# A Quasi-Trefftz Domain Decomposition Method for the Iterative Solution of Time-Harmonic Wave Problems

Sébastien Pernet, Matthias Rivet, Sébastien Tordeux

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## To cite this version:

Sébastien Pernet, Matthias Rivet, Sébastien Tordeux. A Quasi-Trefftz Domain Decomposition Method for the Iterative Solution of Time-Harmonic Wave Problems. DD 2025 - 29th International Conference on Domain Decomposition Methods, Jun 2025, Milan, Italy. hal-05317539

HAL Id: hal-05317539

https://inria.hal.science/hal-05317539v1

Submitted on 16 Oct 2025

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Distributed under a Creative Commons Attribution 4.0 International License
A Quasi-Trefftz Domain Decomposition Method for the
Iterative Solution of Time-Harmonic Wave Problems

Sébastien Pernet¹, Matthias Rivet¹,², Sébastien Tordeux²

¹DTIS, ONERA, Université de Toulouse, 31000, Toulouse, France

²EPI Makutu, INRIA, Université de Pau et des Pays de l'Adour, TotalEnergies, CNRS UMR 5142

29th International Conference on Domain Decomposition Methods

24th June 2025
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Context of the work

* Increasing need of 3D simulations of time-harmonic electromagnetic waves: **high-frequency, heterogeneous environments, wide domains** (in terms of wavelengths)...

<figure><img src="image_6.png" /><figcaption>Figure: FDTD simulation on a Manhattan mesh of 700 × 600 × 300m, Thibault Volpert (DEMR ONERA)</figcaption></figure>

Example of computation challenge: field radiated by two antennas in Manhattan

* Large simulation domain.

* Wide range of frequencies (1-18 GHz).

⇒ Need of **efficient methods** to solve such ambitious cases!
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Context of the work

* Increasing need of 3D simulations of time-harmonic electromagnetic waves: **high-frequency, heterogeneous environments, wide domains** (in terms of wavelengths)...

<figure><img src="image_10.png" /><figcaption>Figure: FDTD simulation on a Manhattan mesh of 700 × 600 × 300m, Thibault Volpert (DEMR ONERA)</figcaption></figure>

Orders of magnitude:

* **5G N28 in Manhattan**: 1800 λ.

* **5G N78 at 50m**: 600 λ.

* **Fighter size for a military radar**: 500 λ.

⇒ Need of **efficient methods** to solve such ambitious cases!
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Context of the work

* Increasing need of 3D simulations of time-harmonic electromagnetic waves: **high-frequency, heterogeneous environments, wide domains** (in terms of wavelengths)...

<figure><img src="image_14.png" /><figcaption>Figure: FDTD simulation on a Manhattan mesh of 700 × 600 × 300m, Thibault Volpert (DEMR ONERA)</figcaption></figure>

Limits of classic methods:

* Wide domains lead to **very large linear systems**.

* Traditional direct solvers induce a **prohibitive memory consumption**.

* Usual numerical methods are **not well adapted to iterative resolution**.

⇒ Need of **efficient methods** to solve such ambitious cases!
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# Plan of the talk

1. 3D Maxwell: limitations and (quasi-)Trefftz method

2. Quasi-Trefftz numerical analysis

3. Information transfer optimisation
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The 3D Maxwell problem in wide domains:
current limitations and (quasi-)Trefftz method
interests
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Current issues: memory and iterative resolution

Time-harmonic Maxwell equations for the electromagnetic field
$\mathbf{Y} := (\mathbf{E}, \mathbf{H}) \in [\text{H}(\text{curl}, \Omega)]^6$:

$$i\kappa\mathbf{M}\mathbf{Y} + \sum_{j=1}^{3} \frac{\partial \mathbf{F}^j \mathbf{Y}}{\partial x_j} = 0 \text{ in } \Omega, \quad (1)$$

with impedance Boundary Conditions (BCs)

$$(\mathbf{n}_{\partial\Omega} \times \mathbf{E}) \times \mathbf{n}_{\partial\Omega} + Z_{\partial\Omega} \mathbf{n}_{\partial\Omega} \times \mathbf{H} = \mathbf{g} \text{ on } \partial\Omega. \quad (2)$$
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Considered numerical method: the Trefftz approach

1. Belongs to the Discontinuous Galerkin methods, with the Galerkin space $X := \prod_{T \in \mathcal{T}_h} X_T$ made up of local solutions (i.e. in each mesh cell $T \in \mathcal{T}_h$) of the Maxwell equations.
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Considered numerical method: the Trefftz approach

1. Belongs to the Discontinuous Galerkin methods, with the Galerkin space $X := \prod_{T \in \mathcal{T}_h} X_T$ made up of local solutions (i.e. in each mesh cell $T \in \mathcal{T}_h$) of the Maxwell equations.

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Considered numerical method: the Trefftz approach

1. Belongs to the Discontinuous Galerkin methods, with the Galerkin space $X := \prod_{T \in \mathcal{T}_h} X_T$ made up of local solutions (i.e. in each mesh cell $T \in \mathcal{T}_h$) of the Maxwell equations.

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<figure><img src="image_38.png" /></figure>

Considered numerical method: the Trefftz approach

1. Belongs to the Discontinuous Galerkin methods, with the Galerkin space $X := \prod_{T \in \mathcal{T}_h} X_T$ made up of local solutions (i.e. in each mesh cell $T \in \mathcal{T}_h$) of the Maxwell equations.

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## Considered numerical method: the Trefftz approach

1. Belongs to the Discontinuous Galerkin methods, with the Galerkin space $X := \prod_{T \in \mathcal{T}_h} X_T$ made up of local solutions (i.e. in each mesh cell $T \in \mathcal{T}_h$) of the Maxwell equations.

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## Considered numerical method: the Treffzt approach

1. Belongs to the Discontinuous Galerkin methods, with the Galerkin space $X := \prod_{T \in \mathcal{T}_h} X_T$ made up of local solutions (i.e. in each mesh cell $T \in \mathcal{T}_h$) of the Maxwell equations.

2. Use of the reciprocity formula verified in each cell

$$ \forall T \in \mathcal{T}_h, \quad \int_{\partial T} \gamma_{\times}^{T} \mathbf{H}^{T} \cdot \gamma_{t} \overline{\mathbf{E}^{T}} + \gamma_{t} \mathbf{E}^{T} \cdot \overline{\gamma_{\times}^{T} \mathbf{H}^{T}} = 0, \quad (3) $$

in addition to the introduction of numerical traces [Sirley 2022]:

Find $\mathbf{Y} = (\mathbf{E}, \mathbf{H}) \in X$ such that $\forall \mathbf{Y}' = (\mathbf{E}', \mathbf{H}') \in X$,

$$ \sum_{T \in \mathcal{T}_h} \int_{\partial T} \gamma_{\times}^{T} \overline{\mathbf{H}^{T}} \cdot \overline{\gamma_{t} \mathbf{E}^{T}} + \overline{\gamma_{t} \mathbf{E}^{T}} \cdot \overline{\gamma_{\times}^{T} \mathbf{H}^{T}} = \ell(\mathbb{E}'), $$

where $\hat{\times}$ stands for upwind (or Riemann) numerical traces and with

$$ \gamma_t \mathbf{u}^T = \mathbf{u}^T - (\mathbf{u}^T \cdot \mathbf{n}_T) \mathbf{n}_T \quad \text{and} \quad \gamma_{\times}^T \mathbf{u}^T = \mathbf{n}_T \times \mathbf{u}^T. $$
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## The Trefftz approach: properties and classic limits

1. The formulation is posed on the mesh skeleton and adaptable to an iterative resolution (**contraction property**) [Cessenat, Després 1998].

2. A relevant choice of the numerical traces naturally implies formulation **coercivity**.

3. Basis functions have a **physical meaning**: leads to a reduced numerical pollution [Ihlenburg, Babuška 1995].

→ **Matrix-free approach**: use only a few interaction blocks without matrix assembly!

Limits of the classic **Plane Waves (PWs)** choice:

* **Numerical dependence phenomena:** ill-conditioned basis [Congreve et al. 2019].

* **No adaptability of the basis** to the local properties: expected singularities, complex interference phenomena...

<figure><img src="image_50.png" /><figcaption>Figure: Eigenvalues of the mass matrix for 196 PWs.</figcaption></figure>

⇒ **Possible algebraic adaptations** [Barucq et al. 2021].

⇒ **Introduction of well-conditioned basis** by parameterising the space of local solutions by **polynomial boundary conditions**.
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# A Quasi-Trefftz approach

## Definition of the local problem

A local solution $\mathbf{Y}^T$ in $T$ is parameterised by a tangential field $g^T \in \mathcal{V}^T$ such that

$$ \gamma_t \mathbf{E}^T + Z_{\partial T} \gamma_{\times}^T \mathbf{H}^T = \mathbf{g}^T \quad \text{on } \partial T. \quad (4) $$

⇒ Consider a finite-dimensional subspace $\mathcal{V}_h^T \subset \mathcal{V}^T$ and the associated Maxwell solutions as local basis [Fure et al. 2020].

### Construction of $\mathcal{V}_h$:

* Consider a cell $\mathcal{V}_h^T$

* Define a mesh of $T_h^T$

* Consider piecewise polynomial $g^T$ of degree k on $T_h^T$

<figure><img src="image_54.png" /><figcaption>Figure: 2D example of the mesh of $\partial T$ for $k_{QT} = 1$.</figcaption></figure>
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<figure><img src="image_57.png" /></figure>

# A Quasi-Trefftz approach

## Definition of the local problem

A local solution $\mathbf{Y}^T$ in $T$ is parameterised by a tangential field $g^T \in \mathcal{V}^T$ such that

$$ \gamma_t \mathbf{E}^T + Z_{\partial T} \gamma_t^T \mathbf{H}^T = \mathbf{g}^T \quad \text{on } \partial T. \qquad (4) $$

⇒ Consider a finite-dimensional subspace $\mathcal{V}_h^T \subset \mathcal{V}^T$ and the associated Maxwell solutions as local basis [Fure et al. 2020].

## Construction of $\mathcal{V}_h$:

* Consider a cell $T \in \mathcal{T}_h^T$.

* Define a mesh of $\mathcal{T}(\partial T)$.

* Consider a piecewise polynomial $g^T$ of degree $k_{QT}$ on $\mathcal{T}(\partial T)$.

<figure><img src="image_58.png" /><figcaption>Figure: 2D example of the mesh of $\partial T$ for $k_{QT} = 1$.</figcaption></figure>
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<figure><img src="image_61.png" /></figure>

# A Quasi-Trefftz approach

## Definition of the local problem

A local solution $\mathbf{Y}^T$ in $T$ is parameterised by a tangential field $g^T \in \mathcal{V}^T$ such that

$$ \gamma_t \mathbf{E}^T + Z_{\partial T} \gamma_t^T \mathbf{H}^T = \mathbf{g}^T \quad \text{on } \partial T. \quad (4) $$

⇒ Consider a finite-dimensional subspace $\mathcal{V}_h^T \subset \mathcal{V}^T$ and the associated Maxwell solutions as local basis [Fure et al. 2020].

Construction of $\mathcal{V}_h$:

* Consider a cell $T \in \mathcal{T}_h^T$.

* Define a mesh of $\mathcal{T}(\partial T)$.

* Consider a piecewise polynomial $g^T$ of degree $k_{QT}$ on $\mathcal{T}(\partial T)$.

<figure><img src="image_62.png" /><figcaption>Figure: 2D example of the mesh of $\partial T$ for $k_{QT} = 1$.</figcaption></figure>
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# A Quasi-Trefftz approach

## Definition of the local problem

A local solution $\mathbf{Y}^T$ in $T$ is parameterised by a tangential field $g^T \in \mathcal{V}^T$ such that

$$ \gamma_t \mathbf{E}^T + Z_{\partial T} \gamma_t^T \mathbf{H}^T = \mathbf{g}^T \quad \text{on } \partial T. \quad (4) $$

⇒ Consider a finite-dimensional subspace $\mathcal{V}_h^T \subset \mathcal{V}^T$ and the associated Maxwell solutions as local basis [Fure et al. 2020].

Construction of $\mathcal{V}_h$:

* Consider a cell $T \in \mathcal{T}_h^T$.

* Define a mesh of $\mathcal{T}(\partial T)$.

* Consider a piecewise polynomial $g^T$ of degree $k_{QT}$ on $\mathcal{T}(\partial T)$.

<figure><img src="image_66.png" /><figcaption>Figure: 2D example of the mesh of $\partial T$ for $k_{QT} = 1$.</figcaption></figure>
<figure><img src="image_67.png" /></figure>

<figure><img src="image_68.png" /></figure>

<figure><img src="image_69.png" /></figure>

# A Quasi-Trefftz approach

## Definition of the local problem

A local solution $\mathbf{Y}^T$ in $T$ is parameterised by a tangential field $g^T \in \mathcal{V}^T$ such that

$$ \gamma_t \mathbf{E}^T + Z_{\partial T} \gamma_{\times}^T \mathbf{H}^T = \mathbf{g}^T \quad \text{on } \partial T. \quad (4) $$

⇒ Consider a finite-dimensional subspace $\mathcal{V}_h^T \subset \mathcal{V}^T$ and the associated Maxwell solutions as local basis [Fure et al. 2020].

Construction of $\mathcal{V}_h$:

* Consider a cell $T \in \mathcal{T}_h^T$.

* Define a mesh of $\mathcal{T}(\partial T)$.

* Consider a piecewise polynomial $g^T$ of degree $k_{QT}$ on $\mathcal{T}(\partial T)$.

<figure><img src="image_70.png" /><figcaption>Figure: 2D example of the mesh of $\partial T$ for $k_{QT} = 1$.</figcaption></figure>
<figure><img src="image_71.png" /></figure>

<figure><img src="image_72.png" /></figure>

<figure><img src="image_73.png" /></figure>

# A Quasi-Trefftz approach

## Definition of the local problem

A local solution $\mathbf{Y}^T$ in $T$ is parameterised by a tangential field $g^T \in \mathcal{V}^T$ such that

$$ \gamma_t \mathbf{E}^T + Z_{\partial T} \gamma_t^T \mathbf{H}^T = \mathbf{g}^T \quad \text{on } \partial T. \quad (4) $$

⇒ Consider a finite-dimensional subspace $\mathcal{V}_h^T \subset \mathcal{V}^T$ and the associated Maxwell solutions as local basis [Fure et al. 2020].

Construction of $\mathcal{V}_h$:

* Consider a cell $T \in \mathcal{T}_h^T$.

* Define a mesh of $\mathcal{T}(\partial T)$.

* Consider a piecewise polynomial $g^T$ of degree $k_{QT}$ on $\mathcal{T}(\partial T)$.

<figure><img src="image_74.png" /><figcaption>Figure: 2D example of the mesh of $\partial T$ for $k_{QT} = 1$.</figcaption></figure>
<figure><img src="image_75.png" /></figure>

<figure><img src="image_76.png" /></figure>

<figure><img src="image_77.png" /></figure>

# A Quasi-Trefftz approach

## Definition of the local problem

A local solution $\mathbf{Y}^T$ in $T$ is parameterised by a tangential field $g^T \in \mathcal{V}^T$ such that

$$ \gamma_t \mathbf{E}^T + Z_{\partial T} \gamma_t^T \mathbf{H}^T = \mathbf{g}^T \quad \text{on } \partial T. \quad (4) $$

⇒ Consider a finite-dimensional subspace $\mathcal{V}_h^T \subset \mathcal{V}^T$ and the associated Maxwell solutions as local basis [Fure et al. 2020].

Construction of $\mathcal{V}_h$:

* Consider a cell $T \in \mathcal{T}_h^T$.

* Define a mesh of $\mathcal{T}(\partial T)$.

* Consider a piecewise polynomial $g^T$ of degree $k_{QT}$ on $\mathcal{T}(\partial T)$.

<figure><img src="image_78.png" /><figcaption>Figure: 2D example of the mesh of $\partial T$ for $k_{QT} = 1$.</figcaption></figure>
<figure><img src="image_79.png" /></figure>

<figure><img src="image_80.png" /></figure>

<figure><img src="image_81.png" /></figure>

# A Quasi-Trefftz approach

## Definition of the local problem

A local solution $\mathbf{Y}^T$ in $T$ is parameterised by a tangential field $g^T \in \mathcal{V}^T$ such that

$$ \gamma_t \mathbf{E}^T + Z_{\partial T} \gamma_{\times}^T \mathbf{H}^T = \mathbf{g}^T \quad \text{on } \partial T. \quad (4) $$

⇒ Consider a finite-dimensional subspace $\mathcal{V}_h^T \subset \mathcal{V}^T$ and the associated Maxwell solutions as local basis [Fure et al. 2020].

Construction of $\mathcal{V}_h$:

* Consider a cell $T \in \mathcal{T}_h^T$.

* Define a mesh of $\mathcal{T}(\partial T)$.

* Consider a piecewise polynomial $g^T$ of degree $k_{QT}$ on $\mathcal{T}(\partial T)$.

<figure><img src="image_82.png" /><figcaption>Figure: 2D example of the mesh of $\partial T$ for $k_{QT} = 1$.</figcaption></figure>

⇒ Associated Maxwell solutions are unknown: need of an auxiliary solver to compute approximations!

* Any solver considering the first-order problem may be chosen (Flux Reconstruction [Huynh 2007] in this study)!

* Only need to compute for different cell configurations!
<figure><img src="image_83.png" /></figure>

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<figure><img src="image_85.png" /></figure>

# A Quasi-Trefftz approach

## Definition of the local problem

A local solution $\mathbf{Y}^T$ in $T$ is parameterised by a tangential field $g^T \in \mathcal{V}^T$ such that

$$ \gamma_t \mathbf{E}^T + Z_{\partial T} \gamma_{\times}^T \mathbf{H}^T = \mathbf{g}^T \quad \text{on } \partial T. \quad (4) $$

⇒ Consider a finite-dimensional subspace $\mathcal{V}_h^T \subset \mathcal{V}^T$ and the associated Maxwell solutions as local basis [Fure et al. 2020].

Construction of $\mathcal{V}_h$:

* Consider a cell $T \in \mathcal{T}_h^T$.

* Define a mesh of $\mathcal{T}(\partial T)$.

* Consider a piecewise polynomial $g^T$ of degree $k_{QT}$ on $\mathcal{T}(\partial T)$.

<figure><img src="image_86.png" /><figcaption>Figure: 2D example of the mesh of $\partial T$ for $k_{QT} = 1$.</figcaption></figure>

⇒ Associated Maxwell solutions are unknown: need of an auxiliary solver to compute approximations!

⇒ Other Quasi-Trefftz approaches: generalised PWs [Imbert-Gérard, Després 2013], polynomial Quasi-Trefftz functions [Imbert-Gérard, Sylvand 2023], Embedded Trefftz DG [Lehrenfeld, Stocker 2023], BEM local solver [Barucq et al. 2017]...
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# A quasi-Trefftz approach based on an auxiliary solver: numerical analysis
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# Summary of this Quasi-Trefftz approach

Variational formulation posed on $X := \prod_{T \in \mathcal{T}_h} X_T$ made up of local solutions.

<figure><img src="image_93.png" /></figure>

Functions $(E^T, H^T)$ of $X_T$ parameterised by $g^T \in \mathcal{V}^T$ such that:

$$ \gamma_t E^T + Z_{\partial T} \gamma_x^T H^T = g^T \quad \text{on } \partial T. \qquad (5) $$

<figure><img src="image_94.png" /></figure>

Polynomial approximation of the trace space: $\sqrt{h}^T = \text{span}(g_i^T)$. For all $i$, the associated solutions $(E_i^T, H_i^T)$ are taken as basis functions of $X_T$.

<figure><img src="image_95.png" /></figure>

Auxiliary solver to compute an approximation of $(E_i^T, H_i^T)$.
<figure><img src="image_96.png" /></figure>

<figure><img src="image_97.png" /></figure>

<figure><img src="image_98.png" /></figure>

# Summary of this Quasi-Trefftz approach

Variational formulation posed on $X := \prod_{T \in \mathcal{T}_h} X_T$ made up of local solutions.

<figure><img src="image_99.png" /></figure>

Functions $(E^T, H^T)$ of $X_T$ parameterised by $g^T \in V^T$ such that:

$$ \gamma_t E^T + Z_{\partial T} \gamma_x^T H^T = g^T \quad \text{on } \partial T. \qquad (5) $$

<figure><img src="image_100.png" /></figure>

Polynomial approximation of the trace space: $\sqrt{h}^T = \text{span}(g_i^T)$. For all $i$, the associated solutions $(E_i^T, H_i^T)$ are taken as basis functions of $X_T$.

<figure><img src="image_101.png" /></figure>

Auxiliary solver to compute an approximation of $(E_i^T, H_i^T)$.
<figure><img src="image_102.png" /></figure>

<figure><img src="image_103.png" /></figure>

<figure><img src="image_104.png" /></figure>

# Summary of this Quasi-Trefftz approach

Variational formulation posed on $X := \prod_{T \in \mathcal{T}_h} X_T$ made up of local solutions.

<figure><img src="image_105.png" /></figure>

Functions $(E^T, H^T)$ of $X_T$ parameterised by $g^T \in V^T$ such that:

$$ \gamma_t E^T + Z_{\partial T} \gamma_{\times}^T H^T = g^T \quad \text{on } \partial T. \qquad (5) $$

<figure><img src="image_106.png" /></figure>

Polynomial approximation of the trace space: $V_h^T = \text{span}(g_i^T)$. For all $i$, the associated solutions $(E_i^T, H_i^T)$ are taken as basis functions of $X_T$.

<figure><img src="image_107.png" /></figure>

Auxiliary solver to compute an approximation of $(E_i^T, H_i^T)$.
<figure><img src="image_108.png" /></figure>

<figure><img src="image_109.png" /></figure>

<figure><img src="image_110.png" /></figure>

# Summary of this Quasi-Trefftz approach

Variational formulation posed on $X := \prod_{T \in \mathcal{T}_h} X_T$ made up of local solutions.

<figure><img src="image_111.png" /></figure>

Functions $(E^T, H^T)$ of $X_T$ parameterised by $g^T \in V^T$ such that:

$$ \gamma_t E^T + Z_{\partial T} \gamma_{\times}^T H^T = g^T \quad \text{on } \partial T. \qquad (5) $$

<figure><img src="image_112.png" /></figure>

Polynomial approximation of the trace space: $V_h^T = \text{span}(g_i^T)$. For all $i$, the associated solutions $(E_i^T, H_i^T)$ are taken as basis functions of $X_T$.

<figure><img src="image_113.png" /></figure>

Auxiliary solver to compute an approximation of $(E_i^T, H_i^T)$.
<figure><img src="image_114.png" /></figure>

<figure><img src="image_115.png" /></figure>

<figure><img src="image_116.png" /></figure>

## Numerical experiments: basis quality

<figure><img src="image_117.png" /><figcaption>Figure: Eigenvalues of the mass matrix for 196 PWs (and reduction for $\epsilon = 10^{-6}$) and 192 Quasi-Trefftz basis functions.</figcaption></figure>

### Basis quality

- Possible reduction techniques to restore the condition number.

- Avoids spurious numerical modes and ill-conditioning of the mass matrix!

- Controlled conditioning, even for increasing number of basis functions.
<figure><img src="image_118.png" /></figure>

<figure><img src="image_119.png" /></figure>

<figure><img src="image_120.png" /></figure>

## Numerical experiments: basis quality

<figure><img src="image_121.png" /><figcaption>Figure: Eigenvalues of the mass matrix for 196 PWs (and reduction for $\epsilon = 10^{-6}$) and 192 Quasi-Trefftz basis functions.</figcaption></figure>

### Basis quality

- Possible reduction techniques to restore the condition number.

- Avoids spurious numerical modes and ill-conditioning of the mass matrix!

- Controlled conditioning, even for increasing number of basis functions.
<figure><img src="image_122.png" /></figure>

<figure><img src="image_123.png" /></figure>

<figure><img src="image_124.png" /></figure>

## Numerical experiments: basis quality

<figure><img src="image_125.png" /><figcaption>Figure: Eigenvalues of the mass matrix for 196 PWs (and reduction for $\epsilon = 10^{-6}$) and 192 Quasi-Trefftz basis functions.</figcaption></figure>

### Basis quality

⇒ Possible **reduction techniques** to restore the condition number.

⇒ Avoids spurious numerical modes and ill-conditioning of the mass matrix!

⇒ Controlled conditioning, even for increasing number of basis functions.
<figure><img src="image_126.png" /></figure>

<figure><img src="image_127.png" /></figure>

<figure><img src="image_128.png" /></figure>

## Numerical experiments: basis quality

<figure><img src="image_129.png" /><figcaption>Figure: Eigenvalues of the mass matrix for 196 PWs (and reduction for $\epsilon = 10^{-6}$) and 192 Quasi-Trefftz basis functions.</figcaption></figure>

### Basis quality

- Possible **reduction techniques** to restore the condition number.

- Avoids spurious numerical modes and ill-conditioning of the mass matrix!

- Controlled **conditioning**, even for increasing number of basis functions.
<figure><img src="image_130.png" /></figure>

<figure><img src="image_131.png" /></figure>

<figure><img src="image_132.png" /></figure>

## Illustration on a 2D diffraction configuration

<figure><img src="image_133.png" /><figcaption>Figure: Error saturation for a Plane Waves basis.</figcaption></figure>

⇒ **Saturation phenomenon** due to condition number: **difficult representation of**
**complex local solutions with PWs.**

⇒ **Reduction techniques** avoid conditioning issues, but **locked convergence** because
of rounding pollution error for an **asked threshold**.

⇒ **Robust approximation by FR basis.**
<figure><img src="image_134.png" /></figure>

<figure><img src="image_135.png" /></figure>

<figure><img src="image_136.png" /></figure>

## Illustration on a 2D diffraction configuration

<figure><img src="image_137.png" /><figcaption>Figure: Error saturation for a Plane Waves basis and robust convergence for quasi-Trefftz.</figcaption></figure>

⇒ **Saturation phenomenon** due to condition number: **difficult representation of**
**complex local solutions with PWs.**

⇒ **Reduction techniques** avoid conditioning issues, but **locked convergence** because
of rounding pollution error for an **asked threshold**.

⇒ **Robust approximation** by FR basis.
<figure><img src="image_138.png" /></figure>

<figure><img src="image_139.png" /></figure>

<figure><img src="image_140.png" /></figure>

# Calibration of the FR auxiliary solver

Trefftz approach rests on the use of **exact solutions** of the Maxwell equations:

## Is using quasi-solutions enough?
How to adapt the FR order to the Trefftz one?

* Uniform Cartesian mesh $\mathcal{T}_H(\Omega)$ of the domain $\Omega$: $N$ macro-cells per direction.

* Uniform Cartesian mesh $\mathcal{T}_h(\partial T)$ of $\partial T$: $M$ micro-faces per direction.

* Piecewise polynomial BCs of degree $k_{QT} = 1$.

<figure><img src="image_141.png" /><figcaption>Figure: Mesh $\mathcal{T}_H(\Omega)$ for $N = 3$.</figcaption></figure>

<figure><img src="image_142.png" /></figure>

⇒ No theoretical *a priori* error estimates in $L^2(\Omega)$-norm: numerical convergence in mesh and number of basis functions [Fure et al. 2020].
<figure><img src="image_143.png" /></figure>

<figure><img src="image_144.png" /></figure>

<figure><img src="image_145.png" /></figure>

# Calibration of the FR auxiliary solver

Trefftz approach rests on the use of *exact solutions* of the Maxwell equations:

## Is using quasi-solutions enough?
How to adapt the FR order to the Trefftz one?

* Uniform Cartesian mesh $\mathcal{T}_H(\Omega)$ of the domain $\Omega$: $N$ macro-cells per direction.

* Uniform Cartesian mesh $\mathcal{T}_h(\partial T)$ of $\partial T$: $M$ micro-faces per direction.

* Piecewise polynomial BCs of degree $k_{QT} = 1$.

<figure><img src="image_146.png" /><figcaption>Figure: Mesh $\mathcal{T}_H(\Omega)$ for $N = 3$.</figcaption></figure>

<figure><img src="image_147.png" /><figcaption>Figure: Mesh $\mathcal{T}_h(\partial T)$ for $M = 2$ and $k_{QT} = 1$.</figcaption></figure>

⇒ No theoretical *a priori* error estimates in $L^2(\Omega)$-norm: numerical convergence in mesh and number of basis functions [Fure et al. 2020].
<figure><img src="image_148.png" /></figure>

<figure><img src="image_149.png" /></figure>

<figure><img src="image_150.png" /></figure>

# Calibration of the FR auxiliary solver

Trefftz approach rests on the use of *exact solutions* of the Maxwell equations:

## Is using quasi-solutions enough?
How to adapt the FR order to the Trefftz one?

* Uniform Cartesian mesh $\mathcal{T}_H(\Omega)$ of the domain $\Omega$: $N$ macro-cells per direction.

* Uniform Cartesian mesh $\mathcal{T}_h(\partial T)$ of $\partial T$: $M$ micro-faces per direction.

* Piecewise polynomial BCs of degree $k_{QT} = 1$.

<figure><img src="image_151.png" /><figcaption>Figure: Mesh $\mathcal{T}_H(\Omega)$ for $N = 3$.</figcaption></figure>

<figure><img src="image_152.png" /><figcaption>Figure: Mesh $\mathcal{T}_h(\partial T)$ for $M = 2$ and $k_{QT} = 1$.</figcaption></figure>
<figure><img src="image_153.png" /></figure>

<figure><img src="image_154.png" /></figure>

<figure><img src="image_155.png" /></figure>

Calibration of the FR auxiliary solver

Trefftz approach rests on the use of **exact solutions** of the Maxwell equations:

Is using quasi-solutions enough?
How to adapt the FR order to the Trefftz one?

* Uniform Cartesian mesh $\mathcal{T}_H(\Omega)$ of the domain $\Omega$: $N$ **macro-cells** per direction.

* Uniform Cartesian mesh $\mathcal{T}_h(\partial T)$ of $\partial T$: $M$ **micro-faces** per direction.

* Piecewise polynomial BCs of degree $k_{QT} = 1$.

<figure><img src="image_156.png" /><figcaption>Figure: Mesh $\mathcal{T}_H(\Omega)$ for $N = 3$.</figcaption></figure>

<figure><img src="image_157.png" /><figcaption>Figure: Mesh $\mathcal{T}_h(\partial T)$ for $M = 2$ and $k_{QT} = 1$.</figcaption></figure>
<figure><img src="image_158.png" /></figure>

<figure><img src="image_159.png" /></figure>

<figure><img src="image_160.png" /></figure>

Calibration of the FR auxiliary solver

Trefftz approach rests on the use of **exact solutions** of the Maxwell equations:

Is using quasi-solutions enough?
How to adapt the FR order to the Trefftz one?

* Uniform Cartesian mesh $\mathcal{T}_H(\Omega)$ of the domain $\Omega$: $N$ **macro-cells** per direction.

* Uniform Cartesian mesh $\mathcal{T}_h(\partial T)$ of $\partial T$: $M$ **micro-faces** per direction.

* Piecewise polynomial BCs of degree $k_{QT} = 1$.

<figure><img src="image_161.png" /><figcaption>Figure: Mesh $\mathcal{T}_H(\Omega)$ for $N = 3$.</figcaption></figure>

<figure><img src="image_162.png" /><figcaption>Figure: Mesh $\mathcal{T}_h(\partial T)$ for $M = 2$ and $k_{QT} = 1$.</figcaption></figure>

⇒ No theoretical *a priori* error estimates in $L^2(\Omega)$-norm: **numerical convergence in mesh and number of basis functions** [Fure et al. 2020].
# Calibration of the auxiliary solver (2D Helmholtz)

## Is using quasi-solutions enough?

## How to adapt the auxiliary solver's order to the Trefftz one?

⇒ Mesh-convergence: refinement of the macro-mesh $T_H(\Omega)$ for $M = 2$.

<figure><img src="image_163.png" /><figcaption>Figure: Evolution of the error for 1 nano-cell per micro-face.</figcaption></figure>

## Influence of the local solver precision (through the number of nano-cells):

* Regime 1 ('imperfect' local solver): quasi-optimal for $k_{FR} \ge k_{QT} + 2$ and the higher the order and the more refined the nano-mesh is, the later the transition.

* Regime 2 ('perfect' local solver): common super-convergence for $k_{FR} \ge k_{QT} + 1$.
<figure><img src="image_164.png" /></figure>

<figure><img src="image_165.png" /></figure>

<figure><img src="image_166.png" /></figure>

# Calibration of the auxiliary solver (2D Helmholtz)

## Is using quasi-solutions enough?

## How to adapt the auxiliary solver's order to the Trefftz one?

⇒ Mesh-convergence: refinement of the macro-mesh $\mathcal{T}_H(\Omega)$ for $M = 2$.

<figure><img src="image_167.png" /><figcaption>Figure: Evolution of the error for 2 nano-cells per micro-face.</figcaption></figure>

## Influence of the local solver precision (through the number of nano-cells):

* Regime 1 ('imperfect' local solver): quasi-optimal for $k_{FR} \ge k_{QT} + 2$ and the higher the order and the more refined the nano-mesh is, the later the transition.

* Regime 2 ('perfect' local solver): common super-convergence for $k_{FR} \ge k_{QT} + 1$.
# Calibration of the auxiliary solver (2D Helmholtz)

## Is using quasi-solutions enough?

## How to adapt the auxiliary solver's order to the Trefftz one?

⇒ Mesh-convergence: refinement of the macro-mesh $T_H(\Omega)$ for $M = 2$.

<figure><img src="image_168.png" /><figcaption>Figure: Evolution of the error for 3 nano-cells per micro-face.</figcaption></figure>

## Influence of the local solver precision (through the number of nano-cells):

* Regime 1 ('imperfect' local solver): quasi-optimal for $k_{FR} \ge k_{QT} + 2$ and the higher the order and the more refined the nano-mesh is, the later the transition.

* Regime 2 ('perfect' local solver): common super-convergence for $k_{FR} \ge k_{QT} + 1$.
<figure><img src="image_169.png" /></figure>

<figure><img src="image_170.png" /></figure>

<figure><img src="image_171.png" /></figure>

# Calibration of the auxiliary solver (2D Helmholtz)

## Is using quasi-solutions enough?

## How to adapt the auxiliary solver's order to the Trefftz one?

⇒ Mesh-convergence: refinement of the macro-mesh $\mathcal{T}_H(\Omega)$ for $M = 2$.

<figure><img src="image_172.png" /><figcaption>Figure: Evolution of the error for 4 nano-cells per micro-face.</figcaption></figure>

## Influence of the local solver precision (through the number of nano-cells):

* Regime 1 ('imperfect' local solver): quasi-optimal for $k_{FR} \ge k_{QT} + 2$ and the higher the order and the more refined the nano-mesh is, the later the transition.

* Regime 2 ('perfect' local solver): common super-convergence for $k_{FR} \ge k_{QT} + 1$.
<figure><img src="image_173.png" /></figure>

<figure><img src="image_174.png" /></figure>

<figure><img src="image_175.png" /></figure>

# Calibration of the auxiliary solver (2D Helmholtz)

## Is using quasi-solutions enough?

## How to adapt the auxiliary solver's order to the Trefftz one?

⇒ Mesh-convergence: refinement of the macro-mesh $\mathcal{T}_H(\Omega)$ for $M = 2$.

<figure><img src="image_176.png" /><figcaption>Figure: Evolution of the error for 5 nano-cells per micro-face.</figcaption></figure>

## Influence of the local solver precision (through the number of nano-cells):

* Regime 1 ('imperfect' local solver): quasi-optimal for $k_{FR} \ge k_{QT} + 2$ and the higher the order and the more refined the nano-mesh is, the later the transition.

* Regime 2 ('perfect' local solver): common super-convergence for $k_{FR} \ge k_{QT} + 1$.
<figure><img src="image_177.png" /></figure>

<figure><img src="image_178.png" /></figure>

<figure><img src="image_179.png" /></figure>

# Calibration of the auxiliary solver (2D Helmholtz)

## Is using quasi-solutions enough?

## How to adapt the auxiliary solver's order to the Trefftz one?

⇒ Mesh-convergence: refinement of the macro-mesh $\mathcal{T}_H(\Omega)$ for $M = 2$.

<figure><img src="image_180.png" /><figcaption>Figure: Evolution of the error for 6 nano-cells per micro-face.</figcaption></figure>

## Influence of the local solver precision (through the number of nano-cells):

* Regime 1 ('imperfect' local solver): quasi-optimal for $k_{FR} \ge k_{QT} + 2$ and the higher the order and the more refined the nano-mesh is, the later the transition.

* Regime 2 ('perfect' local solver): common super-convergence for $k_{FR} \ge k_{QT} + 1$.
# Calibration of the auxiliary solver (2D Helmholtz)

## Is using quasi-solutions enough?

## How to adapt the auxiliary solver's order to the Trefftz one?

⇒ Mesh-convergence: refinement of the macro-mesh $T_H(\Omega)$ for $M = 2$.

<figure><img src="image_181.png" /><figcaption>Figure: Evolution of the error for 7 nano-cells per micro-face.</figcaption></figure>

## Influence of the local solver precision (through the number of nano-cells):

* Regime 1 ('imperfect' local solver): quasi-optimal for $k_{FR} \ge k_{QT} + 2$ and the higher the order and the more refined the nano-mesh is, the later the transition.

* Regime 2 ('perfect' local solver): common super-convergence for $k_{FR} \ge k_{QT} + 1$.
<figure><img src="image_182.png" /></figure>

<figure><img src="image_183.png" /></figure>

<figure><img src="image_184.png" /></figure>

# Calibration of the auxiliary solver (2D Helmholtz)

## Is using quasi-solutions enough?

## How to adapt the auxiliary solver's order to the Trefftz one?

⇒ Mesh-convergence: refinement of the macro-mesh $\mathcal{T}_H(\Omega)$ for $M = 2$.

<figure><img src="image_185.png" /><figcaption>Figure: Evolution of the error for 7 nano-cells per micro-face.</figcaption></figure>

## Influence of the local solver precision (through the number of nano-cells):

* Regime 1 ('imperfect' local solver): quasi-optimal for $k_{FR} \ge k_{QT} + 2$ and the higher the order and the more refined the nano-mesh is, the later the transition.

* Regime 2 ('perfect' local solver): common super-convergence for $k_{FR} \ge k_{QT} + 1$.
<figure><img src="image_186.png" /></figure>

<figure><img src="image_187.png" /></figure>

<figure><img src="image_188.png" /></figure>

# Calibration of the auxiliary solver (3D Maxwell)

## Is using quasi-solutions enough?

## How to adapt the auxiliary solver's order to the Trefftz one?

⇒ Mesh-convergence: refinement of the macro-mesh $T_H(\Omega)$ for $M = 2$.

<figure><img src="image_189.png" /></figure>

* Need of $k_{FR} \ge k_{QT} + 2$ for quasi-optimal orders, but **no interest in over-resolving**!
<figure><img src="image_190.png" /></figure>

<figure><img src="image_191.png" /></figure>

<figure><img src="image_192.png" /></figure>

# Calibration of the auxiliary solver (3D Maxwell)

## Is using quasi-solutions enough?

## How to adapt the auxiliary solver's order to the Trefftz one?

⇒ Local basis enrichment through refinement of the micro-mesh $T_h(\partial T)$ for $N = 2$.

<figure><img src="image_193.png" /></figure>

* Need of $k_{FR} \ge k_{QT}$ to be sufficiently 'quasi-solution'.

* Improvement for $k_{FR} \ge k_{QT} + 1$, but limited interest in over-resolving!
<figure><img src="image_194.png" /></figure>

<figure><img src="image_195.png" /></figure>

<figure><img src="image_196.png" /></figure>

# Conclusions on the quasi-Trefftz approach

* Classic numerical schemes are **not well adapted to wide domain simulations**, contrary to the *Trefftz method*.

* The classic choice of *Plane Waves* leads to limitations: we introduce a *Quasi-Trefftz approach*, in which a FR solver computes approximate Maxwell solutions associated to polynomial BCs in each cell.

* Good numerical *independence properties* of the basis functions.

* Numerical calibration of the local solver: $k_{FR} \ge k_{QT} + 2$ for asymptotic quasi-optimality, but **no need to over-resolve!**

* **2 convergence regimes** according to the local solver resolution: **as refined as possible!**
<figure><img src="image_197.png" /></figure>

<figure><img src="image_198.png" /></figure>

<figure><img src="image_199.png" /></figure>

# Improvement of the information transfer: DtN approximation
<figure><img src="image_200.png" /></figure>

<figure><img src="image_201.png" /></figure>

<figure><img src="image_202.png" /></figure>

## Problematic: reduce the number of iterations to convergence

Iterative solution of the linear system thanks to a Krylov method, as GMRES: improving the information transfer would allow to reduce the number of iterations to convergence.

<figure><img src="image_203.png" /><figcaption>(i) Solution evolution</figcaption></figure>

<figure><img src="image_204.png" /><figcaption>(ii) Solution modification</figcaption></figure>

Figure: Iterative convergence: information propagation through iterations.
<figure><img src="image_205.png" /></figure>

<figure><img src="image_206.png" /></figure>

<figure><img src="image_207.png" /></figure>

# Trefftz formulation generalisation

## Principle of the proposed generalisation for the 2D Helmholtz equation:

* Introduction of the **outgoing trace** from a cell $T$ ($\mathcal{O}^T = i\kappa Id$ classically):

$$ \gamma_{out}^{T} \mathbf{y}^{T} = -v^{T} + \mathcal{O}^{T} (u^{T}) \quad \text{with} \quad v^{T} = \frac{\kappa}{\kappa_{T}} \partial_{n_{T}} u^{T}. $$

* Consistent numerical traces $\hat{u}$ and $\hat{v}^T$ defined from $\gamma_{out}^T \mathbf{y}^T$ and $\gamma_{out}^K \mathbf{y}^K$.

⇒ Need to have $\mathcal{O}^T$ in a set of operators $\mathcal{C}$ which ensures the formulation conserves classic properties (coercivity and contraction):

$\mathcal{C} \subset \{\mathcal{O}, \text{ there exists a definite linear operator } \Lambda, \mathcal{O} = i\Lambda\Lambda^* \text{ for each face}\}.$
Approximation of the Dirichlet-to-Neumann operator

Treffzt method can be seen as the variational formulation of

$$v^T + \mathcal{O}^K(u^T) = -v^K + \mathcal{O}^K(u^K) \quad \text{on} \quad \partial T \cap \partial K.$$

The theory of Domain Decomposition Methods (DDM) ensures the optimal operator $\mathcal{O}^T$ is the exterior Dirichlet-to-Neumann (DtN) one.

3 approximation types :

* Exact operator for classic outgoing waves with respect to the cell centre $x_0$:

$$\mathcal{O}^T = i\kappa \mathbf{d} \cdot \mathbf{n}_{\partial\Omega} \text{ (directive PWs) and } \mathcal{O}^T = -\kappa \frac{H_1^{(1)}(\kappa r)}{H_0^{(1)}(\kappa r)} \text{ (Green kernel)},$$

for $\mathbf{d} = (\mathbf{x} - x_0)/r$ and $r = |\mathbf{x} - x_0|$.

* Operator with same principal symbol as the classic approximation [El Bouajaji et al. 2014]

$$\mathcal{O}^T = i\kappa \sqrt{Id + \frac{\Delta_F}{\kappa^2}},$$

as Padé-type approximations [Després et al. 2021]

$$\mathcal{O}^T = i\kappa \left( Id - \frac{\Delta_F}{2\kappa^2} \right)^{-1}.$$

→ Face point of view: BCs for the Laplace-Beltrami operator?

* Approximation thanks to neural networks (NN) : ($\kappa_T$, $h_F$) $\mapsto \mathcal{O}^T$, which allows to enforce $\mathcal{O}^T \in \mathcal{C}$.
<figure><img src="image_208.png" /></figure>

<figure><img src="image_209.png" /></figure>

<figure><img src="image_210.png" /></figure>

Approximation of the Dirichlet-to-Neumann operator

<figure><img src="image_211.png" /><figcaption>Figure: Influence of DtN approximations on the GMRES convergence for different mesh sizes (with respect to the wavelength λ) for N=50 macro-cells.</figcaption></figure>

Promising preliminary results to reduce the number of iterations to convergence:

* Analytic approximations work especially for large enough cells.

* Robust approximation by NN: 25% iterations gain.
<figure><img src="image_212.png" /></figure>

<figure><img src="image_213.png" /></figure>

<figure><img src="image_214.png" /></figure>

## Perspectives and actual work

<figure><img src="image_215.png" /></figure>

Figure: FDTD simulation on a Manhattan mesh of 700 × 600 × 300m, Thibault Volpert (DEMR ONERA)

* Theoretical a priori error estimate of the quasi-Trefftz approach.

* Incorporation of unstructured meshes on the object boundaries: keep the Cartesian matrix-free framework as much as possible, with precise boundary effects.

* Local operator optimisation at each iteration, depending on the current solution.
<figure><img src="image_216.png" /></figure>

<figure><img src="image_217.png" /></figure>

<figure><img src="image_218.png" /></figure>

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<figure><img src="image_221.png" /></figure>

<figure><img src="image_222.png" /></figure>

[Sirdey 2022] M. Sirdey. 'Méthode itérative de Trefftz pour la simulation d'ondes électromagnétiques en trois dimensions', PHD, Université de Pau et des Pays de l'Adour, 2022.

<figure><img src="image_223.png" /></figure>

[Yee 1966] Kane Yee. 'Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media', IEEE Transactions on Antennas and Propagation, 14(3):302–307, 1966.

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[Fezoui et al. 2005] L. Fezoui, S. Lanteri, S. Lohrengel, and S. Piperno, 'Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes', ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, vol. 39, no. 6, pp. 1149–1176, 2005.

<figure><img src="image_226.png" /></figure>

[Cessenat, Després 1998] O. Cessenat, B. Després. 'Application of an Ultra Weak Variational Formulation of Elliptic PDEs to the Two-Dimensional Helmholtz Problem', SIAM Journal on Numerical Analysis, vol. 35, pp. 255-299, 1998.

<figure><img src="image_227.png" /></figure>

[Ihlenburg, Babuška 1995] F. Ihlenburg, I. Babuška. 'Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM', Computers & Mathematics with Applications, vol. 30, pp. 9-37, 1995.
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[Barucq et al. 2021] H. Barucq, A. Bendali, J. Diaz, and S. Tordeux, 'Local strategies for improving the conditioning of the plane-wave Ultra-Weak Variational Formulation', Journal of Computational Physics, vol. 441, p. 110449, 2021.

<figure><img src="image_233.png" /></figure>

[Fure et al. 2020] H. Fure, S. Pernet, M. Sirdey, S. Tordeux, 'A discontinuous Galerkin Trefftz type method for solving the two dimensional Maxwell equations', SN Partial Differential Equations and Applications, vol. 1, pp. 1-25, 2020.

<figure><img src="image_234.png" /></figure>

[Imbert-Gérard, Després 2013] L.-M. Imbert-Gérard, B. Després. 'A generalized plane wave numerical method for smooth non constant coefficients'. IMA Journal of Numerical Analysis 34, 2024.

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[Imbert-Gérard, Sylvand 2023] L.-M. Imbert-Gérard, G. Sylvand. 'Three types of quasi-Trefftz functions for the 3D convected Helmholtz equation: construction and approximation properties'. arXiv preprint, 2023.

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<figure><img src="image_245.png" /></figure>

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<figure><img src="image_246.png" /></figure>

<figure><img src="image_247.png" /></figure>

<figure><img src="image_248.png" /></figure>

# Appendix
<figure><img src="image_249.png" /></figure>

<figure><img src="image_250.png" /></figure>

<figure><img src="image_251.png" /></figure>

# The harmonic system and current approaches

We obtain the harmonic version by looking for e and h as

$$e(\mathbf{x}, t) = e_0 \Re (\mathbf{E}(\mathbf{x}) e^{i\omega t}) \quad \text{and} \quad \mathbf{h}(\mathbf{x}, t) = h_0 \Re (\mathbf{H}(\mathbf{x}) e^{i\omega t}), \quad (6)$$

denoting their amplitudes by $e_0$ and $h_0 = \sqrt{\frac{\epsilon_0}{\mu_0}} e_0$, the wave number $\kappa = \frac{\omega}{c_0}$ and the speed of light in a vacuum $c_0 = (\epsilon_0 \mu_0)^{-1/2}$.

Normalised harmonic Maxwell equations for $\mathbf{E}, \mathbf{H} \in [L^2(\Omega)]^3$:

$$\left\{
\begin{array}{l}
\nabla \times \mathbf{H} = i\kappa\epsilon_r\mathbf{E} \quad \text{in } \Omega, \\
\nabla \times \mathbf{E} = -i\kappa\mu_r\mathbf{H} \quad \text{in } \Omega, \\
\gamma_t\mathbf{E} + Z_{\partial\Omega}\mathbf{n}_{\partial\Omega} \times \gamma_t\mathbf{H} = \mathbf{G} \quad \text{on } \partial\Omega,
\end{array}
\right.
\quad (7)$$

with

$$\mathbf{G} \in \mathbf{L}_t^2(\partial\Omega).$$
<figure><img src="image_252.png" /></figure>

<figure><img src="image_253.png" /></figure>

<figure><img src="image_254.png" /></figure>

## Quasielliptic interpretation of the Maxwell system

$$ \left\{ \begin{array}{ll} \nabla \times (\nabla \times \mathbf{E}) - \kappa^2 \mathbf{E} = 0 & \text{in } \Omega, \\ \gamma_t \mathbf{E} - i(\kappa \mu_r)^{-1} Z_{\partial\Omega} \mathbf{n}_{\partial\Omega} \times (\nabla \times \mathbf{E}) = \mathbf{G} & \text{on } \partial\Omega. \end{array} \right. \qquad (8) $$

Find $\mathbf{E} \in H_{imp}(\Omega)$ such that for all $\mathbf{E}' \in H_{imp}(\Omega)$

$$ \int_{\Omega} \nabla \times \mathbf{E} \cdot \overline{\nabla \times \mathbf{E}'}_{\uparrow} - \kappa^2 \int_{\Omega} \mathbf{E} \cdot \mathbf{E}' + \int_{\partial\Omega} \frac{i\kappa}{Z_{\partial\Omega}} \gamma_t \mathbf{E} \cdot \overline{\gamma_t \mathbf{E}'} = \int_{\Omega} \frac{i\kappa}{Z_{\partial\Omega}} \mathbf{G} \cdot \overline{\gamma_t \mathbf{E}'} \quad (9) $$

posed on the Monk space

$$ \mathbf{E} \in H_{imp}(\Omega) \iff \mathbf{E} \in H(\text{curl}, \Omega), \gamma_t \mathbf{E} \in [\mathbf{L}^2(\partial\Omega)]^3. $$

**Theorem (Monk 03)** The problem is well-posed and the FEM is convergent.

### Advantages

* Reduction of the number of degrees of freedom,

* Well established mathematical background.

### Drawbacks

* Numerical dispersion issue at low order,

* Memory cost issue for large linear systems,

* Difficult post-processing: $\mathbf{H} = -\frac{1}{i\kappa\mu_r}\nabla \times \mathbf{E}$.
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## Hyperbolic interpretation of the Maxwell system

$$i\kappa \mathbf{D}\mathbf{E} = \frac{\partial}{\partial x}(\mathbf{F}_x\mathbf{E}) + \frac{\partial}{\partial y}(\mathbf{F}_y\mathbf{E}) + \frac{\partial}{\partial z}(\mathbf{F}_z\mathbf{E}) \quad (\text{flux divergence } \nabla \cdot \mathbf{F}(\mathbf{E})), \quad (10)$$

with the electromagnetic field

$$\mathbf{E} = \begin{bmatrix} E_x \\ E_y \\ E_z \\ H_x \\ H_y \\ H_z \end{bmatrix}$$

Electromagnetic metric and hyperbolic flux operator

$$\mathbf{D} = \begin{bmatrix} \varepsilon_r & 0 & 0 & 0 & 0 & 0 \\ 0 & \varepsilon_r & 0 & 0 & 0 & 0 \\ 0 & 0 & \varepsilon_r & 0 & 0 & 0 \\ 0 & 0 & 0 & \mu_r & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu_r & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu_r \end{bmatrix} \quad \mathbf{F}_x = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \end{bmatrix}$$
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# Current issues: memory and iterative resolution

Many classic methods as Finite Differences [Yee 1966], Finite Elements [Nédélec 1980], Discontinuous Galerkin [Fezoui et al. 2005], but limitations for current applications:

* Wide domains lead to **very large linear systems**.

* Traditional direct solvers induce a prohibitive memory consumption.

* Usual numerical methods are not well adapted to iterative resolution.
<figure><img src="image_261.png" /></figure>

<figure><img src="image_262.png" /></figure>

<figure><img src="image_263.png" /></figure>

# Current issues: memory and iterative resolution

Many classic methods as Finite Differences [Yee 1966], Finite Elements [Nédélec 1980], Discontinuous Galerkin [Fezoui et al. 2005], but limitations for current applications:

* Wide domains lead to **very large linear systems**.

* Traditional direct solvers induce a **prohibitive memory consumption**.

* Usual numerical methods are not well adapted to iterative resolution.

<figure><img src="image_264.png" /><figcaption>Figure: Memory consumption of a FEM w.r.t. the domain size (in wavelengths) [Sirdey 2022].</figcaption></figure>
<figure><img src="image_265.png" /></figure>

<figure><img src="image_266.png" /></figure>

<figure><img src="image_267.png" /></figure>

# Current issues: memory and iterative resolution

Many classic methods as Finite Differences [Yee 1966], Finite Elements [Nédélec 1980], Discontinuous Galerkin [Fezoui et al. 2005], but limitations for current applications:

* Wide domains lead to **very large linear systems**.

* Traditional direct solvers induce a **prohibitive memory consumption**.

* Usual numerical methods are **not well adapted to iterative resolution**.

<figure><img src="image_268.png" /><figcaption>Figure: Memory consumption of a FEM w.r.t. the domain size (in wavelengths) [Sirdey 2022].</figcaption></figure>

<figure><img src="image_269.png" /><figcaption>Figure: GMRES convergence of DG and FEM methods [Sirdey 2022].</figcaption></figure>
<figure><img src="image_270.png" /></figure>

<figure><img src="image_271.png" /></figure>

<figure><img src="image_272.png" /></figure>

The reciprocity formula and Trefftz formulation

Due to the integration by parts formula, one has for every $\mathbf{Y}^T = (\mathbf{E}^T, \mathbf{H}^T)$ and $\mathbf{Y}'^T = (\mathbf{E}'^T, \mathbf{H}'^T)$ solutions of the Maxwell equations in $T$:

$$ \sum_{F \in \mathcal{F}_T} \int_F \gamma_t \mathbf{E}^T \cdot \overline{\gamma_t \mathbf{H}'^T} + \gamma_t \mathbf{H}^T \cdot \overline{\gamma_t \mathbf{E}'^T} = 0. \quad (11) $$

To ensure the continuity of the tangential component and trace across the interfaces and the respect of the BCs, we use consistent numerical traces [Sirdey 2022]:

$$ \sum_{T \in \mathcal{T}_h} \int_{\partial T} \gamma_{\times}^{T} \overline{\mathbf{H}^T} \cdot \overline{\gamma_t \mathbf{E}'^T} + \gamma_t \overline{\mathbf{E}^T} \cdot \overline{\gamma_{\times}^{T} \mathbf{H}'^T} = \ell(\mathbf{Y}'). \quad (12) $$

These numerical traces can be obtained in three different ways:

* By adding weighted consistent forms to the reciprocity formula.

* By using a Riemann solver in the case of a homogeneous material.

* As a linear combination of outgoing traces
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The equivalent Ultra Weak Variational Formulation

Trefftz formulation:

$$a(\mathbf{Y}, \mathbf{Y}') = \sum_{T \in \mathcal{T}_h} \int_{\partial T} \widehat{\gamma_{\times}^T \mathbf{H}^T} \cdot \overline{\gamma_t \mathbf{E}'^T} + \widehat{\gamma_t \mathbf{E}^T} \cdot \overline{\gamma_{\times}^T \mathbf{H}'^T} = \ell(\mathbf{Y}').$$

Equivalent [Sirdey 2022] Ultra Weak Variational Formulation (UWVF) [Cessenat, Després 1998]:

$$a_{CD}(\mathbf{x}, \mathbf{x}') = \sum_{T \in \mathcal{T}_h} \int_{\partial T} \frac{1}{Z_T} (\mathbf{x}^T \cdot \overline{\mathbf{x}'^T} - \widehat{u\mathbf{x}^T} \cdot \overline{u\mathbf{x}'^T}) = \ell(\mathbf{x}'),$$

for the outgoing and ingoing traces:

$$\mathbf{x}^T = \gamma_t \mathbf{E}^T - Z_T \gamma_{\times}^T \mathbf{H}^T \quad \text{and} \quad u\mathbf{x}^T = \gamma_t \mathbf{E}^T + Z_T \gamma_{\times}^T \mathbf{H}^T. \qquad (13)$$
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## Natural coercivity of the Trefftz formulation

Trefftz formulation:

$$a(\mathbf{Y}, \mathbf{Y}') = \sum_{T \in \mathcal{T}_h} \int_{\partial T} \widehat{\gamma_T^T \mathbf{H}^T} \cdot \overline{\gamma_t \mathbf{E}'^T} + \widehat{\gamma_t \mathbf{E}^T} \cdot \overline{\gamma_T^T \mathbf{H}'^T} = \ell(\mathbf{Y}').$$

We introduce the norm

$$\|\mathbf{Y}\|_{DG}^2 = \|\mathbf{Y}\|_{int}^2 + \|\mathbf{Y}\|_{ext}^2, \quad (14)$$

with

$$\|\mathbf{Y}\|_{int}^2 = \sum_{F \in \mathcal{F}_{int}} \int_F \left( \frac{1}{Z_T + Z_K} \left| \|\gamma_t \mathbf{E}\|_{T,F} \right|^2 + \frac{Z_T Z_K}{Z_T + Z_K} \left| \|\gamma_T \mathbf{H}\|_F \right|^2 \right), \quad (15)$$

and

$$\|\mathbf{Y}\|_{ext}^2 = \sum_{F \in \mathcal{F}_{ext}} \int_F \left( \frac{1}{Z_T + Z_{\partial\Omega}} |\gamma_t \mathbf{E}|^2 + \frac{Z_T Z_{\partial\Omega}}{Z_T + Z_{\partial\Omega}} |\gamma_T \mathbf{H}|^2 \right), \quad (16)$$

and the jumps defined as

$$\|\gamma_t \mathbf{E}\|_{T,F} = \gamma_t \mathbf{E}^T - \gamma_t \mathbf{E}^K \quad \text{and} \quad \|\gamma_T \mathbf{H}\|_F = \gamma_T^T \mathbf{H}^T + \gamma_T^K \mathbf{H}^K. \quad (17)$$

Then, one has [Sirdey 2022]:

$$\mathrm{Re}[a(\mathbf{Y}, \mathbf{Y}')] = \|\mathbf{Y}\|_{DG}^2. \quad (18)$$
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Description of the FR local solver

Choice of the local solver: **the Flux Reconstruction method** [Huynh 2007]

* Based on the first-order strong formulation (solves both E and H): no dependence on a quadrature rule.

* Idea: copy the **strong equation in each cell** of the mesh, which is solved on **piecewise polynomials**: the flux is corrected at interfaces thanks to **numerical traces** and **correction polynomial functions**.

$$i\kappa y + \frac{d\phi}{dx} = 0 \quad \text{with} \quad \phi = \mathcal{F}(y) \implies \forall n \in [1, N], \ i\kappa y_h + \frac{d\tilde{\phi}_h}{dx} = 0 \quad \text{in} \ [X_{n-1}, X_n].$$

* Natural high-order method and **adaptable** to unstructured meshes.

* Choice of correction polynomials allows to **retrieve usual methods**: nodal DG (for Radau polynomials), Spectral Differences (for Lagrange polynomials)...

* Choice of the correction polynomials to the user: possibility to **optimise them for wave propagation problems**, especially in pre-asymptotic regime [Rivet, Pernet, Tordeux 2024].

* For identical cells, the **inversion has to be realised only once**!

⇒ **Any numerical method, solving the first-order system, may be used!**
<figure><img src="image_282.png" /></figure>

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# FR properties and 3D generalisation

Main properties:

* $\mathbf{y}_h$ of degree $k_{FR}$ and $\tilde{\phi}_h$ of degree $k+1$ in each cell: compatibility of the ODE.

* Strong formulation: **no dependence on a quadrature rule.**

* Choice of the numerical trace computation and the correction polynomials: allows to **retrieve classic formulations** (nodal DG, SD) and possibility to get an optimised choice?

Easy 3D generalisation in a tensorisation idea:

$$ \forall \mathbf{n} := (n_1, n_2, n_3), \quad i\kappa \begin{pmatrix} \varepsilon_r \mathbf{I}_3 & \mathbf{0}_3 \\ \mathbf{0}_3 & \mu_r \mathbf{I}_3 \end{pmatrix} \mathbf{Y}_h + \sum_{j=1}^{3} \frac{\partial \tilde{\phi}_h^j}{\partial x_j} = \mathbf{0} \text{ in } \mathbf{K}_n. \quad (19) $$

1. Approximate solution $\mathbf{Y}_h$ as piecewise polynomial.

2. Flux numerical traces $\tilde{\phi}_h^j$ on faces orthogonal to $e_j$.

3. Choose correction polynomials $P^{j,\rightarrow}$ and $P^{j,\leftarrow}$ for each direction.

4. Correction of each flux in its direction.

<figure><img src="image_285.png" /></figure>

More complex on an unstructured mesh!
<figure><img src="image_286.png" /></figure>

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## Choice of the numerical flux of the FR method

Split the flux operator F according to information propagation towards right and left:

$$F = F^{\rightarrow} + F^{\leftarrow}. \qquad (20)$$

→ Diagonalise F to highlight its positive and negative parts:

$$F = P\Lambda P^{-1} \Rightarrow \begin{cases} F^{\rightarrow} = P\Lambda^{+}P^{-1}, \\ F^{\leftarrow} = P\Lambda^{-}P^{-1}. \end{cases} \qquad (21)$$

Recalling that $\phi = Fy$, we define

$$\hat{\phi}_n = F^{\rightarrow}y_h^{-}(X_n) + F^{\leftarrow}y_h^{+}(X_n), \qquad (22)$$

with on the left boundary

$$F^{\rightarrow}y_h^{-}(0) = R^{\leftarrow}F^{\leftarrow}y_h^{+}(0) + G_1. \qquad (23)$$

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Correction polynomials

* Symmetry property:

$$ \forall x \in [0, 1], P^{\leftarrow}(x) = P^{\rightarrow}(1-x). \qquad (24) $$

* Choices allow to retrieve classic numerical methods [Huynh 2007]:

* Lagrange polynomials with symmetric nodes in the interval allow to retrieve the Spectral Difference method: roots of the Legendre polynomials (SD.IG)...

* Radau polynomials (FR.Rad) allow to retrieve a nodal Discontinuous Galerkin method.

* G2 polynomials (FR.G2) [Huynh 2007].

<figure><img src="image_293.png" /></figure>
<figure><img src="image_294.png" /></figure>

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Quasi-Trefftz approach: BCs and basis functions illustration

Piecewise polynomial BCs and the associated basis functions:

<figure><img src="image_297.png" /><figcaption>Table: Piecewise polynomial BCs</figcaption></figure>
<figure><img src="image_298.png" /></figure>

<figure><img src="image_299.png" /></figure>

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## Quasi-Trefftz approach: BCs and basis functions illustration

### Piecewise polynomial BCs and the associated basis functions:

<figure><img src="image_301.png" /><figcaption>Table: Associated solutions taken as basis functions</figcaption></figure>
<figure><img src="image_302.png" /></figure>

<figure><img src="image_303.png" /></figure>

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## Local basis quality

Natural scalar product

$$ 
\begin{aligned}
(\mathbf{x}^T, \mathbf{x}'^T)_{L_t^2(\partial T)}
&= \int_{\partial T} \mathbf{x}^T \cdot \overline{\mathbf{x}'^T} \\
&= \int_{\partial T} \gamma_t \mathbf{E}^T \cdot \overline{\gamma_t \mathbf{E}^T} + (Z_T)^2 \gamma_\times^T \mathbf{H}^T \cdot \overline{\gamma_\times^T \mathbf{H}^T} \quad (25) \\
&= (\mathbf{Y}^T, \mathbf{Y}'^T)_{L_t^2(\partial T)}.
\end{aligned}
$$

Mass matrix $M = (\mathbf{Y}_i^T, \mathbf{Y}_j^T)_{L_t^2(\partial T)}$ to evaluate its conditioning.

<figure><img src="image_305.png" /><figcaption>Figure: Eigenvalues of the scalar product matrix for local basis functions: Trefftz method based on 196 plane waves and 192 Quasi-Trefftz basis functions.</figcaption></figure>

⇒ No more numerical dependence of the basis functions!
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# Recovery of the Trefftz properties under local solver convergence

Trefftz approach rests on the use of *exact solutions* of the Maxwell equations:

## Is using quasi-solutions enough?

### Quasi-Trefftz properties at convergence

**Hypothesis :** the local solver verifies the convergence property for $\gamma_T \mathbf{E}^T + Z_{\partial T} \gamma_\times^T \mathbf{H}^T = g^T$ on $\partial T$:

$$ \|\gamma_t(\mathbf{E}^T - \mathbf{E}_h^T)\|_{L^2(\partial T)} + \|\gamma_\times^T(\mathbf{H}^T - \mathbf{H}_h^T)\|_{L^2(\partial T)} \le \varepsilon_{ls}^T \|\mathbf{g}^T\|_{L^2(\partial T)} \text{ with } \varepsilon_{ls}^T \to 0, $$

when the local solver mesh is refined.

**If the local solver mesh is sufficiently refined, the quasi-Trefftz formulation verifies** the

* weak-coercivity,

* contraction properties,

of the original Trefftz formulation.