Three types of quasi-Trefftz functions for the 3D convected Helmholtz equation: construction and approximation properties

TL;DR

This work addresses constructing local quasi-Trefftz bases for the 3D convected Helmholtz equation to enable Trefftz-like discretizations in variable-coefficient media. It introduces two generalized plane-wave families (amplitude-based GPWs and phase-based GPWs) and a fully polynomial quasi-Trefftz basis, providing explicit, layer-by-layer algorithms to enforce the quasi-Trefftz property with order $q$ and dimension $p=(n+1)^2$. The paper proves that each space achieves the target local approximation order $O(h^{n+1})$ for solutions of $\mathcal{L}_c u=0$, and shows that the polynomial basis avoids the ill-conditioning typical of wave-like bases while requiring far fewer degrees of freedom than standard polynomials. Numerical results confirm the predicted convergence and demonstrate the polynomial basis remains well-conditioned as $n$ grows, underscoring its practical potential for aero-acoustic simulations and other variable-coefficient wave problems.

Abstract

Trefftz methods are numerical methods for the approximation of solutions to boundary and/or initial value problems. They are Galerkin methods with particular test and trial functions, which solve locally the governing partial differential equation (PDE). This property is called the Trefftz property. Quasi-Trefftz methods were introduced to leverage the advantages of Trefftz methods for problems governed by variable coefficient PDEs, by relaxing the Trefftz property into a so-called quasi-Trefftz property: test and trial functions are not exact solutions but rather local approximate solutions to the governing PDE. In order to develop quassi-Trefftz methods for aero-acoustics problems governed by the convected Helmholtz equation, the present work tackles the question of the definition, construction and approximation properties of three families of quasi-Trefftz functions: two based on generalizations on plane wave solutions, and one polynomial. The polynomial basis shows significant promise as it does not suffer from the ill-conditioning issue inherent to wave-like bases.

Figures (7)

• Figure 1: Illustration of a multi-index layer in $(\mathbb N_0)^3$. Within the layer $\{{\mathfrak n}\in\mathbb N_0^3, |{\mathfrak n}|=\ell\}$ for $\ell =7$, represented in gray, indices with $i_1\in\{0,1\}$ are represented in red while those with $i_1\geq 2$ are represented in blue. Indices with $|i|\leq 1$ are also represented in magenta.
• Figure 2: Illustration of a multi-index layer in $(\mathbb N_0)^3$. The layer $\{{\mathfrak n}\in\mathbb N_0^3, |{\mathfrak n}|=\ell\}$ for $\ell =7$ is represented in blue. All elements in the layer are represented as blue dots, the element ${\mathfrak n}=(3,1,3)$ is highlighted in white.
• Figure 3: Two representations of a layer of indices $i$ of the unknowns of the $\ell$th subsystem for $\ell = 5$, corresponding to the blue layer in \ref{['fig:IndexLayer']}. Each grid point corresponds to one index $i\in(\mathbb N_0)^3$ with $|i|=\ell+2$. Left: Indices of the unknowns involved in equation $\beta = (3,1,1)$ are highlighted with black circles. The index $\beta+2e_1 = (5,1,1)$ is highlighted as a blue diamond. Right: Indices corresponding to $\beta+2e_1$ for all $\beta\in(\mathbb N_0)^3$ such that $|\beta|=\ell$.
• Figure 4: Local approximation of an exact solution from quasi-Trefftz bases: convergence results for the first test case, where PW functions are exact solutions, for $n$ from $1$ to $8$. For each value of $n$, the expected order of convergence, namely $n+1$, is observed and the error decreases until it reaches a threshold. Comparison of a PW and GPW bases using the same initialization, for amplitude-based (top) and phase-based (bottom) GPWs.
• Figure 5: Local approximation of an exact solution from quasi-Trefftz bases: convergence results for the first test case, where PW functions are exact solutions, for $n$ from $1$ to $8$. For each value of $n$, the expected order of convergence, namely $n+1$, is observed. Comparison of a PW and the polynomial quasi-Trefftz bases using the same initialization, evidencing the absence of conditioning problem with the polynomial basis.

Theorems & Definitions (44)

• Remark 1
• Definition 1
• Remark 2
• Remark 3
• Lemma 1
• Remark 4
• Remark 5
• Lemma 2
• proof
• Lemma 3