A quasi-Trefftz space for a second order time-harmonic Maxwell's equation

TL;DR

This paper introduces a novel local Taylor-based polynomial quasi-Trefftz space tailored for electromagnetic wave propagation in inhomogeneous media, enabling explicit construction and dimension analysis for Maxwell's equations.

Contribution

It is the first to develop and analyze a quasi-Trefftz space for a system of PDEs, specifically for second order Maxwell's equations with variable coefficients.

Findings

Explicit dimension of the quasi-Trefftz space derived

Procedure for constructing quasi-Trefftz functions established

Application to electromagnetic wave propagation in inhomogeneous media

Abstract

Quasi-Trefftz methods are a family of Discontinuous Galerkin methods relying on equation-dependent function spaces. This work is the first study of the notion of local Taylor-based polynomial quasi-Trefftz space for a system of Partial Differential Equations (PDEs). These discrete spaces are introduced here for electro-magnetic wave propagation in inhomogeneous media, governed by a second order formulation of Maxwell's equation with variable coefficients. Thanks to an adequate Helmholtz decomposition for spaces of homogeneous polynomial vector fields, the outcome is the explicit dimension of the proposed quasi-Trefftz space as well as a procedure to construct quasi-Trefftz functions.

Equations

$$e^{\text{i}\kappa {\mathbf{d}}_1\cdot \mathbf{x}}-e^{\text{i}\kappa {\mathbf{d}}_2\cdot \mathbf{x}}=2\text{i}\exp \left(\frac{\text{i}\kappa ({\mathbf{d}}_1+{\mathbf{d}}_2)\cdot \mathbf{x}}{2}\right)\sin \frac{\kappa ({\mathbf{d}}_1-{\mathbf{d}}_2)\cdot \mathbf{x}}{2}\to 0\text{ as }|{\mathbf{d}}_1-{\mathbf{d}}_2|\to 0,$$

$$\nabla \times \nabla \times \mathbf{V}-ϵ\mathbf{V}=0,$$

$$\nabla \cdot (ϵ\mathbf{V})=\sum _{g^{\prime }}\sum _g{\partial }_g({ϵ}_{g{g}^{\prime }})V_{g^{\prime }}+\sum _g\sum _{g^{\prime }}{ϵ}_{g{g}^{\prime }}{\partial }_g(V_{g^{\prime }}).$$

$$\forall \mathbf{i}\in ({\mathbb{N}}_0{)}^3,{\mathbf{X}}^{\mathbf{i}}=\prod _{g=1}^3(X_g{)}^{i_g}\text{ and }\forall l\in {\mathbb{N}}_0,{\tilde{\mathbb{P}}}_l=Span\left\{{\mathbf{X}}^{\mathbf{i}},\mathbf{i}\in ({\mathbb{N}}_0{)}^3,|\mathbf{i}|=l\right\}.$$

$$\forall l\in {\mathbb{N}}_0,{\mathbb{P}}_p=\underset{l=0}{\overset{p}{⨁}}{\tilde{\mathbb{P}}}_l\text{ and }({\mathbb{P}}_p{)}^3=\underset{l=0}{\overset{p}{⨁}}({\tilde{\mathbb{P}}}_l{)}^3.$$

$$\forall p\in {\mathbb{N}}_0,\dim {\tilde{\mathbb{P}}}_p=\frac{1}{2}(p+1)(p+2)\text{ and }\dim {\mathbb{P}}_p=\frac{1}{6}(p+1)(p+2)(p+3).$$

$$T_k[\phi ]=\sum _{l=0}^k\sum _{|\mathbf{i}|=l}\frac{1}{\mathbf{i}!}{\partial }^{\mathbf{i}}\phi ({\mathbf{x}}_0)(\mathbf{X}-{\mathbf{x}}_0{)}^{\mathbf{i}}.$$

$$T_q[\mathcal{L}\phi ]=0.$$

$$\mathbb{Q}\mathbb{T}_{p}=\{\Pi\in\mathbb{P}_{p}|T_{p-\gamma}[\mathcal{L}\Pi]=0_{\mathbb{P}_{p-\gamma}}\}\text{ and }\begin{array}[]{rll}\mathcal{D}_{p}:&\mathbb{P}_{p}&\to\mathbb{P}_{p-\gamma}\\ &\Pi&\mapsto T_{p-\gamma}[\mathcal{L}\Pi]\end{array}$$

$$\begin{array}[]{rrcl}Grad_{p}:&\mathbb{P}_{p+1}&\to&(\mathbb{P}_{p})^{3},\\ &\Phi&\mapsto&\nabla\Phi,\end{array}\quad\begin{array}[]{rrcl}Div_{p}:&(\mathbb{P}_{p+1})^{3}&\to&\mathbb{P}_{p},\\ &\mathbf{\Phi}&\mapsto&\nabla\cdot\mathbf{\Phi},\end{array}\text{ and }\begin{array}[]{rrcl}Curl_{p}:&(\mathbb{P}_{p+1})^{3}&\to&(\mathbb{P}_{p})^{3},\\ &\mathbf{\Phi}&\mapsto&\nabla\times\mathbf{\Phi}.\end{array}$$

$$\begin{array}[]{rrcl}Lap_{p}:&\mathbb{P}_{p+2}&\to&\mathbb{P}_{p},\\ &\Phi&\mapsto&\Delta\Phi,\end{array}\text{ and }\begin{array}[]{rrcl}\overrightarrow{Lap}_{p}:&(\mathbb{P}_{p+2})^{3}&\to&(\mathbb{P}_{p})^{3},\\ &\mathbf{\Phi}&\mapsto&\overrightarrow{\Delta}\mathbf{\Phi},\end{array}$$

$$\mathcal{T}:\underset{l=0}{\overset{p+\gamma }{⨁}}{\left({\tilde{\mathbb{P}}}_l\right)}^n\to \underset{k=0}{\overset{p}{⨁}}{\left({\tilde{\mathbb{P}}}_k\right)}^m,\text{ for }n,m\in \{1,3\},$$

$$\mathcal{T}\left(\sum _{k=0}^{p+\gamma }{x}_k\right)={\mathcal{T}}_{\ast }\left(\sum _{k=0}^{\gamma -1}{x}_k\right)+\sum _{k=0}^p{\mathcal{T}}_k\left(x_k\right)\text{ with }{\mathcal{T}}_{\ast }:({\mathbb{P}}_{\gamma -1}{)}^n\to \{(0{)}^m\}\text{ and }{\mathcal{T}}_k:{\left({\tilde{\mathbb{P}}}_{k+\gamma }\right)}^n\to {\left({\tilde{\mathbb{P}}}_k\right)}^m$$

$$\mathcal{R}(\mathcal{T})=\underset{0\le k\le p}{⨁}\mathcal{R}({\mathcal{T}}_k)$$

$$\begin{array}[]{rrcl}G_{k}:&\widetilde{\mathbb{P}}_{k+1}&\to&\left(\widetilde{\mathbb{P}}_{k}\right)^{3},\\ &\Phi&\mapsto&\nabla\Phi,\end{array}\text{ and }\begin{array}[]{rrcl}G_{*}:&\widetilde{\mathbb{P}}_{0}&\to&\{\mathbf{0}^{3}\},\\ &\Phi&\mapsto&\mathbf{0}^{3}.\end{array}$$

$$\left\{\left(\begin{array}{c}{i}_1{\mathbf{X}}^{\mathbf{i}-{\mathbf{e}}_1},i_2{\mathbf{X}}^{\mathbf{i}-{\mathbf{e}}_2},i_3{\mathbf{X}}^{\mathbf{i}-{\mathbf{e}}_3}\end{array}\right),\mathbf{i}\in {\mathbb{N}}_0^3,|\mathbf{i}|=k+1\right\},$$

$$\dim \ker (G_k)=0\text{ and }rk(G_k)=\frac{1}{2}(k+2)(k+3).$$

$$\left\{\left(\begin{array}{c}{i}_1{\mathbf{X}}^{\mathbf{i}-{\mathbf{e}}_1},i_2{\mathbf{X}}^{\mathbf{i}-{\mathbf{e}}_2},i_3{\mathbf{X}}^{\mathbf{i}-{\mathbf{e}}_3}\end{array}\right),\mathbf{i}\in {\mathbb{N}}_0^3,|\mathbf{i}|=k+1\right\},$$

$$\sum _{|\mathbf{i}|=k+1}{\alpha }_{\mathbf{i}}\nabla {\mathbf{X}}^{\mathbf{i}}=0\Rightarrow \nabla \left(\sum _{|\mathbf{i}|=k+1}{\alpha }_{\mathbf{i}}{\mathbf{X}}^{\mathbf{i}}\right)=0\Rightarrow \sum _{|\mathbf{i}|=k+1}{\alpha }_{\mathbf{i}}{\mathbf{X}}^{\mathbf{i}}=0\Rightarrow \forall \mathbf{i}\text{ s.t. }|\mathbf{i}|=k+1,{\alpha }_{\mathbf{i}}=0$$

$$\begin{array}[]{rrcl}D_{k}:&\left(\widetilde{\mathbb{P}}_{k+1}\right)^{3}&\to&\widetilde{\mathbb{P}}_{k},\\ &\mathbf{\Phi}&\mapsto&\nabla\cdot\mathbf{\Phi},\end{array}\text{ and }\begin{array}[]{rrcl}D_{*}:&\left(\widetilde{\mathbb{P}}_{0}\right)^{3}&\to&\{\mathbf{0}\},\\ &\mathbf{\Phi}&\mapsto&\mathbf{0}.\end{array}.$$

$$\dim \ker (D_k)=(k+2)(k+4)\text{ and }rk(D_k)=\frac{1}{2}(k+1)(k+2).$$

$$\forall \mathbf{i}\in {\mathbb{N}}_0^3,|\mathbf{i}|=k,\nabla \cdot \left(\begin{array}{c}\frac{1}{i_1+1}{\mathbf{X}}^{\mathbf{i}+{\mathbf{e}}_1},0,0\end{array}\right)={\mathbf{X}}^{\mathbf{i}}\text{ where }\left(\begin{array}{c}\frac{1}{i_1+1}{\mathbf{X}}^{\mathbf{i}+{\mathbf{e}}_1},0,0\end{array}\right)\in {\left({\tilde{\mathbb{P}}}_{k+1}\right)}^3.$$

$$\dim \ker (D_k)=\frac{3}{2}(k+2)(k+3)-\frac{1}{2}(k+1)(k+2)=(k+2)(k+4).$$

$$\begin{array}[]{rrcl}C_{k}:&\left(\widetilde{\mathbb{P}}_{k+1}\right)^{3}&\to&\left(\widetilde{\mathbb{P}}_{k}\right)^{3},\\ &\mathbf{\Phi}&\mapsto&\nabla\times\mathbf{\Phi},\end{array}\text{ and }\begin{array}[]{rrcl}C_{*}:&\left(\widetilde{\mathbb{P}}_{0}\right)^{3}&\to&\{\mathbf{0}^{3}\},\\ &\mathbf{\Phi}&\mapsto&\mathbf{0}^{3}.\end{array}.$$

$$\dim \ker (C_k)=(k+3)(k+4)/2\text{ and }rk(C_k)=(k+1)(k+3).$$

$$\left(\begin{array}{c}{\mathbf{X}}^0,0,0\end{array}\right)=\nabla \times \left(\begin{array}{c}0,0,{\mathbf{X}}^{{\mathbf{e}}_2}\end{array}\right),\left(\begin{array}{c}0,{\mathbf{X}}^0,0\end{array}\right)=\nabla \times \left(\begin{array}{c}{\mathbf{X}}^{{\mathbf{e}}_3,0,0}\end{array}\right),\text{ and }\left(\begin{array}{c}0,0,{\mathbf{X}}^0\end{array}\right)=\nabla \times \left(\begin{array}{c}0,0,{\mathbf{X}}^{{\mathbf{e}}_1}\end{array}\right),$$

$$\dim \ker (C_k)=\frac{3}{2}(k+2)(k+3)-(k+1)(k+3),$$

$$\{0\}\to {\tilde{\mathbb{P}}}_{k+3}\stackrel{G_{k+2}}{\to }{\left({\tilde{\mathbb{P}}}_{k+2}\right)}^3\stackrel{C_{k+1}}{\to }{\left({\tilde{\mathbb{P}}}_{k+1}\right)}^3\stackrel{D_k}{\to }{\tilde{\mathbb{P}}}_k\to \{0\}.$$

$$\ker (G_{k+2})=\{0\},\mathcal{R}(G_{k+2})=\ker (C_{k+1}),\mathcal{R}(C_{k+1})=\ker (D_k)\text{ and }\mathcal{R}(D_k)={\tilde{\mathbb{P}}}_k.$$

$$\begin{array}[]{rrcl}L_{k}:&\widetilde{\mathbb{P}}_{k+2}&\to&\widetilde{\mathbb{P}}_{k},\\ &\Phi&\mapsto&\Delta\Phi,\end{array}\text{ and }\begin{array}[]{rrcl}L_{*}:&{\mathbb{P}}_{1}&\to&\{\mathbf{0}\},\\ &\Phi&\mapsto&\mathbf{0}.\end{array}.$$

Full text

1. 1 Introduction

2. 1.1 Maxwell’s equation

3. 1.2 Notation

4. 1.3 Quasi-Trefftz spaces: scalar equations versus systems of equations

5. 2 Differential operators and exact sequences

6. 2.1 First order operators

7. 2.2 Second order operators

8. 3 Helmholtz decomposition

9. 3.1 Existence of a decomposition

10. 3.2 Harmonic vector fields

11. 3.3 A unique decomposition

12. 4 A polynomial quasi-Trefftz space

13. 4.1 Definition and characterization

14. 4.2 Restricted operators

15. 4.3 Construction of individual quasi-Trefftz functions

16. 4.4 Dimension

17. 5 Conclusion

**Acknowledgments

**This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, under Award Number DE-SC0024246.

**Abstract

**Quasi-Trefftz methods are a family of Discontinuous Galerkin methods relying on equation-dependent function spaces. This work is the first study of the notion of local Taylor-based polynomial quasi-Trefftz space for a system of Partial Differential Equations (PDEs). These discrete spaces are introduced here for electro-magnetic wave propagation in inhomogeneous media, governed by a second order formulation of Maxwell’s equation with variable coefficients. Thanks to an adequate Helmholtz decomposition for spaces of homogeneous polynomial vector fields, the outcome is the explicit dimension of the proposed quasi-Trefftz space as well as a procedure to construct quasi-Trefftz functions.

1 Introduction

In the field of numerical partial differential equations (PDEs), there exist various methods leveraging problem-dependent basis functions [BMS02, DG10, AHP21, Pet14, MB96]. Specifically for problems modeling wave propagation phenomena, different high-order methods incorporate the expected oscillating behavior of the solution to the PDE problem within the discrete basis [FIF01, DVVD11, Des98].

Among them, a specific type of discontinuous Galerkin methods, referred to as Trefftz methods, fundamentally rely on local spaces of exact solutions to the governing PDE [Des94, CD98, HMCK04, Gab07, GD15, CJGP09, HMP11]. For convenience, the Trefftz property for a function refers to being a local exact solutions to the governing PDE. A weak formulation holding only for Trefftz functions is then discretized on local discrete Trefftz spaces. They have been developed in particular for time-harmonic wave propagation problems, mainly thanks to discrete spaces of local Plane Waves (PWs), but also spherical waves or Bessel-based waves (generalized harmonic polynomials) [LHM12]. From the point of view of the size of the discrete system, such methods may need much smaller discrete spaces in order to reach the same order of accuracy compared to standard Galerkin methods, resulting in a considerably smaller linear system. From the point of view of the assembly of the matrix, such methods present two main advantages. First, thanks to the Trefftz property, the weak formulation contains only integrals on the boundary of mesh elements, hence the domains of integration are lower-dimensional compared to standard Galerkin method. Second, there are closed formulas for integrating products of PWs and their derivatives. Both facts reduce considerably to cost of assembly of the matrix. Surveys of some of these methods can be found in [HMP16, DAC*+*14], and some related computational aspects are covered in [LGB16, HMK02].

Trefftz-like methods also have known limitations. From the point of view of wave-like bases, there exists a conditioning problem that can be simply illustrated considering the difference between two PWs propagating respectively in the directions $\mathbf{d}_{1}$ and $\mathbf{d}_{2}$:

[TABLE]

as the linear independence of these two PWs deteriorates as their directions of propagation get closer. Moreover, the range of applications of Trefftz-like methods is limited by the need for discrete spaces of exact solutions to the governing PDEs with good approximation properties and explicit basis functions. For most problems beyond wave propagation in homogeneous media, such discrete spaces are not available.

A class of so-called quasi-Trefftz methods were introduced to circumvent this limitation. These methods relax the Trefftz property into a so-called quasi-Trefftz property: being a local approximate solution to the governing PDE, by imposing that the image of the function by the differential operator (instead of being zero) has a Taylor expansion that vanishes up to a desired order. These methods were first developed for the Helmholtz equation variable wave number based on so-called Generalized PWs basis functions [IGD13, IG15, IGM17], and later for the wave equation [LMIGS23], the Shrödinger equation [GM24], and elliptic problems [IGMPS25]. The methods have so far not been studied for problems governed by systems of PDEs.

1.1 Maxwell’s equation

Given $\epsilon$ is a smoothly varying coefficient, this article focuses on Maxwell’s equation in $\mathbb{R}^{3}$ under the form

[TABLE]

and it is straightforward to verify that $\nabla\cdot(\nabla\times\mathbf{V})=0$. A direct consequence is that any solution $\mathbf{V}$ to Equation (1) necessarily satisfies the so-called divergence condition $\nabla\cdot(\epsilon\mathbf{V})=0$.

Remark 1**.**

In the present work, the variable coefficient $\epsilon$ is assumed to be scalar, yet in the cold plasma model $\epsilon$ is matrix valued and this is key to study the interaction of various propagation modes. Extending this work to the case of a matrix-valued variable coefficient $\epsilon$ is straightforward, and simply requires for $\epsilon$ to be smooth component-wise and for its eigenvalues to be bounded away from zero - with no restriction on their sign. Moreover, for a matrix-valued coefficient, the product rule for the divergence reads

[TABLE]

There exists a rich literature dedicated to the mathematical theory of Maxwell’s equations and their applications, see for instance [Mül69, Ces96a, Ned01, RV10, GNC15, FAL18, CK19, Don20, Max24]. The divergence condition and the Helmholtz decomposition of vector fields as the sum of a divergence-free (or solenoidal) component and a curl-free (or irrotational) component play a fundamental role in this context [AG94, BNPB13], despite the existence of many different formulations of these equations.

Many numerical methods and tools have also been developed specifically for systems of PDEs, and in particular for Maxwell’s equations [IM11, KPL13, Arn18, CHH18, Cot23]. These for instance include the Finite Difference Yee scheme, the Finite Element Exterior Calculus, and various types of wave propagation algorithms for hyperbolic systems. Here again, the divergence condition remains fundamental, and the choice of discrete spaces is crucial for Galerkin methods, for Maxwell’s equations [Mon03, HPS04, BKN11, XWZ13, SLS18, Arn18]. For example, Nédélec finite elements can be used to discretize $H(curl)$ and Raviart-Thomas finite elements can be used to discretize $H(div)$. The dimension of such spaces is also of interest [ST24]. More generally, exact sequences of finite element spaces have been developed to discretize differential complexes, such as the deRham complex for Maxwell problems [DMVR00, Hip02, AFW06, CHH18], in particular for stability [Arn02]. Many versions of discrete Helmholtz decompositions have also been proposed, within the Galerkin framework and beyond [HSZ13, TLHD03, BKS25].

In the case of Trefftz-Discontinuous Galerkin (DG) methods for wave propagation in homogeneous media, the discrete space of plane waves (PWs) includes a compatibility condition between the propagation directions and the polarizations to guarantee that the PWs satisfy the governing PDE [HMP13]. In this first work on quasi-Trefftz spaces for Maxwell’s equation, an adequate Helmholtz decomposition of homogeneous polynomial vector fields, including a harmonic component is proposed. This is the key to the study of polynomial quasi-Trefftz spaces for Maxwell’s equation (1).

1.2 Notation

Throughout the article, while $\mathbb{N}$ refers to the set of positive integers, $\mathbb{N}_{0}$ refers to the set of non-negative integers; the kernel of an operator is denoted $\ker$ and range $\mathcal{R}$. Moreover, $g$ refers to an integer, the dimension of the ambient space is equal to $3$. Then $\mathbf{x}=(x_{g},g\text{ from }1\text{ to }3)\in\mathbb{R}^{3}$ refers to points in the ambient space, $\mathbf{i}=(i_{g},g\text{ from }1\text{ to }3)\in(\mathbb{N}_{0})^{3}$ refers to a multi-index, partial derivatives are denoted $\partial_{x_{g}}$ and $\partial^{\mathbf{i}}=\partial_{x_{1}}^{i_{1}}\partial_{x_{2}}^{i_{2}}\partial_{x_{3}}^{i_{3}}$, and finally the canonical basis of $\mathbb{R}^{3}$ is denoted $\{\mathbf{e}_{g},1\leq g\leq 3\}$. The spaces of scalar and vector-valued polynomials of degree at most equal to $p\in\mathbb{N}_{0}$ are denoted respectively $\mathbb{P}_{p}$ and $(\mathbb{P}_{p})^{3}$. Monomials and spaces of scalar homogeneous polynomials are denoted

[TABLE]

The grade structure of polynomial spaces can be described as

[TABLE]

Remark 2**.**

As a reminder, dimensions of polynomial spaces in three variables are given by

[TABLE]

The work presented here is limited to local properties. More precisely, we only consider quasi-Trefftz spaces defined in the neighborhood of a point $\mathbf{x}_{0}\in\mathbb{R}^{d}$, so $\mathbf{x}_{0}$ will be omitted in the following notation. For $n\in\mathbb{N}_{0}$, the space of functions with continuous derivatives up to order $n$ in a neighborhood of $\mathbf{x}_{0}$ is denoted $\mathcal{C}^{n}$. For a $\mathcal{C}^{k}$ function $\varphi$, its $k$th Taylor polynomial at $\mathbf{x}_{0}$, denoted $T_{k}[\varphi]\in\mathbb{P}_{k}$, is given by

[TABLE]

For convenience, for a $\varphi\in\mathcal{C}^{k}$, then $\varphi_{k}\in\widetilde{\mathbb{P}}_{k}$ will be used to denote the homogeneous component of degree $k$ of $T_{k}[\varphi]\in\mathbb{P}_{k}$.

Definition 1**.**

Assume $q\in\mathbb{N}_{0}$. Given a linear partial differential operator $\mathcal{L}$ with smooth variable coefficients in $\mathcal{C}^{q}$, the Taylor-based quasi-Trefftz property of order $q$ for a function $\varphi$ is

[TABLE]

1.3 Quasi-Trefftz spaces: scalar equations versus systems of equations

Quasi-Trefftz spaces of GPWs for various scalar equations were the focus of [IG15, IGS21, IG21], including a polynomial space in the latter. In [IG25], a general framework was proposed to study polynomial quasi-Trefftz spaces in any dimension for differential operators of order $\gamma\in\mathbb{N}$ of the form $\displaystyle\mathcal{L}:=\sum_{m=0}^{\gamma}\sum_{|\mathbf{j}|=m}c_{\mathbf{j}}\left({\mathbf{x}}\right)\partial_{{\mathbf{x}}}^{\mathbf{j}}$ with smooth variable coefficients in the vicinity of a point $\mathbf{x}_{0}$ with at least one $\mathbf{j}\in(\mathbb{N}_{0})^{3}$ such that $|\mathbf{j}|=\gamma$ and $c_{\mathbf{j}}(\mathbf{x}_{0})\neq 0$. There the quasi-Trefftz space and the quasi-Trefftz operator are defined as

[TABLE]

so $\mathbb{Q}\mathbb{T}_{p}=\ker\mathcal{D}_{p}$. In this scalar case, thanks to the graded structure of polynomial spaces, the operator $\mathcal{D}_{p}|_{\bigoplus_{\ell=\gamma}^{p}\widetilde{\mathbb{P}}_{\ell}}$ was proved to have a block-triangular structure, and these diagonal blocks correspond to the graded operators of degree $-\gamma$ defined by $\displaystyle\sum_{|\mathbf{j}|=\gamma}c_{\mathbf{j}}\left({\mathbf{x}_{0}}\right)\partial_{{\mathbf{x}}}^{\mathbf{j}}$ between spaces of homogeneous polynomials. In the scalar case, such operators were proved to be surjective. From there a natural block forward substitution procedure lead to an algorithm to construct quasi-Trefftz functions and consequently to the dimension of the quasi-Trefftz space. As a reminder, quasi-Trefftz space are smaller than standard polynomial spaces, and the dimension of these discrete spaces with the same order $p$ of approximation properties in dimension $d$, between the polynomial space $\mathbb{P}_{p}$ and some quasi-Trefftz space $\mathbb{Q}\mathbb{T}_{p}$ can be described as follows.

[TABLE]

For systems of PDEs, the challenge comes from the coupling between components within the highest order derivatives in the differential operator. If these components are not coupled, then extending the scalar framework is straightforward, but for Equation (1) this is not the case: the curl-curl operator is not surjective between spaces of homogeneous polynomials. Yet the quasi-Trefftz operator retains an underlying triangular structure, and it is thanks to the quasi-divergence condition that the quasi-Trefftz space will be expressed -as in the scalar case- as the kernel of a different operator with an underlying triangular structure and surjective diagonal blocks. The present work leverages the block structure of this operator for a detailed study of deRham complex operators between spaces of homogeneous polynomials, and introduces a Helmholtz decomposition of homogeneous polynomial vector fields. The outcome is the explicit dimension of the quasi-Trefftz space as well as a procedure to construct quasi-Trefftz functions.

2

Differential operators and exact sequences

Consider the gradient, divergence and curl operators between standard spaces of polynomials, namely:

[TABLE]

These maps are graded linear maps of degree $-1$ on polynomials, and for each of these, the range and kernel, their dimensions and a basis of each are of interest to construct quasi-Trefftz basis for Maxwell’s equation. Also consider two Laplacian operators between standard spaces of polynomials:

[TABLE]

Both of these operators are a scalar-valued graded operator of degree -2. The former operator, as it is defined between spaces of scalar polynomials, falls under the general framework developed in [IG25] . The later is crucial in the study of Equation (1) as $\nabla\times\nabla\times\mathbf{V}=-\overrightarrow{\Delta}\mathbf{V}+\nabla(\nabla\cdot\mathbf{V})$.

Each of these operators is of the generic form

[TABLE]

where $\gamma\in\{1,2\}$ is the order of $\mathcal{T}$, and these graded operator can be decomposed as follows: for all $\left\{x_{l}\in\left(\widetilde{\mathbb{P}}_{l}\right)^{n},l\in[\![0,p+\gamma]\!]\right\}$,

[TABLE]

Moreover, since $\mathcal{R}(\mathcal{T}_{*})=\{\mathbf{0}^{m}\}$, then

[TABLE]

Hence the graded structure of polynomial spaces can be leveraged to study the kernels and ranges of the operators $\mathcal{T}_{k}$ and $\mathcal{T}_{*}$.

This preliminary section focuses on basic properties of these various differential operators that form building blocks for the rest of the article.

2.1 First order operators

For each of the gradient, divergence and curl operators, the operators $\mathcal{T}_{k}$ and $\mathcal{T}_{*}$ are defined, and their kernels and ranges are described by a basis.

For the gradient operator, the input is scalar-valued and the output is vector-valued; in the generic form (2) this corresponds to objects of dimensions $n=1$ for its domain and $m=3$ for its co-domain. The next definition follows the decomposition (3).

Definition 2**.**

For any $k\in\mathbb{N}_{0}$, the gradient operator between spaces of homogeneous polynomials is defined as

[TABLE]

The corresponding kernel and range can be described as follows.

Lemma 1**.**

For any $k\in\mathbb{N}_{0}$, $\ker(G_{k})=\{\mathbf{0}\}$ while the set

[TABLE]

forms a basis of $\mathcal{R}(G_{k})$, so the dimensions of the kernel and range of $G_{k}$ are

[TABLE]

Proof.

Only constant functions have a zero gradient. Moerover, for any $k\in\mathbb{N}_{0}$ $\ker(G_{k})\subset\widetilde{\mathbb{P}}_{k+1}$, hence $\ker(G_{k})=\{\mathbf{0}\}$, and so $\dim\ker(G_{k})=0$.

Since $\dim\widetilde{\mathbb{P}}_{k+1}=\frac{1}{2}(k+2)(k+3)$, the rank-nullity theorem gives $rk(G_{k})=\frac{1}{2}(k+2)(k+3)$. Besides, a basis for $\mathcal{R}(G_{k})$ is given by

[TABLE]

where for any $g\in\{1,2,3\}$, $0\mathbf{X}^{\mathbf{i}-\mathbf{e}_{g}}=\mathbf{0}$ is the zero scalar polynomial. Indeed, it is a set of $12(k+2)(k+3)$ elements belonging to the range, and they are linearly independent because

[TABLE]

since $\{\mathbf{X}^{\mathbf{i}},\mathbf{i}\in\mathbb{N}_{0}^{3},|\mathbf{i}|=k+1\}$ forms a basis of $\widetilde{\mathbb{P}}_{k+1}$. ∎

For the divergence operator, the input is vector-valued and the output is scalar-valued; in the generic form (2) this corresponds to objects of dimensions $n=3$ for its domain and $m=1$ for its co-domain. The next definition follows again the decomposition (3).

Definition 3**.**

For any $k\in\mathbb{N}_{0}$, the divergence operator between spaces of homogeneous polynomials is defined as

[TABLE]

The corresponding kernel and range can be described as follows.

Lemma 2**.**

For any $k\in\mathbb{N}_{0}$, the union of the following sets forms a basis for the kernel $\ker(D_{k})$:

• •

$\{\begin{pmatrix}\mathbf{X}^{\mathbf{i}},\mathbf{0},\mathbf{0}\end{pmatrix},\forall\mathbf{i}\in\mathbb{N}^{3}$* such that $|\mathbf{i}|=k+1,i_{1}=0\}$,*

• •

$\{\begin{pmatrix}\mathbf{0},\mathbf{X}^{\mathbf{i}},\mathbf{0}\end{pmatrix},\forall\mathbf{i}\in\mathbb{N}^{3}$* such that $|\mathbf{i}|=k+1,i_{2}=0\}$,*

• •

$\{\begin{pmatrix}\mathbf{0},\mathbf{0},\mathbf{X}^{\mathbf{i}}\end{pmatrix},\forall\mathbf{i}\in\mathbb{N}^{3}$* such that $|\mathbf{i}|=k+1,i_{3}=0\}$,*

• •

$\{\begin{pmatrix}(i_{3}+1)\mathbf{X}^{\mathbf{i}},\mathbf{0},-i_{1}\mathbf{X}^{\mathbf{i}-\mathbf{e}_{1}+\mathbf{e}_{3}}\end{pmatrix},\forall\mathbf{i}\in\mathbb{N}^{3}$* such that $|\mathbf{i}|=k+1,i_{1}>0\}$,*

• •

$\{\begin{pmatrix}\mathbf{0},(i_{3}+1)\mathbf{X}^{\mathbf{i}},-i_{2}\mathbf{X}^{\mathbf{i}-\mathbf{e}_{2}+\mathbf{e}_{3}}\end{pmatrix}\forall\mathbf{i}\in\mathbb{N}^{3}$* such that $|\mathbf{i}|=k+1,i_{2}>0\}$,*

while the operator $D_{k}$ is surjective, so the dimensions of the kernel and range of $D_{k}$ are

[TABLE]

Proof.

All elements of the canonical basis of $\widetilde{\mathbb{P}}_{k}$, namely $\{\mathbf{X}^{\mathbf{i}},\mathbf{i}\in\mathbb{N}_{0}^{3},|\mathbf{i}|=k\}$, belong to the range since

[TABLE]

Hence the operator is surjective and $rk(D_{k})=\dim\widetilde{\mathbb{P}}_{k}$.

Then from the rank-nullity theorem,

[TABLE]

Consider the candidate union of sets. The cardinals of the first three sets are each equal to $k+2$, while the cardinals of the last two sets are both equal to $\dim\widetilde{\mathbb{P}}_{k+1}-(k+2)=\frac{1}{2}(k+2)(k+3)-(k+2)$. So together they form a set of $3(k+2)+2\frac{1}{2}(k+2)(k+1)=(k+2)(k+4)$ elements that are clearly linearly independent, hence they form a basis of the space they span. Moreover this space is included in $\ker(D_{k})$ and has the same dimension. This proves the claim. ∎

Further useful properties of the divergence operator will be studied in Section 4.2.

For the curl operator, the input is vector-valued and the output is vector-valued; in the generic form (2) this corresponds to objects of dimensions $n=3$ for its domain and $m=3$ for its co-domain. One more, the next definition follows the decomposition (3).

Definition 4**.**

For any $k\in\mathbb{N}_{0}$, the curl operator between spaces of homogeneous polynomials is defined as

[TABLE]

The corresponding kernel and range can be described as follows.

Lemma 3**.**

For any $k\in\mathbb{N}$, the union of the following sets forms a basis for the range $\mathcal{R}(C_{k})$:

• •

$\{\begin{pmatrix}\mathbf{X}^{\mathbf{i}},\mathbf{0},\mathbf{0}\end{pmatrix},\forall\mathbf{i}\in\mathbb{N}^{3}$* such that $|\mathbf{i}|=k,i_{1}=0\}$,*

• •

$\{\begin{pmatrix}\mathbf{0},\mathbf{X}^{\mathbf{i}},\mathbf{0}\end{pmatrix},\forall\mathbf{i}\in\mathbb{N}^{3}$* such that $|\mathbf{i}|=k,i_{2}=0\}$,*

• •

$\{\begin{pmatrix}\mathbf{0},\mathbf{0},\mathbf{X}^{\mathbf{i}}\end{pmatrix},\forall\mathbf{i}\in\mathbb{N}^{3}$* such that $|\mathbf{i}|=k,i_{3}=0\}$,*

• •

$\{\begin{pmatrix}(i_{3}+1)\mathbf{X}^{\mathbf{i}},\mathbf{0},-i_{1}\mathbf{X}^{\mathbf{i}-\mathbf{e}_{1}+\mathbf{e}_{3}}\end{pmatrix},\forall\mathbf{i}\in\mathbb{N}^{3}$* such that $|\mathbf{i}|=k,i_{1}>0\}$,*

• •

$\{\begin{pmatrix}\mathbf{0},(i_{3}+1)\mathbf{X}^{\mathbf{i}},-i_{2}\mathbf{X}^{\mathbf{i}-\mathbf{e}_{2}+\mathbf{e}_{3}}\end{pmatrix}\forall\mathbf{i}\in\mathbb{N}^{3}$* such that $|\mathbf{i}|=k,i_{2}>0\}$;*

$\{\mathbf{X}^{\mathbf{i}},\mathbf{i}\in\mathbb{N}_{0}^{3},|\mathbf{i}|=0\}$* forms a basis of $\mathcal{R}(C_{0})=({\mathbb{P}}_{0})^{3}$, while $\left\{\begin{pmatrix}i_{1}\mathbf{X}^{\mathbf{i}-\mathbf{e}_{1}},i_{2}\mathbf{X}^{\mathbf{i}-\mathbf{e}_{2}},i_{3}\mathbf{X}^{\mathbf{i}-\mathbf{e}_{3}}\end{pmatrix},\mathbf{i}\in\mathbb{N}_{0}^{3},|\mathbf{i}|=k+2\right\}$ defines a basis of $\ker(C_{k})$ for any $k\in\mathbb{N}_{0}$.*

Hence for any $k\in\mathbb{N}_{0}$, the dimensions of the kernel and range of $C_{k}$ are

[TABLE]

Proof.

It is a classical results that for any function $\mathbf{f}\in(\mathcal{C}^{1})^{3}$, $\nabla\cdot(\nabla\times\mathbf{f})=0$, and this in particular implies that $\mathcal{R}(C_{0})\subset\ker(D_{*})$ and $\mathcal{R}(C_{k})\subset\ker(D_{k-1})$ if $k>0$. These inclusions are actual identities, indeed, on the one hand,

• •

$\forall\mathbf{i}\in\mathbb{N}_{0}^{3}$ such that $|\mathbf{i}|=k+1,i_{1}=0$, $\begin{pmatrix}\mathbf{X}^{\mathbf{i}},\mathbf{0},\mathbf{0}\end{pmatrix}=\nabla\times\begin{pmatrix}\mathbf{0},\mathbf{0},\frac{1}{i_{2}+1}\mathbf{X}^{\mathbf{i}+\mathbf{e}_{2}}\end{pmatrix}$,

• •

$\forall\mathbf{i}\in\mathbb{N}_{0}^{3}$ such that $|\mathbf{i}|=k+1,i_{2}=0$, $\begin{pmatrix}\mathbf{0},\mathbf{X}^{\mathbf{i}},\mathbf{0}\end{pmatrix}=\nabla\times\begin{pmatrix}\frac{1}{i_{3}+1}\mathbf{X}^{\mathbf{i}+\mathbf{e}_{3},\mathbf{0},\mathbf{0}}\end{pmatrix}$,

• •

$\forall\mathbf{i}\in\mathbb{N}_{0}^{3}$ such that $|\mathbf{i}|=k+1,i_{3}=0$, $\begin{pmatrix}\mathbf{0},\mathbf{0},\mathbf{X}^{\mathbf{i}}\end{pmatrix}=\nabla\times\begin{pmatrix}\mathbf{0},\mathbf{0},\frac{1}{i_{1}+1}\mathbf{X}^{\mathbf{i}+\mathbf{e}_{1}}\end{pmatrix}$,

• •

$\forall\mathbf{i}\in\mathbb{N}_{0}^{3}$ such that $|\mathbf{i}|=k+1,i_{1}>0$, $\begin{pmatrix}-(i_{3}+1)\mathbf{X}^{\mathbf{i}},\mathbf{0},i_{1}\mathbf{X}^{\mathbf{i}-\mathbf{e}_{1}+\mathbf{e}_{3}}\end{pmatrix}=\nabla\times\begin{pmatrix}\mathbf{0},\mathbf{X}^{\mathbf{i}+\mathbf{e}_{3}},\mathbf{0}\end{pmatrix}$,

• •

$\forall\mathbf{i}\in\mathbb{N}_{0}^{3}$ such that $|\mathbf{i}|=k+1,i_{2}>0$, $\begin{pmatrix}\mathbf{0},(i_{3}+1)\mathbf{X}^{\mathbf{i}},-i_{2}\mathbf{X}^{\mathbf{i}-\mathbf{e}_{2}+\mathbf{e}_{3}}\end{pmatrix}=\nabla\times\begin{pmatrix}\mathbf{X}^{\mathbf{i}+\mathbf{e}_{3},\mathbf{0},\mathbf{0}}\end{pmatrix}$.

So if $k\in\mathbb{N}$ for any $\mathbf{P}$ in the basis of $\ker(D_{k-1})$ provided in Lemma 2, there exists $\mathbf{Q}\in\left(\widetilde{\mathbb{P}}_{k+1}\right)^{3}$ such that $\nabla\times\mathbf{Q}=\mathbf{P}$, or equivalently $\mathbf{P}\in\mathcal{R}(C_{k})$: hence $\ker(D_{k-1})\subset\mathcal{R}(C_{k})$. On the other hand,

[TABLE]

so $\ker(D_{*})=({\mathbb{P}}_{0})^{3}\subset\mathcal{R}(C_{0})$. This proves the claims about ranges.

It is also a classical result that for any function $f\in\mathcal{C}^{1}$, $\nabla\times(\nabla f)=0$, and this in particular implies that $\mathcal{R}(G_{k+1})\subset\ker(C_{k})$. Moreover from the rank-nullity theorem,

[TABLE]

hence, according to Lemma 1, it is equal to $rk(G_{k+1})={(k+3)(k+4)}/2$. This proves the claims about kernels. ∎

Remark 3**.**

Together, these operators form the following exact sequence:

[TABLE]

This directly follows from Lemmas 1, 2 and 3 since

[TABLE]

2.2 Second order operators

For the Laplacian operator, the input is scalar-valued and the output is scalar-valued; in the generic form (2) this corresponds to objects of dimensions $n=1$ for its domain and $m=1$ for its co-domain. The next definition follows again the decomposition (3).

Definition 5**.**

For any $k\in\mathbb{N}_{0}$, the (scalar) Laplacian operator between spaces of homogeneous polynomials is defined as

[TABLE]

The corresponding kernel and range can be described as follows.

Proposition 1**.**

For any $k\in\mathbb{N}_{0}$, the operator $L_{k}$ is surjective, so the dimensions of the kernel and range of $L_{k}$ are

[TABLE]

Proof.

As proved in [IG25], $\mathcal{L}_{*}|_{\mathbb{V}_{k_{2}}}$ is bijective if $\mathbb{V}_{k+2}:=Span\{\mathbf{X}^{2\mathbf{e}_{1}+\mathbf{i}},\mathbf{i}\in\mathbb{N}_{0}^{d},|\mathbf{i}|=k\}$. Therefore $\dim rk(L_{k})=\dim\widetilde{\mathbb{P}}_{k}$, and from the rank-nullity theorem $\dim\ker(L_{k})=\dim\widetilde{\mathbb{P}}_{k+2}-\dim\widetilde{\mathbb{P}}_{k}$. ∎

For the so-called vector Laplacian operator, both the input and the output are vector-valued, and the output in the component-wise (scalar) Laplacian of the input; in the generic form (2) this corresponds to objects of dimensions $n=3$ for its domain and $m=3$ for its co-domain. The next definition follows again the decomposition (3).

Definition 6**.**

For any $k\in\mathbb{N}_{0}$, the vector Laplacian operator between spaces of homogeneous polynomials is defined as

[TABLE]

Remark 4**.**

The operator $\overrightarrow{L}_{k}$ has a block upper-triangular structure. Indeed, from bases of $\ker(D_{k+1})$ and $\ker(D_{k-1})$ introduced in Lemma 2, define

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

*then *

[TABLE]

[TABLE]

Further useful properties of the vector Laplacian operator will also be studied in Section 4.2.

3

Helmholtz decomposition

The spaces of polynomial solenoidal field and polynomial irrotational fields, both described in the previous section, play an important role in the study of 3D polynomial vector fields.

Definition 7**.**

Consider $k\in\mathbb{N}$. Within the space of homogeneous polynomials of degree $k$, the spaces of polynomial solenoidal fields and polynomial irrotational fields are respectively defined as:

[TABLE]

*For the record, all constant vector fields are both solenoidal and irrotational. *

Similarly, consider $p\in\mathbb{N}$. Within the polynomial space $\mathbb{P}^{p}$, the spaces of polynomial solenoidal fields and polynomial irrotational fields are respectively defined as:

[TABLE]

While the previous material is sufficient to prove the existence of a decomposition of vector fields into the sum of a solenoidal component and an irrotational component, see Section 3.1, such a decomposition is not unique as there are non-trivial polynomials that are both solenoidal and irrotational. Thanks to the introduction of these so-called harmonic polynomials, see Section 3.2, polynomial vector fields can be uniquely decomposed into a component that is solenoidal but not irrotational, a component that is irrotational but not solenoidal, and a harmonic component, see Section 3.3

3.1 Existence of a decomposition

Homogeneous polynomial vector fields can be decomposed into the sum of a solenoidal component and an irrotational component as follows.

Lemma 4**.**

Consider any $k\in\mathbb{N}_{0}$. Then

[TABLE]

Proof.

Consider any polynomial vector field $\mathbf{V}\in\left(\widetilde{\mathbb{P}}_{k+1}\right)^{3}$. Since $L_{k}$ is surjective from Proposition 1, there exists $g\in\widetilde{\mathbb{P}}_{k+2}$ such that $L_{k}g=D_{k}\mathbf{V}$, which is equivalent to $D_{k}(G_{k+1}g-\mathbf{V})=0$. Then $\mathbf{F}:=\mathbf{V}-G_{k+1}g$ belongs to $\widetilde{\mathbb{S}}_{k+1}$ since $\ker(D_{k})=\mathcal{R}(C_{k+1})$ according to Lemma 3, $\mathbf{G}:=G_{k+1}g$ belongs to $\widetilde{\mathbb{I}}_{k+1}$, and they satisfy $\mathbf{V}=\mathbf{F}+\mathbf{G}$. ∎

Here is a simple proof of non-uniqueness for this decomposition. Given , any non-trivial $g\in\ker L_{k}$ and any $(\mathbf{F},\mathbf{G})\in\ker(D_{k})\times\ker(C_{k})$, then

[TABLE]

where the two sides of this identity are two distinct Helmholtz decompositions of a single vector field.

Moreover, assuming that $(\mathbf{F}_{\alpha},\mathbf{G}_{\alpha})\in\ker(D_{k})\times\ker(C_{k})$ for $\alpha\in\{1,2\}$ with $\mathbf{F}_{1}+\mathbf{G}_{1}=\mathbf{F}_{2}+\mathbf{G}_{2}$, then the difference $\mathbf{F}_{1}-\mathbf{F}_{2}=\mathbf{G}_{2}-\mathbf{G}_{1}$, denoted hereafter $\mathbf{V}$, is in $\ker(D_{k})\cap\ker(C_{k})\subset\left(\widetilde{\mathbb{P}}_{k+1}\right)^{3}$. This motivates the introduction of the notion of harmonic vector fields.

3.2 Harmonic vector fields

The next step towards a unique polynomial Helmholtz decomposition is to consider the intersection of spaces of solenoidal and irrotational vector fields within the space of homogeneous polynomial vector fields.

Definition 8**.**

Polynomial harmonic vector fields are polynomial vector fields that are both solenoidal and irrotational, and the corresponding vector spaces of homogeneous polynomials are defined as:

[TABLE]

In other words, $\widetilde{\mathbb{H}}_{k}$ is the space of homogeneous polynomial harmonic vector fields of degree $k$. For the record, the space of constant vector fields, ${\mathbb{P}}_{0}^{3}=\widetilde{\mathbb{P}}_{0}^{3}$ is also equal to $\widetilde{\mathbb{H}}_{0}$.

Proposition 2**.**

The dimensions of spaces of polynomial harmonic vector fields are given by:

[TABLE]

Proof.

The space $\widetilde{\mathbb{H}}_{0}$ is simply $\left(\widetilde{\mathbb{P}}_{0}\right)^{3}$, so its dimension is 3. Considering then $k\in\mathbb{N}$, for any polynomial vector field $\mathbf{V}\in(\widetilde{\mathbb{P}}_{k})^{3}$, the condition $\mathbf{V}\in\widetilde{\mathbb{H}}_{k}$ is equivalent to

[TABLE]

So $\widetilde{\mathbb{H}}_{k}$ is the space of harmonic gradients, i.e. the space of gradients of elements of $\ker(L_{k-1})$. Moreover, $\dim\ker(L_{k-1})=2k+3$ from Proposition 1, and for any basis of $\ker(L_{k-1})$ denoted $\{v_{l},l\in[\![1,2k+3]\!]\}\subset\widetilde{\mathbb{H}}_{k+1}$ then $\{\nabla v_{l},l\in[\![1,2k+3]\!]\}\subset\widetilde{\mathbb{H}}_{k}$ is linearly independent since

[TABLE]

Hence $\dim\nabla\ker(L_{k-1})=\dim\ker(L_{k-1})$ so $\dim\widetilde{\mathbb{H}}_{k}=\dim\ker(L_{k-1})$, which proves the first claim. ∎

By definition, the spaces of harmonic vector fields are subspaces of both spaces of solenoidal fields and irrotational fields, but there is more.

3.3 A unique decomposition

This section focuses on a unique Helmholtz decomposition for homogeneous polynomial vector fields.

Definition 9**.**

For any $k\in\mathbb{N}$, $\widetilde{\mathbb{S}}_{k}^{*}$ and $\widetilde{\mathbb{I}}_{k}^{*}$ refer to the complements of $\widetilde{\mathbb{H}}_{k}$ respectively in $\widetilde{\mathbb{S}}_{k}$ and $\widetilde{\mathbb{I}_{k}}$:

[TABLE]

The dimensions of these spaces directly follow from the previous definition.

Corollary 1**.**

For any $k\in\mathbb{N}$, $\dim\widetilde{\mathbb{S}}^{*}_{k}=k(k+2)$ and $\dim\widetilde{\mathbb{I}}^{*}_{k}=k(k+1)/2$.

Proof.

By definition 9,

[TABLE]

Moreover, $\dim\widetilde{\mathbb{S}}^{*}_{k}=\dim\widetilde{\mathbb{S}}_{k}-\dim\widetilde{\mathbb{H}}_{k}$ and $\dim\widetilde{\mathbb{I}}^{*}_{k}=\dim\widetilde{\mathbb{I}}_{k}-\dim\widetilde{\mathbb{H}}_{k}$. The conclusion then follows from Lemma 2, as $\dim\widetilde{\mathbb{S}}_{k}=(k+1)(k+3)$, Lemma 3, as $\dim\widetilde{\mathbb{I}}_{k}=(k+2)(k+3)/2$, and Proposition 2, as $\dim\widetilde{\mathbb{H}}_{k}=2k+3$. ∎

The following proposition states the existence and uniqueness of the desired Helmholtz decomposition.

Proposition 3**.**

*Consider $k\in\mathbb{N}$. Homogeneous polynomial harmonic vector fields can be decomposed as follows: *

[TABLE]

or, equivalently,

[TABLE]

Moreover, $\displaystyle\left(\widetilde{\mathbb{P}}_{0}\right)^{3}=\widetilde{\mathbb{H}}_{0}$.

Proof.

According to Section 2.1 and Proposition 2, for homogeneous polynomials, the dimensions kernels of the divergence and curl operators are related to the dimension of the space of harmonic vector fields by:

[TABLE]

According to Definition 9 and Corollary 1, this concludes the proof.

∎

4 A polynomial quasi-Trefftz space

This section returns to the PDE of interest in this work, namely Maxwell’s equation (1) with the differential operator $\mathcal{L}_{M}=\nabla\times\nabla\times\mathbf{-}\epsilon$. In this case, in the sense introduced in Definition 1, a natural quasi-Trefftz property for a polynomial function $\mathbf{\Pi}$ of degree $p$ reads

[TABLE]

[TABLE]

This equivalence simply leverages the graded structure of polynomial spaces, yet the two equivalent properties imply a corresponding divergence condition. Indeed, for any $\mathbf{\Pi}\in\left({\mathbb{P}}_{p}\right)^{3}$

[TABLE]

which follows form $T_{p-2}\left[\nabla\times\nabla\times\mathbf{\Pi}\right]=\nabla\times\nabla\times\mathbf{\Pi}$ and $\nabla\cdot(\nabla\times\nabla\times\mathbf{\Pi})=0$, or equivalently

[TABLE]

which follows from $\mathcal{R}(C_{k})=\ker D_{k-1}$.

In the scalar case, $T_{p-2}\circ\mathcal{L}|_{\bigoplus_{\ell=2}^{p}\widetilde{\mathbb{P}}_{\ell}}$ has a block-triangular structure and its diagonal blocks are surjective. Here, while the operator $T_{p-2}\circ\mathcal{L}_{M}|_{\bigoplus_{\ell=2}^{p}\widetilde{\mathbb{P}}_{\ell}}$ also has a block-triangular structure, its diagonal blocks, corresponding to the operators $C_{k}\circ C_{k+1}$, are not surjective. However, the corresponding divergence condition can be leveraged to define an operator on $\bigoplus_{\ell=2}^{p}\widetilde{\mathbb{P}}_{\ell}$ that has both a block-triangular structure and surjective diagonal blocks. The Helmholtz decomposition introduced in the previous section then provides a powerful tool to derive an algorithm for the construction of quasi-Trefftz functions.

4.1 Definition and characterization

Besides the natural quasi-Trefftz property (8), the following quasi-Trefftz space definition includes a quasi-divergence condition. As discussed above, the quasi-divergence condition is a consequence of the first one up to a lower order, but this definition imposes the two conditions at a higher order. This divergence condition will be key to study the corresponding quasi-Trefftz space.

Definition 10**.**

Given $p\in\mathbb{N}$ such that $p>2$ and $\mathbf{x}_{0}\in\mathbb{R}^{3}$, consider the second order differential operator $\mathcal{L}_{M}:=\nabla\times\nabla\times-\epsilon$ with a scalar variable coefficient $\epsilon\in\mathcal{C}^{p}(\mathbf{x}_{0})$, the quasi-Trefftz $\mathbb{Q}\mathbb{T}_{p}$ is defined as

[TABLE]

or equivalently, in terms of homogeneous polynomials,

[TABLE]

Remark 5**.**

The equivalence between the two definitions is due to the facts that:

[TABLE]

Furthermore, the space can be conveniently characterized in terms of the different components of the Helmholtz decomposition.

Proposition 4**.**

Given $p\in\mathbb{N}$ such that $p>2$ and $\mathbf{x}_{0}\in\mathbb{R}^{3}$, consider the second order differential operator $\mathcal{L}_{M}:=\nabla\times\nabla\times-\epsilon$ with scalar variable coefficients $\epsilon\in\mathcal{C}^{p}(\mathbf{x}_{0})$, the quasi-Trefftz space can be described as

[TABLE]

Proof.

This directly follows from the Helmholtz decomposition introduced in Proposition 3 since

[TABLE]

together with the fact that

[TABLE]

∎

This characterization of the quasi-Trefftz space provides foundations to construct individual quasi-Trefftz functions thanks to right inverse operators for the vector Laplacian $\overrightarrow{L}_{k}$ and the divergence $D_{k}$ acting on appropriate subspaces of vector-valued polynomials.

4.2 Restricted operators

In Proposition 4, the operators coming into play are $D_{k}|_{\tilde{\mathbb{I}}^{*}_{k+1}}$ and $\overrightarrow{L}_{k}|_{\tilde{\mathbb{S}}^{*}_{k+2}}$. This section focuses on proving that both are surjective. For the divergence operator, it simply follows from the fact that the kernel $\widetilde{\mathbb{S}}_{k+1}$ of the divergence $D_{k}$. As for the Laplacian operator, $\widetilde{\mathbb{I}}_{k+2}$ is not the kernel of $\overrightarrow{L}_{k}$, yet the result can be proved by considering first the operator $\overrightarrow{L}_{k}|_{\tilde{\mathbb{S}}_{k+2}}$.

Proposition 5**.**

The restricted divergence operator defined by

[TABLE]

is bijective.

Proof.

For any $k\in\mathbb{N}_{0}$, the operator $D_{k}$ defined on the domain $\left(\widetilde{\mathbb{P}}_{k+1}\right)^{3}$ is surjective according to Lemma 2. Moreover, according to Proposition 3, $\left(\widetilde{\mathbb{P}}_{k+1}\right)^{3}=\widetilde{\mathbb{S}}_{k+1}\oplus\widetilde{\mathbb{I}}^{*}_{k+1}$. Since, by definition, $\widetilde{\mathbb{S}}_{k+1}$ is precisely the kernel of $D_{k}$, then $D_{k}|_{\tilde{\mathbb{I}}^{*}_{k+1}}$ is also surjective. Besides, from Corollary 1, $\dim\widetilde{\mathbb{I}}^{*}_{k+1}=(k+1)(k+2)/2$, therefore $\dim\widetilde{\mathbb{I}}^{*}_{k+1}=\dim\widetilde{\mathbb{P}}_{k}$. This concludes the proof.

∎

Remark 6**.**

Since $\tilde{\mathbb{I}}_{k+1}=\ker(C_{k})=\mathcal{R}(G_{k+1})$, given $f\in\widetilde{\mathbb{P}}_{k}$ then

[TABLE]

Therefore the surjectivity of the restricted divergence operator $D_{k}|_{\tilde{\mathbb{I}}_{k+1}}$ directly follows from the surjectivity of the Laplacian operator $L_{k}$ stated in Proposition 1.

The vector Laplacian acting between spaces of divergence-free fields has the following property.

Proposition 6**.**

The vector Laplacian operator restricted to solenoidal fields, namely

[TABLE]

*is surjective. As a consequence, *

[TABLE]

For the sake of compactness, consider the following notation for elements of divergence-free bases, for any $(k,\mathbf{i})\in\mathbb{N}_{0}\times\mathbb{N}_{0}^{3}$ with $|\mathbf{i}|=k$:

• •

$\mathbf{\Psi}^{k,1,\mathbf{i}}:=\begin{pmatrix}\mathbf{X}^{\mathbf{i}},\mathbf{0},\mathbf{0}\end{pmatrix}$ if $i_{1}=0$; $\mathbf{\Psi}^{k,2,\mathbf{i}}=\begin{pmatrix}\mathbf{0},\mathbf{X}^{\mathbf{i}},\mathbf{0}\end{pmatrix}$ if $i_{2}=0$; $\mathbf{\Psi}^{k,3,\mathbf{i}}=\begin{pmatrix}\mathbf{0},\mathbf{0},\mathbf{X}^{\mathbf{i}}\end{pmatrix}$ if $i_{3}=0$;

• •

$\mathbf{\Psi}^{k,4,\mathbf{i}}=\begin{pmatrix}(i_{3}+1)\mathbf{X}^{\mathbf{i}},\mathbf{0},-i_{1}\mathbf{X}^{\mathbf{i}-\mathbf{e}_{1}+\mathbf{e}_{3}}\end{pmatrix}$ if $i_{1}>0$; $\mathbf{\Psi}^{k,5,\mathbf{i}}=\begin{pmatrix}\mathbf{0},(i_{3}+1)\mathbf{X}^{\mathbf{i}},-i_{2}\mathbf{X}^{\mathbf{i}-\mathbf{e}_{2}+\mathbf{e}_{3}}\end{pmatrix}$ if $i_{2}>0$.

Even though given $\mathbf{i}\in\mathbb{N}_{0}^{3}$ the value of $k=|\mathbf{i}|$ is fixed, and therefore the $k$ in this notation is somewhat redundant, this notation will enhance readability in what follows.

Proof.

For $k=0$, it is easy to verify that

[TABLE]

and $\{\mathbf{\Psi}^{0,1,\mathbf{0}},\mathbf{\Psi}^{0,2,\mathbf{0}},\mathbf{\Psi}^{0,3,\mathbf{0}}\}$ forms a basis of $\widetilde{\mathbb{P}}_{0}^{3}$. Hence $\widetilde{\mathbb{P}}_{0}^{3}\subset\overrightarrow{L}_{0}\left(\tilde{\mathbb{S}}_{2}\right)$. So the two spaces are equal and the operator is surjective.

For $k=1$,$\overrightarrow{L}_{1}\left(\tilde{\mathbb{S}}_{3}\right)\subset\tilde{\mathbb{S}}_{1}$ since $D_{0}\circ\overrightarrow{L}_{1}=L_{0}\circ D_{2}$. Moreover,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and $\{\mathbf{\Psi}^{1,1,\mathbf{e}_{3}},\mathbf{\Psi}^{1,1,\mathbf{e}_{2}},\mathbf{\Psi}^{1,2,\mathbf{e}_{1}},\mathbf{\Psi}^{1,2,\mathbf{e}_{3}},\mathbf{\Psi}^{1,3,\mathbf{e}_{1}},\mathbf{\Psi}^{1,3,\mathbf{e}_{2}},\mathbf{\Psi}^{1,4,\mathbf{e}_{1}},\mathbf{\Psi}^{1,5,\mathbf{e}_{2}}\}$ forms a basis of $\tilde{\mathbb{S}}_{1}$. Hence $\tilde{\mathbb{S}}_{1}\subset\overrightarrow{L}_{k}\left(\tilde{\mathbb{S}}_{3}\right)$. So again the two spaces are equal and the operator is surjective.

For any $k>1$, $\overrightarrow{L}_{k}\left(\tilde{\mathbb{S}}_{k+2}\right)\subset\tilde{\mathbb{S}}_{k}$ since $D_{k-1}\circ\overrightarrow{L}_{k}=L_{k-1}\circ D_{k+1}$. Furthermore, according to Lemma 2 $\dim\tilde{\mathbb{S}}_{k}=(k+1)(k+3)$, so $\dim\overrightarrow{L}_{k}\left(\tilde{\mathbb{S}}_{k+2}\right)\geq(k+1)(k+3)$. Indeed, the union of the following sets of polynomial functions in $\overrightarrow{L}_{k}\left(\tilde{\mathbb{S}}_{k+2}\right)$ is linearly independent:

• •

$\{\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,1,\mathbf{i}}),\forall\mathbf{i}\in\mathbb{N}_{0}^{3}\text{ such that }|\mathbf{i}|=k+2,i_{1}=0,i_{2}\leq k\}$ with $k+1$ elements,

• •

$\{\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,2,\mathbf{i}}),\forall\mathbf{i}\in\mathbb{N}_{0}^{3}\text{ such that }|\mathbf{i}|=k+2,i_{2}=0,i_{3}\leq k\}$ with $k+1$ elements,

• •

$\{\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,3,\mathbf{i}}),\forall\mathbf{i}\in\mathbb{N}_{0}^{3}\text{ such that }|\mathbf{i}|=k+2,i_{3}=0,i_{1}\leq k\}$ with $k+1$ elements,

• •

$\{\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,4,\mathbf{i}}),\forall\mathbf{i}\in\mathbb{N}_{0}^{3}\text{ such that }|\mathbf{i}|=k+2,i_{1}>2\}$ with $k(k+1)/2$ elements,

• •

$\{\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,5,\mathbf{i}}),\forall\mathbf{i}\in\mathbb{N}_{0}^{3}\text{ such that }|\mathbf{i}|=k+2,i_{2}>2\}$ with $k(k+1)/2$ elements,

as one can easily verify that

• •

$\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,1,(k+2)\mathbf{e}_{3}})=(k+2)(k+1)\mathbf{\Psi}^{k,1,k\mathbf{e}_{3}}$, $\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,1,\mathbf{e}_{2}+(k+1)\mathbf{e}_{3}})=(k+1)k\mathbf{\Psi}^{k,1,\mathbf{e}_{2}+(k-1)\mathbf{e}_{3}}$, and for all $i_{2}\in[\![2,k]\!]$ $\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,1,\mathbf{i}})=i_{2}(i_{2}-1)\mathbf{\Psi}^{k+2,1,\mathbf{i}-2\mathbf{e}_{2}}+i_{3}(i_{3}-1)\mathbf{\Psi}^{k+2,1,\mathbf{i}-2\mathbf{e}_{3}}$, and together they form a linearly independent set since for any $i_{2}\in[\![0,k]\!]$ $\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,1,\mathbf{i}})$ is a linear combination of one or two $\mathbf{\Psi}^{k,1,\mathbf{j}}$ with $j_{2}\leq i_{2}$;

• •

$\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,2,(k+2)\mathbf{e}_{1}})=(k+2)(k+1)\mathbf{\Psi}^{k,2,k\mathbf{e}_{1}}$, $\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,2,\mathbf{e}_{3}+(k+1)\mathbf{e}_{1}})=(k+1)k\mathbf{\Psi}^{k,2,\mathbf{e}_{3}+(k-1)\mathbf{e}_{1}}$, and for all $i_{3}\in[\![2,k]\!]$ $\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,2,\mathbf{i}})=i_{3}(i_{3}-1)\mathbf{\Psi}^{k+2,2,\mathbf{i}-2\mathbf{e}_{3}}+i_{1}(i_{1}-1)\mathbf{\Psi}^{k+2,2,\mathbf{i}-2\mathbf{e}_{1}}$, and together they form a linearly independent set since for any $i_{3}\in[\![0,k]\!]$ $\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,2,\mathbf{i}})$ is a linear combination of one or two $\mathbf{\Psi}^{k,2,\mathbf{j}}$ with $j_{3}\leq i_{3}$;

• •

$\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,3,(k+2)\mathbf{e}_{2}})=(k+2)(k+1)\mathbf{\Psi}^{k,3,k\mathbf{e}_{2}}$, $\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,3,\mathbf{e}_{1}+(k+1)\mathbf{e}_{2}})=(k+1)k\mathbf{\Psi}^{k,3,\mathbf{e}_{1}+(k-1)\mathbf{e}_{1}2}$, and for all $i_{1}\in[\![2,k]\!]$ $\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,3,\mathbf{i}})=i_{1}(i_{1}-1)\mathbf{\Psi}^{k+2,3,\mathbf{i}-2\mathbf{e}_{1}}+i_{2}(i_{2}-1)\mathbf{\Psi}^{k+2,3,\mathbf{i}-2\mathbf{e}_{2}}$, and together they form a linearly independent set since for any $i_{1}\in[\![0,k]\!]$ $\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,3,\mathbf{i}})$ is a linear combination of one or two $\mathbf{\Psi}^{k,3,\mathbf{j}}$ with $j_{1}\leq i_{1}$,

• •

$\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,4,(k+2)\mathbf{e}_{1}})=(k+2)(k+1)\mathbf{\Psi}^{k,4,k\mathbf{e}_{1}}$, $\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,4,(k+1)\mathbf{e}_{1}+\mathbf{e}_{2}})=(k+1)k\mathbf{\Psi}^{k,4,(k-1)\mathbf{e}_{1}+\mathbf{e}_{2}}$,

[TABLE]

[TABLE]

for all $i_{1}\in[\![3,k]\!]$ (with $i_{2}=0$)

[TABLE]

for all $i_{1}\in[\![3,k]\!]$ (with $i_{3}=0$)

[TABLE]

for all $i_{1}\in[\![3,k-1]\!]$ (with $i_{2}=1$)

[TABLE]

[TABLE]

for all $i_{1}\in[\![3,k-1]\!]$ (with $i_{3}=1$)

[TABLE]

[TABLE]

and finally for all $\mathbf{i}$ with $i_{1}>2$, $i_{2}>1$ and $i_{3}>1$

[TABLE]

and together they form a linearly independent set that can be evidenced thanks to the fact that each $\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,4,\mathbf{i}})$ contains a term $\mathbf{\Psi}^{k,4,\mathbf{i}-2\mathbf{e}_{1}}$,

• •

$\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,5,(k+2)\mathbf{e}_{2}})=(k+2)(k+1)\mathbf{\Psi}^{k,5,k\mathbf{e}_{2}}$, $\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,5,(k+1)\mathbf{e}_{2}+\mathbf{e}_{1}})=(k+1)k\mathbf{\Psi}^{k,5,(k-1)\mathbf{e}_{2}+\mathbf{e}_{1}}$,

[TABLE]

[TABLE]

for all $i_{2}\in[\![3,k]\!]$ (with $i_{1}=0$)

[TABLE]

for all $i_{2}\in[\![3,k]\!]$ (with $i_{3}=0$)

[TABLE]

for all $i_{2}\in[\![3,k-1]\!]$ (with $i_{1}=1$)

[TABLE]

[TABLE]

for all $i_{2}\in[\![3,k-1]\!]$ (with $i_{3}=1$)

[TABLE]

[TABLE]

and finally for all $\mathbf{i}$ with $i_{1}>1$, $i_{2}>2$ and $i_{3}>1$

[TABLE]

and together they form a linearly independent set that can be evidenced thanks to the fact that each $\overrightarrow{L}_{k}(\mathbf{\Psi}^{k+2,5,\mathbf{i}})$ contains a term $\mathbf{\Psi}^{k,5,\mathbf{i}-2\mathbf{e}_{2}}$

This proves the claim that $\dim\overrightarrow{L}_{k}\left(\tilde{\mathbb{S}}_{k+2}\right)\geq(k+1)(k+3)$ as the union of the previous sets contains $3(k+1)+2\frac{k(k+1)}{2}=(k+1)(k+3)$ elements. As a consequence $\overrightarrow{L}_{k}\left(\tilde{\mathbb{S}}_{k+2}\right)=\tilde{\mathbb{S}}_{k}$ and the operator is surjective.

As a summary, for any $k\in\mathbb{N}_{0}$, $\dim\overrightarrow{L}_{k}\left({\tilde{\mathbb{S}}_{k+2}}\right)=(k+1)(k+3)$. The final result follows from the rank-nullity theorem combined with Lemma 2 since

[TABLE]

∎

Note that, given any $\mathbf{\Phi}\in\tilde{\mathbb{S}}_{k+2}$, $\overrightarrow{\Delta}\mathbf{\Phi}=-\nabla\times\nabla\times\mathbf{\Phi}$, so that $\nabla\cdot(\overrightarrow{L}_{k}\mathbf{\Phi})=0$. This also confirms the codomains of the operators of interest.

Corollary 2**.**

The vector Laplacian operator restricted to the complement of homogeneous polynomial harmonic fields in the space of solenoidal fields, namely

[TABLE]

is also surjective. As a consequence,

[TABLE]

Proof.

Since for all $\mathbf{\Phi}\in\tilde{\mathbb{S}}_{k+2}$, $\overrightarrow{\Delta}\mathbf{\Phi}=-\nabla\times\nabla\times\mathbf{\Phi}$, then $\widetilde{\mathbb{H}}_{k+2}\subset\ker\left(\overrightarrow{L}_{k}|_{\tilde{\mathbb{S}}_{k+2}}\right)$. Moreover, by definition, for all $k\in\mathbb{N}_{0}$, $\widetilde{\mathbb{S}}_{k+2}=\widetilde{\mathbb{S}}_{k+2}^{*}\oplus\widetilde{\mathbb{H}}_{k+2}$. Therefore, following Proposition 6, $\overrightarrow{L}_{k}|_{\tilde{\mathbb{S}}^{*}_{k+2}}$ is indeed surjective.

Finally, the dimension of the kernel follows from the fact that according to Proposition 2 for all $k\in\mathbb{N}_{0}$, $\dim\widetilde{\mathbb{H}}_{k}=2k+3$:

[TABLE]

∎

4.3 Construction of individual quasi-Trefftz functions

Thanks the characterization of the quasi-Trefftz space from Proposition 4 and the two previous surjective operators from the previous section, individual quasi-Trefftz functions can be constructed one homogeneous polynomial component at a time, as in the scalar case. Besides the graded structure of polynomial spaces, here the key point is that the divergence condition ensures that the right-hand side of the Laplacian condition belongs to the desired spaces.

The proposed procedure to construct $\displaystyle\mathbf{\Pi}_{0}+\sum_{k=1}^{p}\mathbf{F}_{k}+\mathbf{G}_{k}+\mathbf{H}_{k}\in\mathbb{Q}\mathbb{T}_{p}$ is as follows.

Choose $\mathbf{\Pi}_{0}\in({\mathbb{P}}_{0})^{3}$. 2. 2.

Compute $tmp_{D}:=-\epsilon_{0}^{-1}G_{0}\epsilon_{1}\cdot\mathbf{\Pi}_{0}$. 3. 3.

Find $\mathbf{G}_{1}\in\widetilde{\mathbb{I}}^{*}_{1}$ such that $D_{0}\mathbf{G}_{1}=tmp_{D}$. 4. 4.

Choose $(\mathbf{F}_{1},\mathbf{H}_{1})\in\widetilde{\mathbb{S}}^{*}_{1}\times\widetilde{\mathbb{H}}_{1}$. 5. 5.

For increasing values of $k$ from [math] to $p-2$ perform steps 5a to 5c.

1. (a)

Compute \left\{\begin{array}[]{l}\displaystyle tmp_{D}:=-\epsilon_{0}^{-1}\left(\sum_{k^{\prime}=0}^{k+1}G_{k+1-k^{\prime}}\epsilon_{k+2-k^{\prime}}\cdot(\mathbf{F}_{k^{\prime}}+\mathbf{G}_{k^{\prime}}+\mathbf{H}_{k^{\prime}})+\sum_{k^{\prime}=1}^{k+1}\epsilon_{k+2-k^{\prime}}D_{k^{\prime}-1}\mathbf{G}_{k^{\prime}}\right),\\ \displaystyle tmp_{L}:=\sum_{k^{\prime}=0}^{k}\epsilon_{k-k^{\prime}}(\mathbf{F}_{k^{\prime}}+\mathbf{G}_{k^{\prime}}+\mathbf{H}_{k^{\prime}}).\end{array}\right. 2. (b)

Find $(\mathbf{G}_{k+2},\mathbf{F}_{k+2})\in\widetilde{\mathbb{I}}^{*}_{k+2}\times\widetilde{\mathbb{S}}^{*}_{k+2}$ such that \left\{\begin{array}[]{l}D_{k+1}\mathbf{G}_{k+2}=tmp_{D},\\ \overrightarrow{L}_{k}\mathbf{F}_{k+2}=tmp_{L}.\end{array}\right. 3. (c)

Choose $\mathbf{H}_{k+2}\in\widetilde{\mathbb{H}}_{k+2}$.

In order to verify that this procedure indeed constructs a quasi-Trefftz function, there are three points to justify.

• •

First, the temporary quantities can only be computed if they depend only on known quantities. In step 2, this is the case as $\mathbf{\Pi}_{0}$ in known from step 1. In step 5a, this is the case since the iterations are performed for increasing values of $k$. Indeed, given $k\in[\![0,p-2]\!]$, all quantities

[TABLE]

are already known from previous steps.

• •

Second, finding a solution to a divergence equation is always possible since all operators $D_{k}|_{\widetilde{\mathbb{I}}^{*}_{k+1}}$ are surjective according respectively to Lemma 2 for $k=0$ and to Proposition 5 for $k>0$.

• •

Third, finding a solution to a Laplace equation is always possible. Indeed, according to Corollary 2 $\overrightarrow{L}_{k}({\tilde{\mathbb{S}}_{k+2}})={\tilde{\mathbb{S}}_{k}}$, and the right hand side of $tmp_{L}$ computed at step 5a is precisely guaranteed to belong to ${\tilde{\mathbb{S}}_{k}}$ from the divergence condition as $\nabla\cdot\left(tmp_{L}\right)=\mathbf{0}$ is equivalent to

[TABLE]

Remark 7**.**

For the computation of an individual quasi-Trefftz function, steps 5b and 5c from the previous procedure can be replaced by a single step by seeking solutions to both equations in the full solenoidal and irrotational spaces.

[TABLE]

Such solutions exist according to Lemma 2 and Proposition 6. In this case $\mathbf{G}_{k+2}$ might have an $\widetilde{\mathbb{S}}_{k+2}$ component and $\mathbf{F}_{k+2}$ might have a harmonic component.

This can be implemented as follows. Following the proof of Lemma 2, given $tmp_{D}$ expressed in the canonical basis as $tmp_{D}=\sum_{|\mathbf{i}|=k+1}\lambda_{\mathbf{i}}\mathbf{X}^{\mathbf{i}}$, then

[TABLE]

Following the proof of Proposition 6, this can be performed similarly for the vector Laplacian equation given $tmp_{L}$ expressed in

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

or if $k>1$ in the basis of $\tilde{\mathbb{S}}_{k}$ defined as the union of

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By comparison, implementing the procedure as presented earlier requires right inverse operators for $D_{k+1}|_{\tilde{\mathbb{I}}^{*}_{k+2}}$ and $\overrightarrow{L}_{k}|_{\tilde{\mathbb{S}}^{*}_{k+2}}$.

4.4 Dimension

One can now establish the dimension of the quasi-Trefftz space.

Proposition 7**.**

Given $p\in\mathbb{N}$ such that $p>2$ and $\mathbf{x}_{0}\in\mathbb{R}^{3}$, consider the second order differential operator $\mathcal{L}_{M}:=\nabla\times\nabla\times-\epsilon$ with scalar variable coefficients $\epsilon\in\mathcal{C}^{p}(\mathbf{x}_{0})$, the dimension of quasi-Trefftz space is

[TABLE]

Proof.

Following the procedure from the previous section, combined with Proposition 2, Corollary 1, Proposition 5 and Proposition 6, the dimension of the quasi-Trefftz space can be computed as

[TABLE]

which concludes the proof. ∎

Remark 8**.**

In [Ces96b] (see Section II.9.1), a discrete PW space of dimension $(p+3)(p+1)$ is proposed to approximate solutions of a single first order equation $\nabla\times\mathbf{V}=\omega\mathbf{V}$. This would correspond to a space of dimension $2(p+3)(p+1)$ to approximate solutions of two first order equations or equivalently a second order equation $\nabla\times\nabla\times\mathbf{V}=\omega^{2}\mathbf{V}$. For comparison, a quasi-Trefftz space

[TABLE]

would precisely be of dimension $2(p+3)(p+1)=2p^{2}+8p+6$, while the proposed quasi-Trefftz space $\mathbb{Q}\mathbb{T}_{p}$ introduced in Definition 10 is smaller due to the choice to impose the divergence condition to a higher order. It is expected that this choice of a smaller space will be sufficient to achieve desired best approximation properties.

5 Conclusion

The general framework of the work presented here is the following. Consider a vector space $\mathbb{V}$, as well as an exact sequence of differential operators (each one of them being of order one)

[TABLE]

While it is more standard to study these operators defined between standard spaces of polynomials, this article strongly leverages the graded structure of polynomial spaces and properties of homogeneous polynomial spaces. In the general context of polynomial spaces, the differential operator for Maxwell’s equation is $\mathcal{L}_{M}=\mathcal{L}_{2}\circ\mathcal{L}_{2}-\epsilon Id$ where $\mathcal{L}_{2}$ is the curl operator, $\epsilon$ is a variable coefficient and $Id$ is the identity operator. This operator acts between spaces of vector fields, and this

The goal of this work is to introduce a notion of polynomial quasi-Trefftz space for this differential operator $\mathcal{L}_{M}$, to study its underlying structure, propose a procedure to build elements of this space and finally finds the dimension of the space.

Back to the general framework, assuming that the operator $\mathcal{L}_{3}\circ\mathcal{L}_{1}$ is surjective leads to the existence of a Helmholtz decomposition. Then defining $\mathbb{H}:=\ker\mathcal{L}_{2}\cap\ker\mathcal{L}_{3}\subset\mathbb{V}^{3}$, and assuming that $\dim\ker\mathcal{L}_{2}+\dim\ker\mathcal{L}_{3}-\dim\mathbb{H}=3\dim\mathbb{V}$ leads to the uniqueness of the decomposition. To be more explicit, one can define ${\mathbb{S}}^{*}$ and ${\mathbb{I}}^{*}$ such that

[TABLE]

and this then implies that $\mathbb{V}^{3}={\mathbb{I}}^{*}\oplus\mathbb{H}\oplus{\mathbb{S}}^{*}$. Moreover, as in the scalar case, the highest-order derivative terms in $\mathcal{L}_{M}$ play a crucial role in this work. Here the component-wise version of the operator $\mathcal{L}_{3}\circ\mathcal{L}_{1}$, denoted $\overrightarrow{\mathcal{L}_{3}\circ\mathcal{L}_{1}}$, comes into play to leverage the unique decomposition assuming that $\mathcal{L}_{2}\circ\mathcal{L}_{2}=-\overrightarrow{\mathcal{L}_{3}\circ\mathcal{L}_{1}}+\mathcal{L}_{1}\circ\mathcal{L}_{3}$. However, unlike in the scalar case, differential operators between spaces of polynomial vector fields are not necessarily surjective, and in particular $\mathcal{L}_{2}\circ\mathcal{L}_{2}$ is not surjective. The well documented divergence condition is a consequence of the exact sequence definition: $\mathcal{R}(\mathcal{L}_{2})=\ker\mathcal{L}_{3}$, therefore $\mathcal{L}_{M}\varphi=0$ implies that $\epsilon\varphi\in\ker\mathcal{L}_{3}$. This fact, together with the unique decomposition, can then be used to express $\mathcal{L}_{M}\varphi=0$ equivalently as the combination of a divergence-free condition for the full vector field $\varphi$ and a differential equation in which the only second order term is the $\overrightarrow{\mathcal{L}_{3}\circ\mathcal{L}_{1}}$ operator acting only on the $\mathbb{S}^{*}$ component. This combined expression now shares with the scalar case the fact that the highest-order derivative operators, acting between well-chosen spaces, are surjective.

This fact, leveraged in the quasi-Trefftz setting, is precisely the corner stone of the present study of the quasi-Trefftz space $\mathbb{Q}\mathbb{T}_{p}$ for Maxwell’s equation $\mathcal{L}_{M}\varphi=0$.

The work presented in Section 2, Section 3 and Section 4.2 is independent of the variable coefficient $\epsilon$ and could be used to study different formulations of Maxwell’s equations. One particular direction of interest for future development is the study of polynomial quasi-Trefftz spaces for a first order formulation of Maxwell’s equations.