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A roadmap for Generalized Plane Waves and their interpolation properties

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Abstract

This work focuses on the study of partial differential equation (PDE) based basis function for Discontinuous Galerkin methods to solve numerically wave-related boundary value problems with variable coefficients. To tackle problems with constant coefficients, wave-based methods have been widely studied in the literature: they rely on the concept of Trefftz functions, i.e. local solutions to the governing PDE, using oscillating basis functions rather than polynomial functions to represent the numerical solution. Generalized Plane Waves (GPWs) are an alternative developed to tackle problems with variable coefficients, in which case Trefftz functions are not available. In a similar way, they incorporate information on the PDE, however they are only approximate Trefftz functions since they don’t solve the governing PDE exactly, but only an approximated PDE. Considering a new set of PDEs beyond the Helmholtz equation, we propose to set a roadmap for the construction and study of local interpolation properties of GPWs. Identifying carefully the various steps of the process, we provide an algorithm to summarize the construction of these functions, and establish necessary conditions to obtain high order interpolation properties of the corresponding basis.

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Notes

  1. http://www.waveller.com/Waveller_Acoustics/waveller_acoustics.shtml.

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Acknowledgements

This material is based upon work supported by the National Science Foundation under Grants No. DMS-1818747 and DMS-2105487.

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Authors and Affiliations

  1. Department of Mathematics, The University of Arizona, Tucson, AZ, USA

    Lise-Marie Imbert-Gérard

  2. Central R&T, Airbus, AIRBUS Central R&T / XRV, 22 Rue du Gouverneur Général EBOUÉ, 92130 , Issy Les Moulineaux, France

    Guillaume Sylvand

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  1. Lise-Marie Imbert-Gérard
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Correspondence to Lise-Marie Imbert-Gérard.

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Appendices

Chain rule in dimension 1 and 2

For the sake of completeness, this section is dedicated to describing the formula to derive a composition of two functions, in dimensions one and two. A wide bibliography about this formula is to be found in [26]. It is linked to the notion of partition of an integer or the one of a set. The 1D version is not actually used in this work but is displayed here as a comparison with a 2D version, mainly concerning this notion of partition.

1.1 Faa Di Bruno formula

Faa Di Bruno formula gives the mth derivative of a composite function with a single variable. It is named after Francesco Faa Di Bruno, but was stated in earlier work of Louis F.A. Arbogast around 1800, see [7].

If f and g are functions with sufficient derivatives, then

$$\begin{aligned} { \frac{d^m}{dx^m}f(g(x)) = m!\sum f^{(\sum _k b_k)} (g(x)) \prod _{k=1}^{m} \frac{1}{b_k!}\left( \frac{g^{(k)}(x)}{k!} \right) ^{b_k}, } \end{aligned}$$

where the sum is over all different solutions in nonnegative integers \((b_k)_{k\in [\![1,m]\!]}\) of \(\sum _k k b_k = m\). These solutions are actually the partitions of m.

1.2 Bivariate version

The multivariate formula has been widely studied, the version described here is the one from [6] applied to dimension 2. A linear order on \({\mathbb {N}}^2\) is defined by: \(\forall (\mu ,\nu )\in \left( {\mathbb {N}}^2\right) ^2\), the relation \(\mu \prec \nu \) holds provided that

  1. 1.

    \(\mu _1+\mu _2<\nu _1+\nu _2 \); or

  2. 2.

    \(\mu _1+\mu _2=\nu _1+\nu _2 \) and \(\mu _1<\nu _1\).

If f and g are functions with sufficient derivatives, then

$$\begin{aligned}&\partial _x^i \partial _y^j f(g(x,y)) \\&\quad =i!j! \sum _{1\le \mu \le i+j} f^{\mu }(g(x,y)) \sum _{s=1}^{i+j} \sum _{p_s((i,j),\mu )} \prod _{l=1}^s \frac{1}{k_l!}\left( \frac{1}{i_l!j_l!} \partial _x^{i_l}\partial _y^{j_l} (g(x,y))\right) ^{k_l}, \end{aligned}$$

where the partitions of (i, j) are defined by the following sets: \(\forall \mu \in [\![1,i+j]\!]\), \(\forall s\in [\![1,i+j]\!]\), \(p_s((i,j),\mu )\) is equal to

$$\begin{aligned}&\left\{ (k_1,\ldots ,k_s;(i_1,j_1),\ldots ,(i_s,j_s)){:}\,k_i>0,0\prec (i_1,j_1)\right. \\&\quad \left. \prec \cdots \prec (i_s,j_s), \sum _{l=1}^s k_l=\mu ,\sum _{l=1}^s k_li_l=i,\sum _{l=1}^s k_lj_l=j\right\} . \end{aligned}$$

See [12] for a proof of the formula interpreted in terms of collapsing partitions.

Faa di Bruno

The multivariate formula has been widely studied, the version described here is the one from [6] applied to dimension 2. A linear order on \({\mathbb {N}}^2\) is defined by: \(\forall (\mu ,\nu )\in \left( {\mathbb {N}}^2\right) ^2\), the relation \(\mu \prec \nu \) holds provided that

  1. 1.

    \(\mu _1+\mu _2<\nu _1+\nu _2 \); or

  2. 2.

    \(\mu _1+\mu _2=\nu _1+\nu _2 \) and \(\mu _1<\nu _1\).

If f and g are functions with sufficient derivatives, then

$$\begin{aligned} \partial _x^i \partial _y^j f(g(x,y))= & {} i!j! \sum _{1\le \mu \le i+j} f^{(\mu )}(g(x,y)) \sum _{s=1}^{i+j} \sum _{p_s((i,j),\mu )} \prod _{l=1}^s \frac{1}{k_l!}\\&\quad \left( \frac{1}{i_l!j_l!} \partial _x^{i_l}\partial _y^{j_l} (g(x,y))\right) ^{k_l},\\ \partial _x^k \partial _y^{\ell -k} e^{P(x,y)}= & {} k!{(\ell -k)}! \sum _{1\le \mu \le \ell } e^{P(x,y)} \sum _{s=1}^{\ell } \sum _{p_s((k,\ell -k),\mu )} \prod _{m=1}^s \frac{1}{k_m!}\\&\quad \left( \frac{1}{i_m!j_m!} \partial _x^{i_m}\partial _y^{j_m} P(x,y)\right) ^{k_m}, \end{aligned}$$

where the partitions of (i, j) are defined by the following sets: \(\forall \mu \in [\![1,i+j]\!]\), \(\forall s\in [\![1,i+j]\!]\), \(p_s((i,j),\mu )\) is equal to

$$\begin{aligned}&\left\{ (k_1,\ldots ,k_s;(i_1,j_1),\cdots ,(i_s,j_s)):k_i>0,0\prec (i_1,j_1)\right. \\&\quad \left. \prec \cdots \prec (i_s,j_s), \sum _{l=1}^s k_l=\mu ,\sum _{l=1}^s k_li_l=i,\sum _{l=1}^s k_lj_l=j\right\} . \end{aligned}$$

Note that s is the number of different terms appearing in the product, while \(\mu \) is the number of terms in the product counting multiplicity, \(k_m\) is the multiplicity of the mth term in the product, while \(p_s\) represents the possible partitions of (i, j).

Note that since \(k_m>0\), the condition \(\sum _{m=1}^s k_m=\mu \) implies that \(\mu = \sum _{m=1}^s k_m\ge \sum _{m=1}^s 1 = s\).

Polynomial formulas

Here are two important comments. The first one concerns the product of polynomials. Assume that \(\min (D_1,D_2)\ge q\). Then the product of two polynomials, respectively of degree \(D_1\) and \(D_2\), satisfies:

$$\begin{aligned}&\left( \sum _{i_1=0}^{D_1}\sum _{j_1=0}^{D_1-i_1} p_{i_1,j_1}x^{i_1}y^{j_1}\right) \left( \sum _{i_2=0}^{D_2}\sum _{j_2=0}^{D_2-i_2} q_{i_2,j_2}x^{i_2}y^{j_2}\right) \\&\quad = \sum _{i=0}^{q-1}\sum _{j=0}^{q-1-i} \left( \sum _{{\tilde{i}}=0}^i\sum _{{\tilde{j}}=0}^j p_{i-{\tilde{i}},j-{\tilde{j}}}q_{{\tilde{i}},{\tilde{j}}} \right) x^iy^j+O(h^q). \end{aligned}$$

Since in particular the summation indices are such that \(0\le {\tilde{i}}\le i\), \(0\le i-{\tilde{i}}\le i\), \(0\le {\tilde{j}}\le j\), and \(0\le j-{\tilde{j}}\le j\), the only coefficients \(p_{i,j}\) and \(q_{i,j}\) appearing in the \((I_0,J_0)\) coefficient of the product have a length of the multi-index \(i+j\le I_0+J_0\). As a consequence, the only coefficients of several polynomials appearing in the \((I_0,J_0)\) coefficient of the product these several polynomials have a length of the multi-index \(i+j\le I_0+J_0\). The second comment turns to the derivative of a polynomial:

$$\begin{aligned} \partial _x^I\partial _y^J\left( \sum _{i=0}^{D}\sum _{j=0}^{D-i} p_{i,j}x^{i}y^{j}\right) = \sum _{i=0}^{D-I-J}\sum _{j=0}^{D-I-J-i} \frac{(i+I)!}{i!}\frac{(j+J)!}{j!} p_{i+I,j+J} x^{i}y^{j}. \end{aligned}$$

In particular the only coefficients \(p_{i,j}\) appearing in the \((I_0,J_0)\) coefficient of the derivative has a length of the multi-index \(i+j = I+J+I_0+J_0\).

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Imbert-Gérard, LM., Sylvand, G. A roadmap for Generalized Plane Waves and their interpolation properties. Numer. Math. 149, 87–137 (2021). https://doi.org/10.1007/s00211-021-01220-9

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