the standard Schatz argument.
Let
By definition of 𝜉, recalling (5.5), we have
so that
It follows that
Now, we write
and simplifying by
Recalling that
Hence, assuming that
and (5.8) follows from (5.10). Finally, (5.9) follows from (5.8) and (5.5).
The uniqueness of
6 Numerical Example
We consider one numerical example that illustrates our key findings. The domain is the revolution cone
of opening
belongs to the kernel of Laplace operator. Similarly, the function
is in the kernel of the Helmholtz operator with frequency 𝜔. For our example, the analytic solution will be
where 𝜒 is a smooth radial cutoff function that equals one close to 0 and zero close to 1.
Since 𝑆 is the solution to the Helmholtz problem (without the boundary condition on
Notice that, since the supports of
We consider a series of meshes generated by the software package gmsh [13]. Contrary to our theoretical analysis, these meshes consist of straight elements, with nodes exactly lying on the boundary of Ω. However, the resulting “variational crime” is expected to result in an error smaller than the best-approximation error since we are working with lowest-order Lagrange elements.
Figure 2 reports the convergence histories of the finite element solutions
For more details, we also refer the reader to the numerical experiments in [4]. There, the 2D case is considered so that finer meshes and higher frequencies may be easily considered, leading to an even better illustration of this phenomenon.
7 Conclusion
Following the lines of our analysis of polygonal domains [4], we analyse high-frequency Helmholtz problems in three-dimensional domains with conical points.
We derive sharp estimates on the stress intensity factors associated with each corner singularity.
Besides, we show that finite element discretisations with lowest-order Lagrange elements are stable under the usual condition that “
Dedicated to Professor Thomas Apel on the occasion of his 60th birthday.
A An Embedding and Green’s Formula
We recall that, for all
that is a Hilbert space equipped with its natural inner product; see [6, Appendix AA].
We first recall the following result; see [6, Theorem AA.7].
The embedding
Let us now introduce the space
where, as usual, for an open subset
For a smooth function
can be continuously extended from
But, according to [14], this mapping is not surjective and its range is not closed.
We then introduce the space
As
The mapping 𝑇 from
Proof
Let us fix
Now fix a radial and smooth cutoff function 𝜂 such that
Let us now introduce the space (see [15, 23])
that is a Banach space with the natural norm
Since the boundary of
The space
The trace mapping 𝑇 from
Proof
As Theorem A.2 guarantees that 𝑇 is continuous and surjective from
Now, for
Splitting the left integral into
Now fix
Now, for a fixed
By estimate (A.2), we find that
This means that
For all
where the first duality bracket has to be understood as a duality between
(note that this bracket is conjugate linear in 𝑢), and similarly for the other brackets.
Proof
By Lemma A.3, there exists a sequence of
We then apply Green’s formula (A.1) to the pair
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