# An Optimised Quasi-Trefftz Method for the Iterative Solution of Time-Harmonic Wave problems

Sébastien Pernet, Matthias Rivet, Sébastien Tordeux

► To cite this version:

Sébastien Pernet, Matthias Rivet, Sébastien Tordeux. An Optimised Quasi-Trefftz Method for the Iterative Solution of Time-Harmonic Wave problems. ICOSAHOM 2025 - 15th International Conference on Spectral and High Order Methods, Jul 2025, Montréal, Canada. hal-05317562

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Submitted on 16 Oct 2025

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# An Optimised Quasi-Trefftz 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

ICOSAHOM 2025

15th July 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_5.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!
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

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_9.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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# 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
$\mathbf{Y} := (\mathbf{E}, \mathbf{H}) \in [\mathrm{H}(\mathrm{curl}, \Omega)]^6$:

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

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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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_22.png" /></figure>

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<figure><img src="image_24.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_25.png" /><figcaption>Figure: Memory consumption of a FEM w.r.t. the domain size (in wavelengths) [Sirdey 2022].</figcaption></figure>
<figure><img src="image_26.png" /></figure>

<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.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_29.png" /><figcaption>Figure: Memory consumption of a FEM w.r.t. the domain size (in wavelengths) [Sirdey 2022].</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>Figure: GMRES convergence of DG and FEM methods [Sirdey 2022].</figcaption></figure>
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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.
<figure><img src="image_34.png" /></figure>

<figure><img src="image_35.png" /></figure>

<figure><img src="image_36.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.

<figure><img src="image_37.png" /></figure>
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<figure><img src="image_40.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.

<figure><img src="image_41.png" /></figure>
<figure><img src="image_42.png" /></figure>

<figure><img src="image_43.png" /></figure>

<figure><img src="image_44.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.

<figure><img src="image_45.png" /></figure>
<figure><img src="image_46.png" /></figure>

<figure><img src="image_47.png" /></figure>

<figure><img src="image_48.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.

<figure><img src="image_49.png" /></figure>
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<figure><img src="image_51.png" /></figure>

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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].

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_56.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 $\mathcal{V}_h^T$

* Consider piecewise polynomial $g^T$ of degree $k$ on $\mathcal{V}_h^T$

<figure><img src="image_60.png" /><figcaption>Figure: 2D example of the mesh of $\partial T$ for $k_{QT} = 1$.</figcaption></figure>
<figure><img src="image_61.png" /></figure>

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<figure><img src="image_63.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_64.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_67.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_68.png" /><figcaption>Figure: 2D example of the mesh of $\partial T$ for $k_{QT} = 1$.</figcaption></figure>
<figure><img src="image_69.png" /></figure>

<figure><img src="image_70.png" /></figure>

<figure><img src="image_71.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_72.png" /><figcaption>Figure: 2D example of the mesh of $\partial T$ for $k_{QT} = 1$.</figcaption></figure>
<figure><img src="image_73.png" /></figure>

<figure><img src="image_74.png" /></figure>

<figure><img src="image_75.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_76.png" /><figcaption>Figure: 2D example of the mesh of $\partial T$ for $k_{QT} = 1$.</figcaption></figure>
<figure><img src="image_77.png" /></figure>

<figure><img src="image_78.png" /></figure>

<figure><img src="image_79.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_80.png" /><figcaption>Figure: 2D example of the mesh of $\partial T$ for $k_{QT} = 1$.</figcaption></figure>
<figure><img src="image_81.png" /></figure>

<figure><img src="image_82.png" /></figure>

<figure><img src="image_83.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_84.png" /><figcaption>Figure: 2D example of the mesh of $\partial T$ for $k_{QT} = 1$.</figcaption></figure>
<figure><img src="image_85.png" /></figure>

<figure><img src="image_86.png" /></figure>

<figure><img src="image_87.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_88.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_89.png" /></figure>

<figure><img src="image_90.png" /></figure>

<figure><img src="image_91.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_92.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]...
<figure><img src="image_93.png" /></figure>

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

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

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**.

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

* 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_96.png" /></figure>

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

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

# A quasi-Trefftz approach based on
a Flux Reconstruction auxiliary solver:
numerical analysis
<figure><img src="image_99.png" /></figure>

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

<figure><img src="image_101.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_102.png" /></figure>

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

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

<figure><img src="image_103.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_104.png" /></figure>

FR auxiliary solver to compute an approximation of $(E_i^T, H_i^T)$.
<figure><img src="image_105.png" /></figure>

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

<figure><img src="image_107.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_108.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_109.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_110.png" /></figure>

FR auxiliary solver to compute an approximation of $(E_i^T, H_i^T)$.
<figure><img src="image_111.png" /></figure>

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

<figure><img src="image_113.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_114.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_115.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_116.png" /></figure>

FR auxiliary solver to compute an approximation of $(E_i^T, H_i^T)$.
<figure><img src="image_117.png" /></figure>

<figure><img src="image_118.png" /></figure>

<figure><img src="image_119.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_120.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_121.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_122.png" /></figure>

FR auxiliary solver to compute an approximation of $(E_i^T, H_i^T)$.
<figure><img src="image_123.png" /></figure>

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

<figure><img src="image_125.png" /></figure>

## Numerical experiments: basis quality

<figure><img src="image_126.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_127.png" /></figure>

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

<figure><img src="image_129.png" /></figure>

## Numerical experiments: basis quality

<figure><img src="image_130.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_131.png" /></figure>

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

<figure><img src="image_133.png" /></figure>

## Numerical experiments: basis quality

<figure><img src="image_134.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_135.png" /></figure>

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

<figure><img src="image_137.png" /></figure>

## Numerical experiments: basis quality

<figure><img src="image_138.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_139.png" /></figure>

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

<figure><img src="image_141.png" /></figure>

Approximation properties of the basis for a smooth solution:
sum of random PWs

<figure><img src="image_142.png" /><figcaption>Figure: Projection error on the different basis for a sum of random PWs.</figcaption></figure>

⇒ Saturation phenomenon due to condition number.

⇒ 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_143.png" /></figure>

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

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

Approximation properties of the basis for a smooth solution:
sum of random PWs

<figure><img src="image_146.png" /><figcaption>Figure: Projection error on the different basis for a sum of random PWs.</figcaption></figure>

⇒ Saturation phenomenon due to condition number.

⇒ **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_147.png" /></figure>

<figure><img src="image_148.png" /></figure>

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

Approximation properties of the basis for a smooth solution:
sum of random PWs

<figure><img src="image_150.png" /><figcaption>Figure: Projection error on the different basis for a sum of random PWs.</figcaption></figure>

⇒ Saturation phenomenon due to condition number.

⇒ **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_151.png" /></figure>

<figure><img src="image_152.png" /></figure>

<figure><img src="image_153.png" /></figure>

Approximation properties of the basis for a 'non-smooth' solution:
sum of random dipoles

<figure><img src="image_154.png" /><figcaption>Figure: Projection error on the different basis for a sum of random dipoles.</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_155.png" /></figure>

<figure><img src="image_156.png" /></figure>

<figure><img src="image_157.png" /></figure>

Approximation properties of the basis for a 'non-smooth' solution:
sum of random dipoles

<figure><img src="image_158.png" /><figcaption>Figure: Projection error on the different basis for a sum of random dipoles.</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_159.png" /></figure>

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

<figure><img src="image_161.png" /></figure>

Approximation properties of the basis for a 'non-smooth' solution:
sum of random dipoles

<figure><img src="image_162.png" /><figcaption>Figure: Projection error on the different basis for a sum of random dipoles.</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_163.png" /></figure>

<figure><img src="image_164.png" /></figure>

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

# 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 $\gamma_T \mathbf{E}^T + Z_{\partial T} \gamma_T^T \mathbf{H}^T = g^T$ 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.
<figure><img src="image_166.png" /></figure>

<figure><img src="image_167.png" /></figure>

<figure><img src="image_168.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 $T_H(\Omega)$ of the domain $\Omega$: $N$ macro-cells per direction.

* Uniform Cartesian mesh $T_h(\partial T)$ of $\partial T$: $M$ micro-faces per direction.

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

<figure><img src="image_169.png" /><figcaption>Figure: Mesh $T_H(\Omega)$ for $N = 3$.</figcaption></figure>

<figure><img src="image_170.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_171.png" /></figure>

<figure><img src="image_172.png" /></figure>

<figure><img src="image_173.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_174.png" /><figcaption>Figure: Mesh $\mathcal{T}_H(\Omega)$ for $N = 3$.</figcaption></figure>

<figure><img src="image_175.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_176.png" /></figure>

<figure><img src="image_177.png" /></figure>

<figure><img src="image_178.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_179.png" /><figcaption>Figure: Mesh $\mathcal{T}_H(\Omega)$ for $N = 3$.</figcaption></figure>

<figure><img src="image_180.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_181.png" /></figure>

<figure><img src="image_182.png" /></figure>

<figure><img src="image_183.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_184.png" /><figcaption>Figure: Mesh $\mathcal{T}_H(\Omega)$ for $N = 3$.</figcaption></figure>

<figure><img src="image_185.png" /><figcaption>Figure: Mesh $\mathcal{T}_h(\partial T)$ for $M = 2$ and $k_{QT} = 1$.</figcaption></figure>
<figure><img src="image_186.png" /></figure>

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

<figure><img src="image_188.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_189.png" /><figcaption>Figure: Mesh $\mathcal{T}_H(\Omega)$ for $N = 3$.</figcaption></figure>

<figure><img src="image_190.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_191.png" /></figure>

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

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

# Calibration of the FR auxiliary solver (2D Helmholtz)

## Is using quasi-solutions enough?

## How to adapt the FR order to the Trefftz one?

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

<figure><img src="image_194.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_195.png" /></figure>

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

<figure><img src="image_197.png" /></figure>

# Calibration of the FR auxiliary solver (2D Helmholtz)

## Is using quasi-solutions enough?

## How to adapt the FR order to the Trefftz one?

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

<figure><img src="image_198.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 FR auxiliary solver (2D Helmholtz)

## Is using quasi-solutions enough?

## How to adapt the FR order to the Trefftz one?

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

<figure><img src="image_199.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_200.png" /></figure>

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

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

Calibration of the FR auxiliary solver (2D Helmholtz)

Is using quasi-solutions enough?

How to adapt the FR order to the Trefftz one?

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

<figure><img src="image_203.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_204.png" /></figure>

<figure><img src="image_205.png" /></figure>

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

# Calibration of the FR auxiliary solver (2D Helmholtz)

## Is using quasi-solutions enough?

## How to adapt the FR order to the Trefftz one?

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

<figure><img src="image_207.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_208.png" /></figure>

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

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

# Calibration of the FR auxiliary solver (2D Helmholtz)

## Is using quasi-solutions enough?

## How to adapt the FR order to the Trefftz one?

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

<figure><img src="image_211.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 $k_{FR} \ge k_{QT} + 1$.
<figure><img src="image_212.png" /></figure>

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

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

# Calibration of the FR auxiliary solver (2D Helmholtz)

## Is using quasi-solutions enough?

## How to adapt the FR order to the Trefftz one?

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

<figure><img src="image_215.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_216.png" /></figure>

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

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

Calibration of the FR auxiliary solver (2D Helmholtz)

Is using quasi-solutions enough?

How to adapt the FR order to the Trefftz one?

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

<figure><img src="image_219.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_220.png" /></figure>

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

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

# Calibration of the FR auxiliary solver

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

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

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

* Need of $k_{FR} \ge k_{QT} + 2$ for quasi-optimal orders, but **no interest in over-resolving**!
<figure><img src="image_224.png" /></figure>

<figure><img src="image_225.png" /></figure>

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

# Numerical experiments: convergence in the number of basis functions

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

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

<figure><img src="image_227.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_228.png" /></figure>

<figure><img src="image_229.png" /></figure>

<figure><img src="image_230.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.

* Robust approximation properties of the basis, even for complex local solutions.

* 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_231.png" /></figure>

<figure><img src="image_232.png" /></figure>

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

# Improvement of the transfer of information: DtN approximation
<figure><img src="image_234.png" /></figure>

<figure><img src="image_235.png" /></figure>

<figure><img src="image_236.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_237.png" /><figcaption>(i) Solution evolution</figcaption></figure>

<figure><img src="image_238.png" /><figcaption>(ii) Solution modification</figcaption></figure>

Figure: Iterative convergence: information propagation through iterations.
<figure><img src="image_239.png" /></figure>

<figure><img src="image_240.png" /></figure>

<figure><img src="image_241.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):

$\mathcal{C} \subset \{\mathcal{O}, \text{ there exists a definite linear operator } \Lambda, \mathcal{O} = i\Lambda\Lambda^* \text{ for each face}\}.$
<figure><img src="image_242.png" /></figure>

<figure><img src="image_243.png" /></figure>

<figure><img src="image_244.png" /></figure>

Approximation of the Dirichlet-to-Neumann operator

Trefftz 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 $x_0$:

$$\mathcal{O}^T = i\kappa \mathbf{d} \cdot \mathbf{n}_{\partial\Omega} \text{ (directive PWs) and } \mathcal{O}^T = -\kappa \frac{\mathbf{H}_1^{(1)}(\kappa r)}{\mathbf{H}_0^{(1)}(\kappa r)} \text{ (Green kernel),}$$

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)**: allows to enforce $\mathcal{O}^T \in \mathcal{C}$.
<figure><img src="image_245.png" /></figure>

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

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

Approximation of the Dirichlet-to-Neumann operator

<figure><img src="image_248.png" /><figcaption>Figure: Influence of DtN approximations on the GMRES convergence for different mesh sizes (with respect to the wavelength $\lambda$) for N=50 macro-cells.</figcaption></figure>

* We proposed a generalisation of the Trefftz method, thanks to general numerical fluxes keeping the original properties of the classic formulation.

* Trefftz approach can be interpreted as a DDM one: we introduced approximations of the Dirichlet-to-Neumann operator, which is supposed to be optimal.

* Analytic approximations work especially for large enough cells.

* Promising preliminary results to reduce the number of iterations to convergence: about 25% reduction by using NNs.
<figure><img src="image_249.png" /></figure>

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

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

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<figure><img src="image_252.png" /></figure>

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

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

<figure><img src="image_255.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_256.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.

<figure><img src="image_257.png" /></figure>

[Nédélec 1980] J. C. Nedelec. 'Mixed finite elements in $\mathbb{R}^3$, Numer. Math., vol. 35, no. 3, pp. 315–341, 1980.

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