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Variational Principles for the Helmholtz equation: application to Finite Element and Neural Network approximations

G. Makrakis, C. Makridakis, D. Mitsoudis, M. Plexousakis, T. Pryer

TL;DR

This work introduces variational principles for the Helmholtz equation with impedance boundary conditions by deriving time-harmonic energies from Hamilton’s principle and augmenting the indefinite physical energy with a least-squares residual term to obtain strong coercivity. The authors prove coercivity using Rellich/Morawetz identities and provide a unified framework that supports conforming finite-element discretisations and neural-network-based approximations, with explicit parameter choices that guarantee stability independent of the wavenumber $k$. The key contributions include a coercive regularised energy $\mathcal E_\gamma$ (and its boundary-penalised variant $\mathcal F_\gamma$), rigorous coercivity bounds, and prototype discretisations demonstrating robustness for high-$k$ regimes. The results have practical impact by offering a stable variational basis for efficient FE and NN solvers for oscillatory Helmholtz problems, addressing a long-standing challenge in accurate high-$k$ approximations.

Abstract

In this paper, we investigate whether Variational Principles can be associated with the Helmholtz equation subject to impedance (absorbing) boundary conditions. This model has been extensively studied in the literature from both mathematical and computational perspectives. It is classical with wide applications, yet accurate approximation at high wavenumbers remains challenging. We address the question of whether there exist energy functionals with a clear physical interpretation whose stationary points, the zeros of their first variation, correspond to solutions of the Helmholtz problem. Starting from Hamilton's principle for the wave equation, we derive time-harmonic energies. The resulting functionals are generally indefinite. As a next step, we construct strongly coercive augmentations of these indefinite functionals that preserve their physical interpretation. Finally, we show how these variational principles lead to practical numerical methods based on finite element spaces and neural network architectures.

Variational Principles for the Helmholtz equation: application to Finite Element and Neural Network approximations

TL;DR

This work introduces variational principles for the Helmholtz equation with impedance boundary conditions by deriving time-harmonic energies from Hamilton’s principle and augmenting the indefinite physical energy with a least-squares residual term to obtain strong coercivity. The authors prove coercivity using Rellich/Morawetz identities and provide a unified framework that supports conforming finite-element discretisations and neural-network-based approximations, with explicit parameter choices that guarantee stability independent of the wavenumber . The key contributions include a coercive regularised energy (and its boundary-penalised variant ), rigorous coercivity bounds, and prototype discretisations demonstrating robustness for high- regimes. The results have practical impact by offering a stable variational basis for efficient FE and NN solvers for oscillatory Helmholtz problems, addressing a long-standing challenge in accurate high- approximations.

Abstract

In this paper, we investigate whether Variational Principles can be associated with the Helmholtz equation subject to impedance (absorbing) boundary conditions. This model has been extensively studied in the literature from both mathematical and computational perspectives. It is classical with wide applications, yet accurate approximation at high wavenumbers remains challenging. We address the question of whether there exist energy functionals with a clear physical interpretation whose stationary points, the zeros of their first variation, correspond to solutions of the Helmholtz problem. Starting from Hamilton's principle for the wave equation, we derive time-harmonic energies. The resulting functionals are generally indefinite. As a next step, we construct strongly coercive augmentations of these indefinite functionals that preserve their physical interpretation. Finally, we show how these variational principles lead to practical numerical methods based on finite element spaces and neural network architectures.

Paper Structure

This paper contains 23 sections, 12 theorems, 128 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Helmholtz equation and Variational Principles
  3. Energy principles
  4. The basic model: Transverse vibrations of a string
  5. Variational derivation of the wave equation
  6. Variational derivation of the Helmholtz equation
  7. Impedance boundary conditions
  8. Variational principles for the Helmholtz equation
  9. Notation and preliminaries
  10. Variational Principles
  11. Proofs for strongly coercive variational principles
  12. Design strategy
  13. Rellich and Morawetz Identities
  14. Energies and coercivity bounds
  15. Energy with Weakly Enforced Boundary Conditions
  16. ...and 8 more sections

Key Result

Proposition 1

Consider the energy $\mathcal{E}_P$ defined in Energy_H_main_s2. The stationary points of $\mathcal{E}_P$ defined on ${\mathscr H}$ satisfy the Helmholtz equation Helm_distr in the sense of distributions. Furthermore, if the energy $\mathcal{E}_P$ restricted on ${\mathscr V}_{BC}$ has a stationary p

Figures (2)

  • Figure 1: Real part of the $H^2$-conforming Argyris finite element solution on $\Omega = [0,1]^2$ with $h = 2^{-7}$ for wave numbers $k \in \{10,50,100,200\}$.
  • Figure 2: Plane-wave neural network approximation on $\Omega=[0,1]^2$ at $k=100$ with two Gaussian sources at $(\tfrac{1}{2},\tfrac{1}{2})$ and $(\tfrac{3}{4},\tfrac{1}{4})$ (relative amplitude $\tfrac{4}{5}$). Panels show $\Re(u)$ at early, mid and late training iterations.

Theorems & Definitions (20)

  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Remark 4
  • Proposition 5: Rellich/Morawetz Identity
  • Remark 6
  • Proposition 7: Low-order Morawetz Identity
  • Remark 8
  • Remark 9
  • Lemma 10: Energy Identity with Penalty
  • ...and 10 more