Polynomial Quasi-Trefftz Bases

• Polynomial quasi-Trefftz bases are local polynomial systems that cancel Taylor residuals, bridging classical Trefftz methods and full polynomial spaces.
• They yield reduced-dimensional approximation spaces with optimal high-order convergence in DG formulations for variable-coefficient and complex PDEs.
• Their construction employs explicit recursions or nullspace extractions, ensuring linear independence and controlled approximation properties.

Searching arXiv for recent and foundational papers on polynomial quasi-Trefftz bases. Polynomial quasi-Trefftz bases are local polynomial function systems tailored to a differential operator by enforcing that the PDE residual vanishes in a truncated Taylor sense at a chosen point, typically an element barycenter or center. They arise when classical Trefftz methods become unavailable because exact elementwise PDE solutions are inaccessible, notably for variable-coefficient problems. In this setting, the discrete functions are not exact local solutions but approximate solutions of controlled order, and the resulting spaces remain substantially smaller than full polynomial spaces while retaining the same local approximation order and high-order convergence in discontinuous Galerkin formulations (Imbert-Gérard et al., 2020, Imbert-Gérard et al., 2024, Imbert-Gerard, 24 May 2025).

1. Definition and conceptual position

Trefftz methods use discrete spaces made of elementwise exact solutions of the governing PDE. This is practical for many linear homogeneous equations with piecewise-constant coefficients, but it generally fails for smoothly varying coefficients because exact local polynomial solutions need not exist (Imbert-Gérard et al., 2020, Perinati, 2023, Imbert-Gérard et al., 2024). Polynomial quasi-Trefftz bases relax the Trefftz property by replacing exact local solvability with a Taylor-based residual cancellation condition.

For the acoustic wave equation in space-time, on each space-time element $K$, the scalar quasi-Trefftz space of degree $p$ is defined by

$${}^p(K) := \{ f\in\mathcal{P}^p(K) \mid D^\alpha(\Box_{\rho,G}f)(x_K,t_K)=0 \ \forall |\alpha|\le p-2 \},$$

where $$\Box_{\rho,G}=\nabla\cdot(\rho^{-1}\nabla)-G\partial_t^2$$, $\mathcal{P}^p(K)$ denotes polynomials of total degree $\le p$, and $(x_K,t_K)$ is the element barycenter (Imbert-Gérard et al., 2020). This enforces that the Taylor polynomial of the residual vanishes through order $p-2$.

The same idea extends to general scalar linear operators. For a linear differential operator

$L=\sum_{|\alpha|\le m}\alpha_\alpha(x)D^\alpha$

of order $m$, with $p$0, the inhomogeneous local polynomial quasi-Trefftz space is

$p$1

and the homogeneous space is $p$2 when $p$3 (Imbert-Gérard et al., 2024). A closely related abstract formulation defines the quasi-Trefftz operator

$p$4

for an operator $p$5 of order $p$6, and then sets

$p$7

making the quasi-Trefftz space explicitly the kernel of a linear map between polynomial spaces (Imbert-Gerard, 24 May 2025).

This framework places polynomial quasi-Trefftz bases between exact Trefftz spaces and full polynomial spaces. They are equation-dependent like Trefftz spaces, but they remain available for smooth variable coefficients because they are defined through local Taylor jets rather than exact closed-form solutions (Imbert-Gérard et al., 2020, Imbert-Gerard, 24 May 2025). A plausible implication is that they should be understood less as a special basis family for one PDE class than as a general local residual-annihilation mechanism for high-order discretization.

2. Local spaces, dimension, and graded structure

A recurring property is that quasi-Trefftz spaces are much smaller than the ambient polynomial spaces of the same degree. For second-order scalar operators in $p$8 dimensions, both the general elliptic framework and the diffusion-advection-reaction construction give

$p$9

so in two dimensions $${}^p(K) := \{ f\in\mathcal{P}^p(K) \mid D^\alpha(\Box_{\rho,G}f)(x_K,t_K)=0 \ \forall |\alpha|\le p-2 \},$$0 (Perinati, 2023, Lederer et al., 2024, Imbert-Gerard, 24 May 2025). This yields $${}^p(K) := \{ f\in\mathcal{P}^p(K) \mid D^\alpha(\Box_{\rho,G}f)(x_K,t_K)=0 \ \forall |\alpha|\le p-2 \},$$1 local dimension rather than $${}^p(K) := \{ f\in\mathcal{P}^p(K) \mid D^\alpha(\Box_{\rho,G}f)(x_K,t_K)=0 \ \forall |\alpha|\le p-2 \},$$2 for the full degree-$${}^p(K) := \{ f\in\mathcal{P}^p(K) \mid D^\alpha(\Box_{\rho,G}f)(x_K,t_K)=0 \ \forall |\alpha|\le p-2 \},$$3 polynomial space (Imbert-Gerard, 24 May 2025).

For the space-time wave equation, the local space has dimension

$${}^p(K) := \{ f\in\mathcal{P}^p(K) \mid D^\alpha(\Box_{\rho,G}f)(x_K,t_K)=0 \ \forall |\alpha|\le p-2 \},$$4

with $${}^p(K) := \{ f\in\mathcal{P}^p(K) \mid D^\alpha(\Box_{\rho,G}f)(x_K,t_K)=0 \ \forall |\alpha|\le p-2 \},$$5 for $${}^p(K) := \{ f\in\mathcal{P}^p(K) \mid D^\alpha(\Box_{\rho,G}f)(x_K,t_K)=0 \ \forall |\alpha|\le p-2 \},$$6, and $${}^p(K) := \{ f\in\mathcal{P}^p(K) \mid D^\alpha(\Box_{\rho,G}f)(x_K,t_K)=0 \ \forall |\alpha|\le p-2 \},$$7 for $${}^p(K) := \{ f\in\mathcal{P}^p(K) \mid D^\alpha(\Box_{\rho,G}f)(x_K,t_K)=0 \ \forall |\alpha|\le p-2 \},$$8 (Imbert-Gérard et al., 2020). For the time-dependent Schrödinger equation, the polynomial Trefftz space of anisotropic maximal degree $${}^p(K) := \{ f\in\mathcal{P}^p(K) \mid D^\alpha(\Box_{\rho,G}f)(x_K,t_K)=0 \ \forall |\alpha|\le p-2 \},$$9 satisfies

$$\Box_{\rho,G}=\nabla\cdot(\rho^{-1}\nabla)-G\partial_t^2$$0

reflecting the mixed derivative structure of the PDE and the use of anisotropic degree $$\Box_{\rho,G}=\nabla\cdot(\rho^{-1}\nabla)-G\partial_t^2$$1 (Gómez et al., 2023).

The linear-algebraic structure behind these dimensions is graded by homogeneous polynomial degree. In the general scalar framework, $$\Box_{\rho,G}=\nabla\cdot(\rho^{-1}\nabla)-G\partial_t^2$$2, and the quasi-Trefftz operator has block-triangular form with diagonal blocks given by the constant-coefficient principal part

$$\Box_{\rho,G}=\nabla\cdot(\rho^{-1}\nabla)-G\partial_t^2$$3

acting from $$\Box_{\rho,G}=\nabla\cdot(\rho^{-1}\nabla)-G\partial_t^2$$4 to $$\Box_{\rho,G}=\nabla\cdot(\rho^{-1}\nabla)-G\partial_t^2$$5 (Imbert-Gerard, 24 May 2025). Surjectivity of these principal blocks yields surjectivity of the full operator onto $$\Box_{\rho,G}=\nabla\cdot(\rho^{-1}\nabla)-G\partial_t^2$$6, and hence the dimension formula $$\Box_{\rho,G}=\nabla\cdot(\rho^{-1}\nabla)-G\partial_t^2$$7 (Imbert-Gerard, 24 May 2025).

For systems, the structure can be more elaborate. In second-order time-harmonic Maxwell, the local Taylor-based quasi-Trefftz space $$\Box_{\rho,G}=\nabla\cdot(\rho^{-1}\nabla)-G\partial_t^2$$8 is defined by

$$\Box_{\rho,G}=\nabla\cdot(\rho^{-1}\nabla)-G\partial_t^2$$9

and its dimension is

$\mathcal{P}^p(K)$0

obtained via a Helmholtz decomposition of homogeneous vector polynomial fields (Imbert-Gérard, 29 Aug 2025). This suggests that dimension formulas remain explicit even for systems, provided one can exploit an exact-sequence or graded decomposition structure.

3. Basis construction algorithms

Polynomial quasi-Trefftz bases are constructed locally, either by explicit recurrence on monomial coefficients or by nullspace extraction from a constraint matrix. The explicit-recursion route is the defining algorithmic feature of most polynomial quasi-Trefftz constructions.

For the wave equation in space-time, one prescribes Cauchy data at $\mathcal{P}^p(K)$1 from two spatial polynomial bases,

$\mathcal{P}^p(K)$2

or

$\mathcal{P}^p(K)$3

and writes

$\mathcal{P}^p(K)$4

The constraints $\mathcal{P}^p(K)$5 for $\mathcal{P}^p(K)$6 become a linear recurrence among the coefficients $\mathcal{P}^p(K)$7, allowing one to solve for $\mathcal{P}^p(K)$8 from already known lower-time-derivative coefficients (Imbert-Gérard et al., 2020). Proposition 4.3 in that work proves that the resulting functions are linearly independent and span the local quasi-Trefftz space (Imbert-Gérard et al., 2020).

For general scalar equations, the explicit algorithm of the elliptic framework assumes a non-degeneracy condition on the principal part, namely that the coefficient of $\mathcal{P}^p(K)$9 does not vanish at the expansion point: $\le p$0 One then chooses $\le p$1 Cauchy-data polynomials $\le p$2 for $\le p$3, which determine all monomial coefficients with $\le p$4, and recursively computes the remaining coefficients $\le p$5 from derivatives of $\le p$6 and of the operator coefficients at $\le p$7 (Imbert-Gérard et al., 2024). Proposition 2.3 states that all coefficients are determined uniquely (Imbert-Gérard et al., 2024).

The systematic scalar theory generalizes this further. It decomposes each homogeneous block into a free-data part $\le p$8 and a complementary part $\le p$9 on which the principal block is invertible, and then applies a forward-substitution procedure degree by degree. The resulting algorithm produces a minimal basis of $(x_K,t_K)$0 in any dimension and for any equation order $(x_K,t_K)$1 (Imbert-Gerard, 24 May 2025). The cost model given there is asymptotically

$(x_K,t_K)$2

which is substantially below a naive dense solve on the full polynomial space (Imbert-Gerard, 24 May 2025).

An alternative is to define the local space as the kernel of a weak constraint operator and compute a basis by SVD. In the embedded Trefftz DG method for Helmholtz, one starts from $(x_K,t_K)$3, enforces

$(x_K,t_K)$4

assembles the local constraint matrix $(x_K,t_K)$5, and extracts the nullspace basis vectors from a thin singular-value decomposition $(x_K,t_K)$6 (Stocker et al., 13 Mar 2026). This avoids explicit symbolic basis construction. Although the paper is framed as an embedded Trefftz strategy rather than a Taylor-based construction, its local nullspace basis functions are explicitly described as quasi-Trefftz functions (Stocker et al., 13 Mar 2026).

4. Approximation theory

The central approximation fact is that the Taylor polynomial of an exact solution belongs to the local quasi-Trefftz space. This gives best-approximation estimates with the same local orders as full polynomials.

For the space-time wave equation, if $(x_K,t_K)$7 solves $(x_K,t_K)$8 smoothly on $(x_K,t_K)$9, then its $p-2$0-st Taylor polynomial $p-2$1 belongs to $p-2$2, and

$p-2$3

with an analogous estimate in a wavespeed-weighted norm $p-2$4 (Imbert-Gérard et al., 2020).

For general elliptic problems, Theorem 2.3 states that if $p-2$5 solves $p-2$6 and $p-2$7 is star-shaped with respect to $p-2$8, then $p-2$9 and, for $L=\sum_{|\alpha|\le m}\alpha_\alpha(x)D^\alpha$0,

$L=\sum_{|\alpha|\le m}\alpha_\alpha(x)D^\alpha$1

This shows that the quasi-Trefftz space retains full polynomial approximation order despite its reduced dimension (Imbert-Gérard et al., 2024).

The same conclusion is stated in the scalar unified theory: if $L=\sum_{|\alpha|\le m}\alpha_\alpha(x)D^\alpha$2 is an exact solution of $L=\sum_{|\alpha|\le m}\alpha_\alpha(x)D^\alpha$3 and is $L=\sum_{|\alpha|\le m}\alpha_\alpha(x)D^\alpha$4 smooth, then $L=\sum_{|\alpha|\le m}\alpha_\alpha(x)D^\alpha$5, with local estimates

$L=\sum_{|\alpha|\le m}\alpha_\alpha(x)D^\alpha$6

The contrast with full polynomials is explicit there: quasi-Trefftz spaces reduce the local dimension to $L=\sum_{|\alpha|\le m}\alpha_\alpha(x)D^\alpha$7 but remain tailored only to solutions of $L=\sum_{|\alpha|\le m}\alpha_\alpha(x)D^\alpha$8 (Imbert-Gerard, 24 May 2025).

For polynomial Trefftz spaces in the time-dependent Schrödinger equation, the situation is exact rather than quasi-Trefftz, but it is directly relevant because it demonstrates the same philosophy with a different degree notion. Using extended averaged Taylor polynomials of total anisotropic degree $L=\sum_{|\alpha|\le m}\alpha_\alpha(x)D^\alpha$9, the Trefftz-DG approximation achieves optimal order $m$0 while the dimension equals that of $m$1 (Gómez et al., 2023). This indicates that degree-counting must match the derivative structure of the PDE; in quasi-Trefftz settings this principle reappears in the choice of residual truncation order.

5. DG formulations and convergence

Polynomial quasi-Trefftz bases are typically inserted into discontinuous Galerkin schemes in which all volume terms are either eliminated or controlled through the residual structure. The resulting methods are well posed, quasi-optimal, and high-order convergent under standard mesh regularity and coefficient smoothness assumptions.

For the first-order acoustic initial-boundary value problem, the space-time quasi-Trefftz DG method of Imbert-Gérard, Moiola, and Stocker proves that the DG bilinear form is coercive in a mesh-dependent norm, continuous in a stronger paired norm, and that the discrete solution $m$2 satisfies the quasi-optimal error bound

$m$3

With $m$4, one obtains

$m$5

and in an $m$6-type final-time norm,

$m$7

The paper states that these are the same $m$8-convergence rates as a full polynomial space or a classical Trefftz space in the constant-coefficient case (Imbert-Gérard et al., 2020).

For variable-coefficient diffusion-advection-reaction equations, the DG formulation combines the usual SIPG diffusion form with upwind advection-reaction terms. Theorem 4.1 proves coercivity, boundedness, and well-posedness, while Theorem 5.2 gives

$m$9

for the quasi-Trefftz DG method (Imbert-Gérard et al., 2024). The earlier diffusion-advection-reaction study states the analogous quasi-optimal estimate in a DG norm and concludes $p$00 in the DG norm and $p$01 in $p$02 for smooth solutions (Perinati, 2023).

The unified Trefftz-like framework of Lederer and collaborators interprets these methods through local/global decomposition. For second-order operators, it gives a Strang-type estimate in a broken energy norm and, under extra $p$03-$p$04 regularity of the dual problem, an Aubin-Nitsche estimate

$p$05

while retaining the local dimension formula

$p$06

The same framework also interprets quasi-Trefftz elimination as a static-condensation-like restriction from the ambient polynomial space to a smaller local kernel space (Lederer et al., 2024).

For embedded Trefftz DG for Helmholtz, the local space is defined by weak constraints rather than Taylor truncation. Nonetheless, the global SIPDG formulation is analyzed by a $p$07-coercivity argument and a Schatz-type duality technique, leading to wavenumber-explicit stability and quasi-optimality under

$p$08

An $p$09-error bound of order $p$10 relative to best DG approximation is also derived (Stocker et al., 13 Mar 2026). This is not a Taylor-based polynomial quasi-Trefftz basis in the strict sense, but it clarifies that locally constrained polynomial subspaces can support Trefftz-like DG analysis even when no exact polynomial solutions exist.

6. Variants, related constructions, and scope

Polynomial quasi-Trefftz bases form one branch of a broader class of Trefftz-like spaces for variable-coefficient problems. Related constructions include generalized plane waves, polynomial Trefftz spaces with anisotropic degree, and first-order or vector-valued quasi-Trefftz spaces.

Amplitude-based generalized plane waves for variable-coefficient Helmholtz use an ansatz

$p$11

with a polynomial amplitude $p$12, and impose a quasi-Trefftz property of order $p$13 by demanding that the Taylor expansion of the residual vanish through total degree $p$14 (Imbert-Gerard, 2020). The construction is triangular and local, and the interpolation theorem proves $p$15 approximation with $p$16 directions in two dimensions (Imbert-Gerard, 2020). The 2020 space-time quasi-Trefftz wave paper explicitly states that its technique is inspired by generalized plane waves previously developed for time-harmonic problems with variable coefficients, but that in the time-domain case the approach allows for polynomial basis functions (Imbert-Gérard et al., 2020).

For the 3D convected Helmholtz equation, three quasi-Trefftz families are studied, including a purely polynomial basis. There the polynomial ansatz $p$17 with $p$18 yields a $p$19-dimensional family, and the approximation theorem states that with $p$20 basis functions one obtains

$p$21

for exact solutions of the convected Helmholtz operator (Imbert-Gerard et al., 2022). The paper also emphasizes that the polynomial basis does not suffer from the ill-conditioning inherent to wave-like bases (Imbert-Gerard et al., 2022).

For first-order systems, the 2025 Helmholtz formulation defines a polynomial quasi-Trefftz space of triples $p$22 by Taylor truncation of the first-order residuals. The space is equivalent to a scalar quasi-Trefftz condition for $p$23 combined with the exact relation $p$24, and its dimension is

$p$25

Two explicit $p$26 basis-construction algorithms are given, one coupled and one decoupled (Imbert-Gérard et al., 10 Sep 2025). This indicates that quasi-Trefftz methodology extends naturally from scalar second-order PDEs to first-order systems once compatibility conditions are handled explicitly.

Vector-valued polynomial quasi-Trefftz spaces for second-order time-harmonic Maxwell provide a further extension. Their basis construction depends on an adequate Helmholtz decomposition for homogeneous vector polynomial fields, with arbitrary choices of harmonic components $p$27 carrying $p$28 degrees of freedom at each level (Imbert-Gérard, 29 Aug 2025). A plausible implication is that exact-sequence technology may be central for quasi-Trefftz spaces of PDE systems, just as scalar graded decomposition is central for single-equation operators.

7. Numerical behavior, advantages, and limitations

The main numerical advantage repeatedly reported is higher accuracy for comparable numbers of degrees of freedom, due to the fact that the local space is adapted to the PDE while remaining much smaller than the full polynomial space (Perinati, 2023, Imbert-Gérard et al., 2024). For the 3D diffusion-dominated elliptic test in $p$29, the 2024 elliptic paper reports

$p$30

exactly as for full $p$31-DG, but with $p$32 fewer degrees of freedom and lower condition numbers (Imbert-Gérard et al., 2024). In two-dimensional advection-dominated tests, both full polynomial DG and quasi-Trefftz DG capture internal layers with small oscillations, and on an L-shaped domain the quasi-Trefftz DG solution with $p$33 differs from full-polynomial DG only by $p$34 in most of $p$35 (Imbert-Gérard et al., 2024).

The diffusion-advection-reaction study likewise states that the quasi-Trefftz space has smaller dimension than the full polynomial space of the same degree while yielding the same optimal convergence rates (Perinati, 2023). In the Schrödinger setting, careful scaling can reduce the stiffness-matrix condition number from $p$36 to $p$37, and the reduced Trefftz dimension yields a substantial reduction in the number of degrees of freedom without sacrificing accuracy (Gómez et al., 2023).

Conditioning is a recurrent point of comparison with oscillatory bases. For the 3D convected Helmholtz problem, the condition number of the normal matrix built from the monomial quasi-Trefftz basis grows only algebraically in $p$38, approximately $p$39 at $p$40, whereas GPW or plane-wave bases exhibit exponential ill-conditioning, approximately $p$41 at $p$42 (Imbert-Gerard et al., 2022). The amplitude-based GPW work makes a related comparison: amplitude-based GPWs and phase-based GPWs achieve the same final slope $p$43, but the amplitude-based ansatz is significantly more stable in the pre-asymptotic regime $p$44 (Imbert-Gerard, 2020).

Several limitations are equally clear. The basis construction generally requires smooth coefficients and local Taylor jets of the operator coefficients, and explicit recursive construction may rely on a non-degeneracy condition such as $p$45 (Imbert-Gérard et al., 2024). The approximation theory is PDE-specific: quasi-Trefftz spaces preserve optimal order only for exact solutions of the underlying operator, not for arbitrary smooth functions (Imbert-Gerard, 24 May 2025). For non-homogeneous problems, one may need a local quasi-Trefftz particular solution and then solve for the difference (Imbert-Gérard et al., 2024). Finally, while exact Trefftz formulations often eliminate all volume terms, quasi-Trefftz formulations generally require control of residual contributions; in the unified viewpoint, this is handled through local constraints and DG coupling rather than exact elementwise annihilation of the operator (Lederer et al., 2024).

Taken together, the literature presents polynomial quasi-Trefftz bases as a general local Taylor-based technology for constructing reduced high-order spaces adapted to variable-coefficient PDEs. The core ingredients are a kernel definition via residual truncation, a graded or recursive basis construction, explicit dimension formulas, Taylor-based best approximation, and DG formulations that recover the same $p$46-convergence rates as much larger ambient polynomial spaces (Imbert-Gérard et al., 2020, Imbert-Gérard et al., 2024, Imbert-Gerard, 24 May 2025).