Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds

Abstract

We develop exterior calculus approaches for partial differential equations on radial manifolds. We introduce numerical methods that approximate with spectral accuracy the exterior derivative $\mathbf {d}$, Hodge star $\star$, and their compositions. To achieve discretizations with high precision and symmetry, we develop hyperinterpolation methods based on spherical harmonics and Lebedev quadrature. We perform convergence studies of our numerical exterior derivative operator $\overline{\mathbf {d}}$ and Hodge star operator $\overline{\star }$ showing each converge spectrally to $\mathbf {d}$ and $\star$. We show how the numerical operators can be naturally composed to formulate general numerical approximations for solving differential equations on manifolds. We present results for the Laplace–Beltrami equations demonstrating our approach.

Appendices

Appendix

Appendix A: Spherical Harmonics

The spherical harmonics are given by

$$\begin{aligned} Y^m_n(\theta , \phi ) = \sqrt{\frac{(2n+1)(n-m)!}{4\pi (n+m)!}}P^m_n\left( \cos (\phi )\right) \exp \left( {im\theta }\right) \end{aligned}$$

(19)

where m denotes the order and n the degree for $n \ge 0$ and $m \in \{-n, \dots , n\}$. The $P^m_n$ denote the Associated Legendre Polynomials. In our notation, $\theta$ denotes the azimuthal angle and $\phi$ the polar angle of the spherical coordinates [17].

Since we work throughout only with real-valued functions, we have that the modes are self-conjugate and we use that $Y^m_n = \overline{Y^{-m}_n}$. We have found it convenient to represent the spherical harmonics as

$$\begin{aligned} Y^m_n(\theta , \phi ) = X^m_n(\theta , \phi )+ iZ^m_n(\theta , \phi ) \end{aligned}$$

(20)

where $X_n^m$ and $Z_n^m$ denote the real and imaginary parts. In our numerical methods we use this splitting to construct a purely real set of basis functions on the unit sphere with maximum degree N. We remark that this consists of $(N+1)^2$ basis elements. In the case of $N=2$ we have the basis elements

$$\begin{aligned}&\tilde{Y}_1 = Y^0_0, \tilde{Y}_2 = Z^1_1, \tilde{Y}_3 = Y^0_1, \tilde{Y}_4 = X^1_1, \tilde{Y}_5 = Z^2_2,\nonumber \\&\quad \tilde{Y}_6 = Z^1_2, \tilde{Y}_7 = Y^0_2, \tilde{Y}_8 = X^1_2, \tilde{Y}_9 = X^2_2. \end{aligned}$$

(21)

We use a similar convention for the basis for the other values of N. We take our final basis elements $Y_i$ to be the normalized as $Y_i = \tilde{Y}_i/\sqrt{\langle \tilde{Y}_i, \tilde{Y}_i \rangle }$.

We compute derivatives of our finite expansions by evaluating analytic formulas for the spherical harmonics in order to try to minimize approximation error [17]. Approximation errors are incurred when sampling the values of expressions involving these derivatives at the Lebedev nodes and performing quadratures. The derivative in the azimuthal coordinate $\theta$ of the spherical harmonics is given by

$$\begin{aligned} \partial _\theta Y^m_n(\theta , \phi ) = \partial _\theta \sqrt{\frac{(2n+1)(n-m)!}{4\pi (n+m)!}}P^m_n(\cos (\phi )) \exp \left( {im\theta }\right) = imY^m_n\left( \theta , \phi \right) . \end{aligned}$$

This maps the spherical harmonic of degree n to again a spherical harmonic of degree n. In our numerics, this derivative can be represented in our finite basis which allows us to avoid projections. This allows for computing the derivative in $\theta$ without incurring an approximation error. For the derivative in the polar angle $\phi$ we have that

$$\begin{aligned} \partial _\phi Y^m_n(\theta , \phi ) = m \cot (\phi )Y^m_n(\theta , \phi )+\sqrt{(n-m)(n+m+1)}\exp \left( {-i\theta }\right) Y^{m+1}_n(\theta , \phi ). \end{aligned}$$

(22)

We remark that the expression can not be represented in terms of a finite expansion of spherical harmonics. We use this expression for $\partial _\phi Y^m_n(\theta , \phi )$ when we compute values at the Lebedev quadrature nodes in Eq. (14). This provides a convenient way to compute derivatives of differential forms following the approach discussed in Sect. 3. We remark that it is the subsequent hyperinterpolation of the resulting expressions where the approximation error is incurred. We adopt the notational convention that $Y^{m}_n = 0$ when $m \ge n+1$. For further discussion of spherical harmonics see [17].

Appendix B: Differential Geometry of Radial Manifolds

We consider throughout manifolds of radial shape. A radial manifold is defined as a surface where each point can be connected by a line segment to the origin without intersecting the surface. In spherical coordinates, any point $\mathbf {x}$ on the radial manifold can be expressed as

$$\begin{aligned} \mathbf {x}(\theta ,\phi ) = \varvec{\sigma }(\theta , \phi ) = r(\theta , \phi )\mathbf {r}(\theta ,\phi ) \end{aligned}$$

(23)

where $\mathbf {r}$ is the unit vector from the origin to the point on the sphere corresponding to angle $\theta ,\phi$ and r is a positive scalar function.

We take an isogeometric approach to representing the manifold M. We sample the scalar function r at the Lebedev nodes and represent the geometry using the finite spherical harmonics expansion $r(\theta ,\phi ) = \sum _i \bar{r}_i Y_i$ up to the order $\lfloor L/2 \rfloor$ where $\bar{r}_i = \langle r, Y_i \rangle _Q$ for a quadrature of order L.

We consider two coordinate charts for our calculations. The first is referred to as Chart A and has coordinate singularities at the north and south pole. The second is referred to as Chart B and has coordinate singularities at the east and west pole. For each chart we use spherical coordinates with $(\theta , \phi ) \in [0 , 2\pi ) \times [0 , \pi ]$ but to avoid singularities only use values in the restricted range $\phi \in [\phi _{min}, \phi _{max}]$, where $0 < \phi _{min} \le \frac{\pi }{4}$, and $\frac{3\pi }{4} \le \phi _{max} < \pi$. In practice, one typically takes $\phi _{min} = 0.8\times \frac{\pi }{4}$ and $\phi _{max} = 0.8 \times \pi$. For Chart A, the manifold is parameterized in the embedding space $\mathbb {R}^3$ as

$$\begin{aligned} \mathbf {x}(\hat{\theta }, \hat{\phi }) = r(\hat{\theta }, \hat{\phi })\mathbf {r}(\hat{\theta }, \hat{\phi }), \mathbf {r}(\hat{\theta }, \hat{\phi }) = \begin{bmatrix}\sin (\hat{\phi })\cos (\hat{\theta }),&\sin (\hat{\phi })\sin (\hat{\theta }),&\cos (\hat{\phi }) \end{bmatrix} \end{aligned}$$

(24)

and for Chart B

$$\begin{aligned} \mathbf {x}(\bar{\theta }, \bar{\phi }) = r(\bar{\theta }, \bar{\phi })\mathbf {r}(\bar{\theta }, \bar{\phi }), \bar{\mathbf {r}}(\bar{\theta }, \bar{\phi }) = \begin{bmatrix} \cos (\bar{\phi }),&\sin (\bar{\phi })\sin (\bar{\theta }),&-\sin (\bar{\phi })\cos (\bar{\theta }) \end{bmatrix}. \quad \end{aligned}$$

(25)

With these coordinate representations, we can derive explicit expressions for geometric quantities associated with the manifold such as the metric tensor and shape tensor. The derivatives used as the basis $\partial _\theta , \partial _\phi$ for the tangent space can be expressed as

$$\begin{aligned} \varvec{\sigma }_{\theta }(\theta , \phi )= & {} r_{\theta }(\theta , \phi )\mathbf {r}(\theta , \phi ) + r(\theta , \phi ) \mathbf {r}_{\theta }(\theta , \phi )\end{aligned}$$

(26)

$$\begin{aligned} \varvec{\sigma }_{\phi }(\theta , \phi )= & {} r_{\phi }(\theta , \phi )\mathbf {r}(\theta , \phi ) + r(\theta , \phi ) \mathbf {r}_{\phi }(\theta , \phi ). \end{aligned}$$

(27)

We have expressions for $\mathbf {r}_{\theta }$ and $\mathbf {r}_{\phi }$ in the embedding space $\mathbb {R}^3$ using Eq. (24) or (25) depending on the chart being used. The first fundamental form $\mathbf {I}$ (metric tensor) and second fundamental form $\mathbf {II}$ (shape tensor) are given by

$$\begin{aligned} \mathbf {I} = \begin{bmatrix} E&F \\ F&G \end{bmatrix} = \begin{bmatrix} \varvec{\sigma }_{\theta } \cdot \varvec{\sigma }_{\theta }&\varvec{\sigma }_{\theta } \cdot \varvec{\sigma }_{\phi } \\ \varvec{\sigma }_{\phi } \cdot \varvec{\sigma }_{\theta }&\varvec{\sigma }_{\phi } \cdot \varvec{\sigma }_{\phi } \end{bmatrix} = \begin{bmatrix} r_{\theta }^2+r^2\sin (\phi )^2&r_{\theta }r_{\phi } \\ r_{\theta }r_{\phi }&r_{\phi }^2+r^2 \end{bmatrix}. \end{aligned}$$

(28)

and

$$\begin{aligned} \mathbf {II} = \begin{bmatrix} L&M \\ N&N \end{bmatrix} = \begin{bmatrix} \varvec{\sigma }_{\theta \theta } \cdot \varvec{n}&\varvec{\sigma }_{\theta \phi } \cdot \varvec{n} \\ \varvec{\sigma }_{\phi \theta } \cdot \varvec{n}&\varvec{\sigma }_{\phi \phi }\cdot \varvec{n} \end{bmatrix}. \end{aligned}$$

(29)

The $\mathbf {n}$ denotes the outward normal on the surface and is computed using

$$\begin{aligned} \varvec{n}(\theta , \phi ) = \frac{\varvec{\sigma }_{\theta }(\theta , \phi ) \times \varvec{\sigma }_{\phi }(\theta , \phi )}{\Vert \varvec{\sigma }_{\theta }(\theta , \phi ) \times \varvec{\sigma }_{\phi }(\theta , \phi ) \Vert }. \end{aligned}$$

(30)

The terms $\varvec{\sigma }_{\theta \theta }$, $\varvec{\sigma }_{\theta \phi }$, and $\varvec{\sigma }_{\phi ,\phi }$ are obtained by further differentiation from Eqs. (26) to (27). We use the notation for the metric tensor $\mathbf {g} = \mathbf {I}$ interchangeably. In practical calculations whenever we need to compute the action of the inverse metric tensor we do so through numerical linear algebra (Gaussian elimination with pivoting) [37, 38]. For notational convenience, we use the tensor notation for the metric tensor $g_{ij}$ and its inverse $g^{ij}$ which has the formal correspondence

$$\begin{aligned} g_{ij} = \left[ \mathbf {I}\right] _{i,j}, g^{ij} = \left[ \mathbf {I}^{-1}\right] _{i,j}. \end{aligned}$$

(31)

For the metric factor we also have that

$$\begin{aligned} \sqrt{|g|} = \sqrt{\det (\mathbf {I})} = r\sqrt{r_{\theta }^2+(r_{\phi }^2+r^2)\sin (\phi )^2} = \Vert \mathbf {\sigma }_{\theta }(\theta , \phi ) \times \mathbf {\sigma }_{\phi }(\theta , \phi ) \Vert . \end{aligned}$$

(32)

To ensure accurate numerical calculations in each of the above expressions the appropriate coordinates either Chart A or Chart B are used to ensure sufficient distance from coordinate singularities at the poles. To compute quantities associated with curvature of the manifold we use the Weingarten map [30] which can be expressed as

$$\begin{aligned} \mathbf {W} = -\mathbf {I}^{-1} \mathbf {II}. \end{aligned}$$

(33)

To compute the Gaussian curvature K, we use

$$\begin{aligned} K(\theta ,\phi ) = \det \left( \mathbf {W}(\theta ,\phi )\right) . \end{aligned}$$

(34)

For further discussion of the differential geometry of manifolds see [29,30,31].