Trefftz methods with evanescent plane waves

1. Problem Statement

The paper addresses the numerical instability inherent in classical Trefftz methods used to solve the Helmholtz equation ($\Delta u + \kappa^2 u = 0$).

• Context: Trefftz methods approximate solutions using basis functions that are exact solutions to the PDE. The standard approach uses Propagative Plane Waves (PPWs) ($e^{i\kappa d \cdot x}$ where $d \in \mathbb{R}^n$).
• The Issue: While PPWs offer fast convergence rates, they suffer from severe numerical instabilities. Theoretical results (Theorem 1) demonstrate that for certain Helmholtz solutions (specifically those involving "evanescent modes"), accurate approximation using PPWs requires coefficients with exponentially large Euclidean norms.
• Consequence: In finite-precision arithmetic, these large coefficients lead to inevitable numerical cancellation and ill-conditioning, preventing the stable representation of the solution regardless of linear algebra techniques (like preconditioning).

2. Methodology

The authors propose replacing the standard PPW basis with Evanescent Plane Waves (EPWs) and implementing a specific sampling strategy within the Ultraweak Variational Formulation (UWVF).

A. Evanescent Plane Waves (EPWs)

Instead of restricting the propagation vector $d$ to the real unit sphere, the method allows $d \in \mathbb{C}^n$ such that $d \cdot d = 1$.

• Definition: An EPW is defined as $e^{i\kappa d \cdot x}$, where $d = \zeta p + i \eta e$.
  • $p$: Propagation direction ($p \in S^{n-1}$).
  • $e$: Evanescence direction ($e \in S^{n-1}$, orthogonal to $p$).
  • $\eta$: Evanescence strength ($\eta \ge 0$).
  • $\zeta = \sqrt{1+\eta^2}$: Apparent wavenumber scaling.
• Behavior: The wave oscillates in direction $p$ with wavenumber $\kappa\zeta$ and decays exponentially in direction $e$ with rate $\kappa\eta$.
• Theoretical Basis: Unlike PPWs, any Helmholtz solution in a bounded domain can be represented as a continuous superposition of EPWs (via a Herglotz transform) with bounded coefficients. This ensures that accurate approximations do not require unstable, large coefficients.

B. Sampling Strategy (2D Recipe)

To construct a discrete EPW space with a fixed "Degrees of Freedom" (DOF) budget $P$, the authors propose a specific sampling recipe:

1. Parameters: Define a Fourier truncation level $L = P/4$.
2. Sampling Domain: Sample points $(\theta, \phi, \xi)$ from a uniform measure on $Y = [0, 2\pi) \times \{\pm 1\} \times [0, 1]$.
   • $\theta$: Polar angle for propagation direction.
   • $\phi$: Sign for the decay direction.
   • $\xi$: Parameter for evanescence strength.
3. Mapping: Convert samples to physical parameters:
   • $\zeta_p = \max\left(1, \frac{2L}{\kappa \text{diam}(K)}\xi_p\right)$
   • $\eta_p = \sqrt{\zeta_p^2 - 1}$
4. Adaptivity: If the DOF budget is low ($2L \le \kappa \text{diam}(K)$), the method defaults to pure PPWs ($\eta=0$). As $P$ increases, the basis automatically includes more evanescent modes to capture complex solution features.

C. Numerical Implementation (UWVF with Regularization)

The method is implemented using the Ultraweak Variational Formulation (UWVF):

• Oversampling: To handle near-linear dependence among basis functions, the system is oversampled (e.g., 10% more test functions than trial functions).
• Regularization: The diagonal block matrix $D$ of the linear system is regularized using Truncated Singular Value Decomposition (SVD). Small singular values (below a threshold $\epsilon$) are discarded, and a pseudo-inverse is computed. This is done locally (element-wise) and in parallel.
• System: The final linear system $(I - D^\dagger_\epsilon C)u_\epsilon = D^\dagger_\epsilon b$ is solved to obtain the solution.

3. Key Contributions

1. Theoretical Justification: The paper reinforces the theoretical limitation of PPWs (Theorem 1) and establishes EPWs as a stable alternative by leveraging the boundedness of the Herglotz transform coefficients.
2. Practical Algorithm: It provides a concrete, simple "recipe" for selecting EPW parameters in 2D (and a reference for 3D) that balances propagative and evanescent modes based on the mesh size and wavenumber.
3. Stabilization Technique: It combines oversampling with element-wise SVD regularization to robustly solve the resulting ill-conditioned linear systems without losing accuracy.

4. Numerical Results

The authors validate the method against three test cases using the UWVF:

• Point Source (Fundamental Solution):
  • Setup: A singularity located near the boundary ($\kappa = 128$).
  • Result: PPW convergence stalls due to instability as DOFs increase. EPWs achieve errors $10^6$ times smaller than PPWs for the same number of DOFs.
• Wavenumber Influence:
  • Setup: Increasing $\kappa$ while scaling the approximation space size linearly ($P = 4\kappa$).
  • Result: EPW error decreases as $\kappa$ increases, whereas PPW error remains high or diverges.
• Scattering Problem:
  • Setup: Scattering by a non-convex, sound-soft obstacle with corners and a cavity ($\kappa = 16$).
  • Result: PPW convergence stagnates early. EPWs successfully resolve the solution with significantly higher accuracy, handling corner singularities and cavity effects effectively.

5. Significance

This work represents a significant advancement in high-frequency wave propagation simulation. By shifting from purely propagative bases to Evanescent Plane Waves, the authors overcome the fundamental numerical barrier that has limited the accuracy of Trefftz methods.

• Stability: The method allows for high-accuracy solutions in regimes where classical Trefftz methods fail due to numerical noise.
• Efficiency: It achieves high accuracy without requiring excessively large meshes or polynomial degrees, maintaining the exponential convergence rates characteristic of Trefftz schemes.
• Applicability: The proposed sampling strategy and regularization make the method robust for complex geometries, high wavenumbers, and singular solutions (like point sources or scattering corners).

In summary, the paper demonstrates that Evanescent Plane Waves are not just a theoretical curiosity but a practical necessity for stable, high-order Trefftz discretizations of the Helmholtz equation.