# Flux Reconstruction Type Auxiliary Solvers for a Trefftz Domain Decomposition Method Dealing with Time-Harmonic Electromagnetism

Sébastien Pernet, Matthias Rivet, Sébastien Tordeux

► To cite this version:

Sébastien Pernet, Matthias Rivet, Sébastien Tordeux. Flux Reconstruction Type Auxiliary Solvers for a Trefftz Domain Decomposition Method Dealing with Time-Harmonic Electromagnetism. ECCO-MAS 2024 - 9th European Congress on Computational Methods in Applied Sciences and Engineering, Jun 2024, Lisbonne, Portugal. [hal-04877044](https://hal-04877044v1)

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

Distributed under a Creative Commons Attribution 4.0 International License
Flux Reconstruction Type Auxiliary Solvers for a Trefftz
Domain Decomposition Method Dealing with Time-Harmonic
Electromagnetism

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

6th June 2024
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_2.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!
# Plan of the talk

1. 3D Maxwell: limitations and Trefftz method

2. FR for wave propagation problems

3. Quasi-Trefftz approach
# The 3D Maxwell problem: current limitations and Trefftz method interests
<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

# Current issues: memory and iterative resolution

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

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

with impedance Boundary Conditions (BCs)

$$(\mathbf{n}_{\partial\Omega} \times \mathbf{E}) \times \mathbf{n}_{\partial\Omega} + Z_{\partial\Omega} \mathbf{n}_{\partial\Omega} \times \mathbf{H} = \mathbf{g} \text{ on } \partial\Omega. \quad (2)$$

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_6.png" /></figure>

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

<figure><img src="image_8.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.

2. Use of the **reciprocity formula** verified in each cell

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

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

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

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

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

$$ \gamma_t \mathbf{u}^T = \mathbf{u}^T - (\mathbf{u}^T \cdot \mathbf{n}_T) \mathbf{n}_T \quad \text{and} \quad \gamma_{\times}^T \mathbf{u}^T = \mathbf{n}_T \times \mathbf{u}^T. $$
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

## 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_12.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**.
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

# Quasi-Trefftz approach

## Definition of the local problem

A local solution $\mathbf{Y}^T$ in $T$ is parameterised by a tangential field $g^T \in \mathcal{V}$ 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 $\mathcal{V}_h \subset \mathcal{V}$ and the associated Maxwell solutions [Fure et al. 2020].

Construction of $\mathcal{V}_h$:

* Consider a cell $T$.

* Define a mesh of $T$.

* Consider a piecewise polynomial $g^T$ of degree one on $T$.

<figure><img src="image_16.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_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

# Quasi-Trefftz approach

## Definition of the local problem

A local solution $Y^T$ in $T$ is parameterised by a tangential field $g^T \in \mathbb{V}$ 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 $\mathbb{V}_h \subset \mathbb{V}$ and the associated Maxwell solutions [Fure et al. 2020].

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

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

* 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_20.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_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

# Quasi-Trefftz approach

## Definition of the local problem

A local solution $\mathbf{Y}^T$ in $T$ is parameterised by a tangential field $g^T \in \mathcal{V}$ 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 $\mathcal{V}_h \subset \mathcal{V}$ and the associated Maxwell solutions [Fure et al. 2020].

Construction of $\mathcal{V}_h$:

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

* 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_24.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_25.png" /></figure>

<figure><img src="image_26.png" /></figure>

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

# Quasi-Trefftz approach

## Definition of the local problem

A local solution $\mathbf{Y}^T$ in $T$ is parameterised by a tangential field $g^T \in \mathcal{V}$ 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 $\mathcal{V}_h \subset \mathcal{V}$ and the associated Maxwell solutions [Fure et al. 2020].

Construction of $\mathcal{V}_h$:

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

* 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_28.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_29.png" /></figure>

<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /></figure>

# Quasi-Trefftz approach

## Definition of the local problem

A local solution $\mathbf{Y}^T$ in $T$ is parameterised by a tangential field $g^T \in \mathcal{V}$ 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 $\mathcal{V}_h \subset \mathcal{V}$ and the associated Maxwell solutions [Fure et al. 2020].

Construction of $\mathcal{V}_h$:

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

* 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_32.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_33.png" /></figure>

<figure><img src="image_34.png" /></figure>

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

# Quasi-Trefftz approach

## Definition of the local problem

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

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

⇒ Consider a finite-dimensional subspace $\mathbb{V}_h \subset \mathbb{V}$ and the associated Maxwell solutions [Fure et al. 2020].

Construction of $\mathbb{V}_h$:

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

* 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_36.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_37.png" /></figure>

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

<figure><img src="image_39.png" /></figure>

# Quasi-Trefftz approach

## Definition of the local problem

A local solution $\mathbf{Y}^T$ in $T$ is parameterised by a tangential field $g^T \in \mathcal{V}$ 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 $\mathcal{V}_h \subset \mathcal{V}$ and the associated Maxwell solutions [Fure et al. 2020].

Construction of $\mathcal{V}_h$:

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

* 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_40.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_41.png" /></figure>

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

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

# Quasi-Trefftz approach

## Definition of the local problem

A local solution $\mathbf{Y}^T$ in $T$ is parameterised by a tangential field $g^T \in \mathcal{V}$ 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 $\mathcal{V}_h \subset \mathcal{V}$ and the associated Maxwell solutions [Fure et al. 2020].

Construction of $\mathcal{V}_h$:

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

* 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_44.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_45.png" /></figure>

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

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

# The Flux Reconstruction method for wave propagation problems
<figure><img src="image_48.png" /></figure>

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

<figure><img src="image_50.png" /></figure>

FR principle

Consider the homogeneous 1D model system on $\Omega = [0, L]$:
Find $y = (u, v) : \Omega \to \mathbb{C}^2$ such that

$$ egin{cases} i\kappa y + \frac{d\phi}{dx} = 0, & \text{in } \Omega \text{ with } \mathbf{F} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}, \\ \phi = \mathbf{F}y, & \end{cases} \quad (5) $$

and the BCs

$$ egin{cases} u(0) - Z_1 v(0) = g_1, \\ u(L) + Z_2 v(L) = g_2. \end{cases} \quad (6) $$

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

Presentation for a **linear** equation!

⇒ Copy this equation at the discrete level on each cell of the mesh [Huynh 2007]:

$$ \forall n \in [1, N], i\kappa y_h + \frac{d\tilde{\phi}_h}{dx} = 0 \text{ in } [X_{n-1}, X_n]. \quad (7) $$
<figure><img src="image_52.png" /></figure>

<figure><img src="image_53.png" /></figure>

<figure><img src="image_54.png" /></figure>

FR construction steps and properties

$$ \forall n \in [1, N], i \kappa y_h + \frac{d\tilde{\phi}_h}{dx} = 0 \text{ in } [X_{n-1}, X_n]. \quad (8) $$

1. Approximation $y_h$ as piecewise polynomial of degree $k_{FR}$.

2. Natural flux approximation $\tilde{\phi}_h = Fy_h$.

3. Numerical fluxes $\tilde{\phi}_n$ as a mean of $\phi_n^-$ and $\phi_n^+$.

4. Correction polynomials $P_n^-$ and $P_n^+$ of degree $k_{FR} + 1$.

5. Correction $\tilde{\phi}_{h,n} = \phi_{h,n} + \delta_{h-1,n} P_n^- + \delta_{h,n} P_n^+$.

<figure><img src="image_55.png" /></figure>
<figure><img src="image_56.png" /></figure>

<figure><img src="image_57.png" /></figure>

<figure><img src="image_58.png" /></figure>

FR construction steps and properties

$$ \forall n \in [1, N], i \kappa y_h + \frac{d\tilde{\phi}_h}{dx} = 0 \text{ in } [X_{n-1}, X_n]. \quad (8) $$

1. Approximation $y_h$ as piecewise polynomial of degree $k_{FR}$.

2. Natural flux approximation $\phi_h = Fy_h$.

3. Numerical fluxes $\tilde{\phi}_n$ as a mean of $\phi_n$ and $\phi_n^+$.

4. Correction polynomials $P_n^-$ and $P_n^+$ of degree $k_{FR} + 1$.

5. Correction $\tilde{\phi}_{h,n} = \phi_{h,n} + \delta_{h-1,n} P_n^- + \delta_{h,n} P_n^+ + \phi_{h,n}^0$.

<figure><img src="image_59.png" /></figure>
<figure><img src="image_60.png" /></figure>

<figure><img src="image_61.png" /></figure>

<figure><img src="image_62.png" /></figure>

FR construction steps and properties

$$ \forall n \in [1, N], i \kappa y_h + \frac{d\tilde{\phi}_h}{dx} = 0 \text{ in } [X_{n-1}, X_n]. \quad (8) $$

1. Approximation $y_h$ as piecewise polynomial of degree $k_{FR}$.

2. Natural flux approximation $\phi_h = Fy_h$.

3. Numerical fluxes $\tilde{\phi}_n$ as a mean of $\phi_n^-$ and $\phi_n^+$.

4. Correction polynomials $P_n^-$ and $P_n^+$ of degree $k_{FR} + 1$.

5. Correction $\tilde{\phi}_{h,n} = \phi_{h,n} + \delta_{h-1,n} P_n^+ + \delta_{h,n} P_n^-$.

<figure><img src="image_63.png" /></figure>
<figure><img src="image_64.png" /></figure>

<figure><img src="image_65.png" /></figure>

<figure><img src="image_66.png" /></figure>

FR construction steps and properties

$$ \forall n \in [1, N], i \kappa y_h + \frac{d\tilde{\phi}_h}{dx} = 0 \text{ in } [X_{n-1}, X_n]. \qquad (8) $$

1. Approximation $y_h$ as piecewise polynomial of degree $k_{FR}$.

2. Natural flux approximation $\phi_h = Fy_h$.

3. Numerical fluxes $\tilde{\phi}_n$ as a mean of $\phi_n^-$ and $\phi_n^+$.

4. Correction polynomials $P_n^-$ and $P_n^+$ of degree $k_{FR} + 1$.

5. Correction $\tilde{\phi}_{h,n} = \phi_{h,n} + \delta_{h,n-1}P_{n-1}^+ + \delta_{h,n}P_{n-1}^-$.

<figure><img src="image_67.png" /></figure>

<figure><img src="image_68.png" /></figure>
<figure><img src="image_69.png" /></figure>

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

<figure><img src="image_71.png" /></figure>

FR construction steps and properties

$$ \forall n \in [1, N], \quad i\kappa y_h + \frac{d\tilde{\phi}_h}{dx} = 0 \text{ in } [X_{n-1}, X_n]. \qquad (8) $$

1. Approximation $y_h$ as piecewise polynomial of degree $k_{FR}$.

2. Natural flux approximation $\phi_h = Fy_h$.

3. Numerical fluxes $\tilde{\phi}_n$ as a mean of $\phi_n^-$ and $\phi_n^+$.

4. Correction polynomials $P_n^-$ and $P_n^+$ of degree $k_{FR} + 1$.

5. Correction $\tilde{\phi}_{h,n} = \phi_{h,n} + \delta_{n-1}^- P_n^- + \delta_n^- P_n^-$.

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

<figure><img src="image_73.png" /></figure>
FR construction steps and properties

$$ \forall n \in [1, N], i \kappa y_h + \frac{d\tilde{\phi}_h}{dx} = 0 \text{ in } [X_{n-1}, X_n]. \quad (8) $$

1. Approximation $y_h$ as piecewise polynomial of degree $k_{FR}$.

2. Natural flux approximation $\phi_h = Fy_h$.

3. Numerical fluxes $\tilde{\phi}_n$ as a mean of $\phi_n^-$ and $\phi_n^+$.

4. Correction polynomials $P_n^-$ and $P_n^+$ of degree $k_{FR} + 1$.

5. Correction $\tilde{\phi}_{h,n} = \phi_{h,n} + \delta_{n-1}^+ P_n^+ + \delta_n^+ P_n^-$.

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

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

* Choice of correction polynomials allows to **retrieve usual methods**: classic choice of Radau polynomials (nodal DG).

* **Natural generalisation to 3D and adaptable to unstructured meshes.**
<figure><img src="image_75.png" /></figure>

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

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

Numerical results

<table><thead><tr><th>k<sub>FR</sub></th><th>k<sub>FR</sub> = 1</th><th>k<sub>FR</sub> = 2</th><th>k<sub>FR</sub> = 3</th></tr></thead><tbody><tr><td rowspan="2">||<sub>ε<sub>h</sub></sub>||<sup>rel</sup><sub>L<sup>2</sup>(Ω)</sub></td><td colspan="3" style="text-align: center; font-weight: bold;">1D</td></tr><tr><td>10<sup>-1</sup></td><td>10<sup>-4</sup></td><td>10<sup>-5</sup></td></tr><tr><td rowspan="2">||<sub>ε<sub>h</sub></sub>||<sup>rel</sup><sub>L<sup>2</sup>(Ω)</sub></td><td colspan="3" style="text-align: center; font-weight: bold;">3D</td></tr><tr><td>10<sup>-1</sup></td><td>10<sup>-2</sup></td><td>10<sup>-3</sup></td></tr></tbody></table>

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

<figure><img src="image_79.png" /></figure>

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

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

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

<figure><img src="image_83.png" /><figcaption>Table: h-convergence of the FR method in 1D and 3D for different correction polynomials (SD_CLo (—★—), SD_IG (—★—), FR_Rad (—★—) and FR_G2 (—★—)).</figcaption></figure>

⇒ Dependence on the choice of the correction polynomials: what is the best choice?
<figure><img src="image_84.png" /></figure>

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

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

Error estimate and optimised correction polynomials

Developed an explicit a priori error estimate with incoming BCs, i.e. $Z_1 = Z_2 = 1$ [Rivet et al. 2023]:

$$ \|\mathbf{y} - \mathbf{y}_h\|_{L^2(\Omega)}^2 \leq C(\kappa, h, L, P) (|g_1|^2 + |g_2|^2) \quad (9) $$

⇒ Choose the correction polynomial $P \to$ as [Rivet et al. 2024]

$$ P \to = \underset{\tilde{P} \in \mathrm{Padm}}{\operatorname{argmin}} C(\kappa, h, L, \tilde{P}). \quad (10) $$
<figure><img src="image_87.png" /></figure>

<figure><img src="image_88.png" /></figure>

<figure><img src="image_89.png" /></figure>

Error estimate and optimised correction polynomials

⇒ Direction-wise optimisation in the 3D case: applicable even for general BCs!

<figure><img src="image_90.png" /><figcaption>Table: General 3D simulation for Radau (---) and optimised (—) polynomials.</figcaption></figure>

⇒ Possible to find **better polynomials than the classic ones!**
<figure><img src="image_91.png" /></figure>

<figure><img src="image_92.png" /></figure>

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

# A quasi-Trefftz approach based on a Flux Reconstruction auxiliary solver
Numerical experiments: basis quality

<figure><img src="image_94.png" /><figcaption>Figure: Eigenvalues of the mass matrix for 196 PWs and 192 Quasi-Trefftz basis functions.</figcaption></figure>

Basis quality

⇒ **Adaptability** to the neighbouring dielectric parameters and BC refinement.

⇒ **Avoids spurious numerical modes and ill-conditioning** of the mass matrix!
<figure><img src="image_95.png" /></figure>

<figure><img src="image_96.png" /></figure>

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

# Numerical experiments: choice of the FR order

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_98.png" /><figcaption>Figure: Mesh $T_H(\Omega)$ for $N = 3$.</figcaption></figure>

<figure><img src="image_99.png" /><figcaption>Figure: Mesh $T_h(\partial T)$ for $M = 6$.</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_100.png" /></figure>

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

<figure><img src="image_102.png" /></figure>

# Numerical experiments: choice of the FR order

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_103.png" /><figcaption>Figure: Mesh $\mathcal{T}_H(\Omega)$ for $N = 3$.</figcaption></figure>

<figure><img src="image_104.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_105.png" /></figure>

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

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

# Numerical experiments: choice of the FR order

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_108.png" /><figcaption>Figure: Mesh $\mathcal{T}_H(\Omega)$ for $N = 3$.</figcaption></figure>

<figure><img src="image_109.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_110.png" /></figure>

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

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

# Numerical experiments: choice of the FR order

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_113.png" /><figcaption>Figure: Mesh $\mathcal{T}_H(\Omega)$ for $N = 3$.</figcaption></figure>

<figure><img src="image_114.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_115.png" /></figure>

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

<figure><img src="image_117.png" /></figure>

# Numerical experiments: choice of the FR order

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_118.png" /><figcaption>Figure: Mesh $\mathcal{T}_H(\Omega)$ for $N = 3$.</figcaption></figure>

<figure><img src="image_119.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].
Numerical experiments: mesh-convergence

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

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

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

• Need of $k_{FR} \ge k_{QT} + 2$ for quasi-optimal orders, but no interest in over-resolving!
# Numerical experiments: convergence in the number of basis functions

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

⇒ Local basis enrichment through refinement of the micro-mesh $J_h(\partial T)$.

<figure><img src="image_121.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!
# Conclusion and perspectives

* 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.

* Numerically, $k_{FR} \ge k_{QT} + 2$ for quasi-optimality, but **no need to over-resolve!**

## Perspectives:

* **Theoretical well-posedness conditions and a priori error estimates** in progress.

* Adaptable for **heterogeneous** configurations.

* Possible **preconditioning** techniques and algebraic adaptations.

* Possible **hybrid approach** by mixing with PW basis.
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