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Green's functions

sies.greens

Green's function of the 2D Laplacian and its derivatives.

The fundamental solution of the Laplace equation in two dimensions is

\[G(x) = \frac{1}{2\pi} \log|x|.\]

All functions in this module are vectorized: point clouds are given as arrays of shape (2, n) and the result is a matrix indexed by the two point clouds.

green2d 

green2d(x, y)

Evaluate the 2D Green's function on two point clouds.

Parameters:

Name Type Description Default
x (ndarray, shape(2, m))

First point cloud.

 required
y (ndarray, shape(2, n))

Second point cloud. Must be disjoint from x (the kernel is singular on the diagonal).

 required

Returns:

Type Description
(ndarray, shape(m, n))

Matrix G with G[i, j] = log(|x_i - y_j|) / (2 pi).
Source code in src/sies/greens.py 
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def green2d(x: NDArray, y: NDArray) -> NDArray:
    """Evaluate the 2D Green's function on two point clouds.

    Parameters
    ----------
    x : ndarray, shape (2, m)
        First point cloud.
    y : ndarray, shape (2, n)
        Second point cloud. Must be disjoint from `x` (the kernel is
        singular on the diagonal).

    Returns
    -------
    ndarray, shape (m, n)
        Matrix ``G`` with ``G[i, j] = log(|x_i - y_j|) / (2 pi)``.
    """
    _check_points(x, y)
    dx = x[0][:, np.newaxis] - y[0][np.newaxis, :]
    dy = x[1][:, np.newaxis] - y[1][np.newaxis, :]
    return np.log(dx**2 + dy**2) / (4 * np.pi)

green2d_grad 

green2d_grad(x, y)

Evaluate the gradient of the 2D Green's function.

Parameters:

Name Type Description Default
x (ndarray, shape(2, m))

First point cloud.

 required
y (ndarray, shape(2, n))

Second point cloud.

 required

Returns:

Name Type Description
gx (ndarray, shape(m, n))

Partial derivative with respect to the first coordinate, evaluated at x_i - y_j.
gy (ndarray, shape(m, n))

Partial derivative with respect to the second coordinate, evaluated at x_i - y_j.
Source code in src/sies/greens.py 
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def green2d_grad(x: NDArray, y: NDArray) -> tuple[NDArray, NDArray]:
    """Evaluate the gradient of the 2D Green's function.

    Parameters
    ----------
    x : ndarray, shape (2, m)
        First point cloud.
    y : ndarray, shape (2, n)
        Second point cloud.

    Returns
    -------
    gx : ndarray, shape (m, n)
        Partial derivative with respect to the first coordinate,
        evaluated at ``x_i - y_j``.
    gy : ndarray, shape (m, n)
        Partial derivative with respect to the second coordinate,
        evaluated at ``x_i - y_j``.
    """
    _check_points(x, y)
    dx = x[0][:, np.newaxis] - y[0][np.newaxis, :]
    dy = x[1][:, np.newaxis] - y[1][np.newaxis, :]
    dist2 = dx**2 + dy**2
    return dx / (2 * np.pi * dist2), dy / (2 * np.pi * dist2)

green2d_dn 

green2d_dn(x, y, normal)

Evaluate the normal derivative of the Green's function on a boundary.

For each source point \(x_i\) and boundary point \(y_j\) with outward normal \(\nu_j\), compute \(\langle \nabla G(y_j - x_i), \nu_j \rangle\).

Parameters:

Name Type Description Default
x (ndarray, shape(2, m))

Evaluation (source) points.

 required
y (ndarray, shape(2, n))

Boundary points.

 required
normal (ndarray, shape(2, n))

Outward unit normal vectors at the boundary points.

 required

Returns:

Type Description
(ndarray, shape(m, n))

The normal derivative matrix.
Source code in src/sies/greens.py 
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def green2d_dn(x: NDArray, y: NDArray, normal: NDArray) -> NDArray:
    r"""Evaluate the normal derivative of the Green's function on a boundary.

    For each source point $x_i$ and boundary point $y_j$ with
    outward normal $\nu_j$, compute
    $\langle \nabla G(y_j - x_i), \nu_j \rangle$.

    Parameters
    ----------
    x : ndarray, shape (2, m)
        Evaluation (source) points.
    y : ndarray, shape (2, n)
        Boundary points.
    normal : ndarray, shape (2, n)
        Outward unit normal vectors at the boundary points.

    Returns
    -------
    ndarray, shape (m, n)
        The normal derivative matrix.
    """
    _check_points(x, y, normal)
    gx, gy = green2d_grad(y, x)
    gn = normal[0][:, np.newaxis] * gx + normal[1][:, np.newaxis] * gy
    return gn.T