Stabilization in relation to wavenumber in HDG methods
 Jay Gopalakrishnan^{1},
 Stéphane Lanteri^{2},
 Nicole Olivares^{1}Email author and
 Ronan Perrussel^{3}
DOI: 10.1186/s403230150032x
© Gopalakrishnan et al. 2015
Received: 20 March 2015
Accepted: 25 May 2015
Published: 25 June 2015
Abstract
Background
Simulation of wave propagation through complex media relies on proper understanding of the properties of numerical methods when the wavenumber is real and complex.
Methods
Numerical methods of the Hybrid Discontinuous Galerkin (HDG) type are considered for simulating waves that satisfy the Helmholtz and Maxwell equations. It is shown that these methods, when wrongly used, give rise to singular systems for complex wavenumbers.
Results
A sufficient condition on the HDG stabilization parameter for guaranteeing unique solvability of the numerical HDG system, both for Helmholtz and Maxwell systems, is obtained for complex wavenumbers. For real wavenumbers, results from a dispersion analysis are presented. An asymptotic expansion of the dispersion relation, as the number of mesh elements per wave increase, reveal that some choices of the stabilization parameter are better than others.
Conclusions
To summarize the findings, there are values of the HDG stabilization parameter that will cause the HDG method to fail for complex wavenumbers. However, this failure is remedied if the real part of the stabilization parameter has the opposite sign of the imaginary part of the wavenumber. When the wavenumber is real, values of the stabilization parameter that asymptotically minimize the HDG wavenumber errors are found on the imaginary axis. Finally, a dispersion analysis of the mixed hybrid Raviart–Thomas method showed that its wavenumber errors are an order smaller than those of the HDG method.
Keywords
HDG Raviart–Thomas Dispersion Dissipation Absorbing material Complex Wave speed Optimal Stabilization Helmholtz Maxwell UnisolvencyBackground
Wave propagation through complex structures, composed of both propagating and absorbing media, are routinely simulated using numerical methods. Among the various numerical methods used, the Hybrid Discontinuous Galerkin (HDG) method has emerged as an attractive choice for such simulations. The easy passage to high order using interface unknowns, condensation of all interior variables, availability of error estimators and adaptive algorithms, are some of the reasons for the adoption of HDG methods.
It is important to design numerical methods that remain stable as the wavenumber varies in the complex plane. For example, in applications like computational lithography, one finds absorbing materials with complex refractive index in parts of the domain of simulation. Other examples are furnished by metamaterials. A separate and important reason for requiring such stability emerges in the computation of resonances by iterative searches in the complex plane. It is common for such iterative algorithms to solve a source problem with a complex wavenumber as its current iterate. Within such algorithms, if the HDG method is used for discretizing the source problem, it is imperative that the method remains stable for all complex wavenumbers.
One focus of this study is on complex wavenumber cases in acoustics and electromagnetics, motivated by the abovementioned examples. Ever since the invention of the HDG method in Ref. [1], it has been further developed and extended to other problems in many works (so many so that it is now impractical to list all references on the subject here). Of particular interest to us are works that applied HDG ideas to wave propagation problems such as [2–8]. We will make detailed comparisons with some of these works in a later section. However, none of these references address the stability issues for complex wavenumber cases. While the choice of the HDG stabilization parameter in the real wave number case can be safely modeled after the wellknown choices for elliptic problems [9], the complex wave number case is essentially different. This will be clear right away from a few elementary calculations in the next section, which show that the standard prescriptions of stabilization parameters are not always appropriate for the complex wave number case. This then raises further questions on how the HDG stabilization parameter should be chosen in relation to the wavenumber, which are addressed in later sections.
Another focus of this study is on the difference in speeds of the computed and the exact wave, in the case of real wavenumbers. By means of a dispersion analysis, one can compute the discrete wavenumber of a wavelike solution computed by the HDG method, for any given exact wavenumber. An extensive bibliography on dispersion analyses for the standard finite element method can be obtained from Refs. [10, 11]. For nonstandard finite element methods however, dispersion analysis is not so common [12], and for the HDG method, it does not yet exist. We will show that useful insights into the HDG method can be obtained by a dispersion analysis. In multiple dimensions, the discrete wavenumber depends on the propagation angle. Analytic computation of the dispersion relation is feasible in the lowest order case. We are thus able to study the influence of the stabilization parameter on the discrete wavenumber and offer recommendations on choosing good stabilization parameters. The optimal stabilization parameter values are found not to depend on the wavenumber. In the higher order case, since analytic calculations pose difficulties, we conduct a dispersion analysis numerically.
We begin, in the next section, by describing the HDG methods. We set the stage for this study by showing that the commonly chosen HDG stabilization parameter values for elliptic problems are not appropriate for all complex wavenumbers. In the subsequent section, we discover a constraint on the stabilization parameter, dependent on the wavenumber, that guarantees unique solvability of both the global and the local HDG problems. Afterward, we perform a dispersion analysis for both the HDG method and a mixed method and discuss the results.
Methods of the HDG type
We borrow the basic methodology for constructing HDG methods from Ref. [1] and apply it to the timeharmonic Helmholtz and Maxwell equations (written as first order systems). While doing so, we set up the notations used throughout, compare the formulation we use with other existing works, and show that for complex wavenumbers there are stabilization parameters that will cause the HDG method to fail.
Undesirable HDG stabilization parameters for the Helmholtz system
Comparison with some HDG formulations in other papers
Reference  Their notations and equations  Connection to our formulation 

Helmholtz case [2]  \({\begin{array}{l} \vec {q}_{[2]} + {\vec \nabla }{u} _{[2]} = \vec {0} \\ \mathop {\vec \nabla \cdot }\vec {q}_{[2]}  k^2 u_{[2]} = 0 \\ \hat{q}_{[2]} \cdot \vec {n} = \vec {q}_{[2]} \cdot \vec {n} + \hat{\imath }\tau _{[2]} (u _{[2]} \hat{u} _{[2]} ) \end{array}}\)  \({ \begin{array}{ll} \tau _{[2]} & = k\, \tau \\ \hat{\imath }k u _{[2]} & = \phi \\ \vec {q} _{[2]} & =\vec {u}\end{array} }\) 
Helmholtz case [4]  \({ \begin{array}{l} \hat{\imath }k \vec {q} _{[4]} + {\vec \nabla }{u} _{[4]} = \vec {0} \\ \hat{\imath }k u _{[4]} + \mathop {\vec \nabla \cdot }\vec {q} _{[4]} = 0 \\ \hat{q} _{[4]} \cdot \vec {n} = \vec {q} _{[4]} \cdot \vec {n} + \tau _{[4]} (u _{[4]} \hat{u} _{[4]} ) \end{array} }\)  \({ \begin{array}{ll} \tau _{[4]} & = \tau \\ u _{[4]} & = \phi \\ \vec {q} _{[4]} & =\vec {u}\end{array} }\) 
2D Maxwell case [6]  \({ \begin{array}{l} \hat{\imath }\omega _{[6]} \varepsilon _r E _{[6]} \mathop {\nabla \times }\vec {H}_{[6]}= 0 \\ \hat{\imath }\omega _{[6]} \mu _r \vec {H}_{[6]} +\mathop {\vec \nabla \times }{E} _{[6]}= \vec {0} \\ \hat{H} _{[6]} = \vec {H}_{[6]} +\tau _{[6]} (E _{[6]} \hat{E} _{[6]})\vec {t} \end{array} }\)  \({ \begin{array}{ll} \tau _{[6]} &= \sqrt{\frac{\varepsilon _r}{\mu _r}}\tau \\ \omega _{[6]} &= \omega \sqrt{\varepsilon _0\mu _0}\\ E _{[6]} &= \frac{1}{\sqrt{\varepsilon _r}}E,~ \vec {H}_{[6]} = \frac{1}{\sqrt{\mu _r}}\vec {H}\\ \end{array} }\) 
Maxwell case [8]  \({ \begin{array}{l} \mu \vec w _{[8]}  \mathop {\vec \nabla \times }\vec {u}_{[8]} = \vec {0} \\ \mathop {\vec \nabla \times }\vec {w}_{[8]} \varepsilon \omega ^2 \vec {u}_{[8]} =\vec {0} \\ \hat{w} _{[8]} = \vec {w}_{[8]} + \tau _{[8]} ( \vec {u}_{[8]}  \hat{u} _{[8]} ) \times \vec {n}\end{array} }\)  \({ \begin{array}{ll} \tau _{[8]} & = \hat{\imath }\,\sqrt{ \frac{\varepsilon \omega ^2}{\mu } } \, \tau \\ \mu \vec {w}_{[8]} & =\hat{\imath }k \vec {H},\text { with }k = \omega \sqrt{\mu \varepsilon }, \\ \vec {u}_{[8]} & =\vec {E}\end{array} }\) 
Intermediate case of the 2D Maxwell system
The 3D Maxwell system
Behavior on tetrahedral meshes
From another perspective, Figure 1e shows the smallest singular value of the element matrix as \(\tau\) is varied in the complex plane, while fixing kh to 1. Figure 1f is similar except that we fixed kh to the value discussed above, approximately 7.49. In both cases, we find that the values of \(\tau\) that yielded the smallest singular values are along the imaginary axis. Finally, in Figure 1g, h, we see the effects of multiplying these real values of kh by \(1+\hat{\imath }.\) The region of the complex plane where such values of \(\tau\) are found changes significantly when kh is complex.
Results on unisolvent stabilization
We now turn to the question of how we can choose a value for the stabilization parameter \(\tau\) that will guarantee that the local matrices are not singular. The answer, given by a condition on \(\tau\), surprisingly also guarantees that the global condensed HDG matrix is nonsingular. These results are based on a tenuous stability inherited from the fact nonzero polynomials are never waves, stated precisely in the ensuing lemma. Then we give the condition on \(\tau\) that guarantees unisolvency, and before concluding the section, present some caveats on relying solely on this tenuous stability.
Before proceeding to the main result, we give a simple lemma, which roughly speaking, says that nontrivial harmonic waves are not polynomials.
Lemma 1
Proof
We use a contradiction argument. If \(E \not \equiv \vec {0}\), then we may assume without loss of generality that at least one of the components of \(\vec {E}\) is a polynomial of degree exactly p. But this contradicts \(k^2 \vec {E}= \mathop {\vec \nabla \times }( \mathop {\vec \nabla \times }\vec {E})\) because all components of \(\mathop {\vec \nabla \times }(\mathop {\vec \nabla \times }\vec {E})\) are polynomials of degree at most \(p2\). Hence \(\vec {E}\equiv \vec {0}\). An analogous argument can be used for the Helmholtz case as well. \(\square\)
Theorem 1
Proof
The proof for the Helmholtz case is entirely analogous. \(\square\)
Note that even with Dirichlet boundary conditions and real k, the theorem asserts the existence of a unique solution for the Helmholtz equation. However, the exact Helmholtz problem (1a, 1b, 1c) is wellknown to be not uniquely solvable when k is set to one of an infinite sequence of real resonance values. The fact that the discrete system is uniquely solvable even when the exact system is not, suggests the presence of artificial dissipation in HDG methods. We will investigate this issue more thoroughly in the next section.
Results of dispersion analysis for real wavenumbers
When the wavenumber k is complex, we have seen that it is important to choose the stabilization parameter \(\tau\) such that (18b) holds. We have also seen that when k is real, the stability obtained by (18a) is so tenuous that it is of negligible practical value. For real wavenumbers, it is safer to rely on stability of the (undiscretized) boundary value problem, rather than the stability obtained by a choice of \(\tau\).
The focus of this section is on real k and the Helmholtz equation (1a, 1b, 1c). In this case, having already separated the issue of stability from the choice of \(\tau\), we are now free to optimize the choice of \(\tau\) for other goals. By means of a dispersion analysis, we now proceed to show that some values of \(\tau\) are better than others for minimizing discrepancies in wavespeed. Since dispersion analyses are limited to the study of propagation of plane waves (that solve the Helmholtz equation), we will not explicitly consider the Maxwell HDG system in this section. However, since we have written the Helmholtz and Maxwell system consistently with respect to the stabilization parameter [see the transition from (3) to (9) via (7)], we anticipate our results for the 2D Helmholtz case to be useful for the Maxwell case also.
The dispersion relation in the onedimensional case
Lowest order twodimensional case
Numerically found values of \(\tau\) that minimize \( kh  k^h(\theta ) h\) for all \(\theta\) in the \(p=0\) case
kh  Optimal \(\tau\),  Optimal \(\tau\), 

\({\mathrm {Im}}(\tau )>0\)  \({\mathrm {Im}}(\tau )<0\)  
\(\pi /4\)  \(0.807\hat{\imath }\)  \(0.931\hat{\imath }\) 
\(\pi /8\)  \(0.837\hat{\imath }\)  \(0.898\hat{\imath }\) 
\(\pi /16\)  \(0.851\hat{\imath }\)  \(0.882\hat{\imath }\) 
\(\pi /32\)  \(0.859\hat{\imath }\)  \(0.874\hat{\imath }\) 
\(\pi /64\)  \(0.863\hat{\imath }\)  \(0.871\hat{\imath }\) 
\(\pi /128\)  \(0.865\hat{\imath }\)  \(0.868\hat{\imath }\) 
\(\pi /256\)  \(0.866\hat{\imath }\)  \(0.867\hat{\imath }\) 
Higher order case
Comparison with dispersion relation for the Hybrid Raviart–Thomas method
The HRT (Hybrid Raviart–Thomas) method is a classical mixed method [14–16] which has a similar stencil pattern, but uses different spaces. Namely, the HRT method for the Helmholtz equation is defined by exactly the same equations as (2a, 2b, 2c) with \(\tau\) set to zero, but with these choices of spaces on square elements: \(V(K) = \mathcal {Q}_{p+1,p}(K) \times \mathcal {Q}_{p,p+1}(K),\) \(W(K) = \mathcal {Q}_p(K)\), and \(M(F) = \mathcal {P}_p(F)\). Here \(Q_{l,m}(K)\) denotes the space of polynomials which are of degree at most l in the first coordinate and of degree at most m in the second coordinate. The general method of dispersion analysis described in the previous subsection can be applied for the HRT method. We proceed to describe our new findings, which in the lowest order case includes an exact dispersion relation for the HRT method.
Conclusions
 1.
There are values of stabilization parameters \(\tau\) that will cause the HDG method to fail in timeharmonic electromagnetic and acoustic simulations using complex wavenumbers. [See Eq. (5) et seq.]
 2.
If the wavenumber k is complex, then choosing \(\tau\) so that \({\text {Re}}(\tau ) {\text {Im}}(k) \le 0\) guarantees that the HDG method is uniquely solvable. (See Theorem 1.)
 3.
If the wavenumber k is real, then even when the exact wave problem is not wellposed (such as at a resonance), the HDG method remains uniquely solvable when \({\text {Re}}(\tau ) \ne 0\). However, in such cases, we found the discrete stability to be tenuous. (See Figure 2 and accompanying discussion.)
 4.
For real wavenumbers k, we found that the HDG method introduces small amounts of artificial dissipation [see Eq. (21)] in general. The artificial dissipation is eliminated [see Eq. (32)] when \({\text {Re}}(\tau) =0\) and kh is sufficiently small, but note that in this case, Theorem 1 no longer guarantees unique solvability. In 1D, the optimal values of \(\tau\) that asymptotically minimize the total error in the wavenumber (that quantifies dissipative and dispersive errors together) are \(\tau = \pm \hat{\imath }\) [see Eq. (22)].
 5.
In 2D, for real wavenumbers k, the best values of \(\tau\) are dependent on the propagation angle. Overall, values of \(\tau\) that asymptotically minimize the error in the discrete wavenumber (considering all angles) is \(\tau = \pm \hat{\imath }\sqrt{3}/2\) [per Eq. (31)]. While dispersive errors dominate the total error for \(\tau = \hat{\imath }\sqrt{3}/2\), dissipative errors dominate when \(\tau =1\) (see Figure 7).
 6.
The HRT method, in both the numerical results and the theoretical asymptotic expansions, gave a total error in the discrete wavenumber that is asymptotically one order smaller than the HDG method. [See (38) and Figure 7.]
Abbreviations
 HDG:

hybrid (or hybridized) discontinuous Galerkin
 1D:

one dimension(al)
 2D:

two dimension(al)
 3D:

three dimension(al)
 HRT:

hybrid (or hybridized) Raviart–Thomas
Declarations
Authors’ contributions
The contributions of all authors are equal. All authors read and approved the final manuscript.
Acknowledgements
JG and NO were supported in part by the NSF Grant DMS1318916 and the AFOSR Grant FA95501210484. NO gratefully acknowledges support in the form of an INRIA internship where discussions leading to this work originated. All authors wish to thank INRIA Sophia Antipolis Méditerranée for hosting the authors there and facilitating this research.
Compliance with ethical guidelines
Competing interests The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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