kernels

Set of kernel functions and their partial derivative usable in the Green cross approximation method. More...

Functions
HEADER_PREFIX real	laplace_kernel (const field *x, const field *y, const uint dim)
	Implementation of the fundamental solution of the negative Laplace operator \(- \Delta\) in 2D and 3D. More...
HEADER_PREFIX real	pdx_laplace_kernel (const field *x, const field *y, const uint dim, const uint i)
	Calculates the partial derivative of the negative Laplace operator \(- \Delta\) in the i th component of the first operand in 2D and 3D. More...
HEADER_PREFIX real	pdy_laplace_kernel (const field *x, const field *y, const uint dim, const uint i)
	Calculates the partial derivative of the negative Laplace operator \(- \Delta\) in the i th component of the second operand in 2D and 3D. More...

Variables
static const real	r_minus_two_pi = -0.159154943091895336
	\(- \frac{1}{2 \pi}\) More...
static const real	r_four_pi = 0.0795774715459476679
	\(\frac{1}{4 \pi}\) More...
static const real	r_minus_pi = -0.318309886183790671
	\(- \frac{1}{\pi}\) More...

Detailed Description

Set of kernel functions and their partial derivative usable in the Green cross approximation method.

Function Documentation

laplace_kernel()

HEADER_PREFIX real laplace_kernel	(	const field *	x,
		const field *	y,
		const uint	dim
	)

Implementation of the fundamental solution of the negative Laplace operator \(- \Delta\) in 2D and 3D.

Calculates the fundamental solution of the negative Laplace operator:

\[ g(x, y) := \begin{cases} -\frac{1}{2 \pi} \log{\left \Vert x - y \right \Vert_2} & \text{if } x \neq y, \ dim = 2 \\ \\ \frac{1}{4 \pi \left \Vert x - y \right \Vert_2} & \text{if } x \neq y, \ dim = 3 \\ \\ 0 & \text{otherwise} \end{cases} \]

Parameters

[in]	x	First operand as an array of dim coefficients.
[in]	y	Second operand as an array of dim coefficients.
[in]	dim	Dimension in which to evaluate the negative Laplace operator.

Returns

The result of the dim dimensional negative Laplace operator in the parameters x and y.

pdx_laplace_kernel()

HEADER_PREFIX real pdx_laplace_kernel	(	const field *	x,
		const field *	y,
		const uint	dim,
		const uint	i
	)

Calculates the partial derivative of the negative Laplace operator \(- \Delta\) in the i th component of the first operand in 2D and 3D.

Calculates the partial derivative of the fundamental solution of the negative Laplace operatorin in the i th component of the first operand:

\[ \frac{\delta g}{\delta x_{i}}(x, y) := \begin{cases} - \frac{x_i - y_i}{\pi \left \Vert x - y \right \Vert_{2}^{2}} & \text{if } x \neq y, \ dim = 2 \\ \\ - \frac{x_i - y_i}{4 \pi \left \Vert x - y \right \Vert_{2}^{3}} & \text{if } x \neq y, \ dim = 3 \\ \\ 0 & \text{otherwise} \end{cases} \]

Parameters

[in]	x	First operand as an array of dim coefficients. Parital derivating the i th component of this operand.
[in]	y	Second operand as an array of dim coefficients.
[in]	dim	Dimension in which to evaluate the partial derivative of the negative Laplace operator
[in]	i	Index of the component of x in which to parital derive.

Returns

The result of the dim dimensional partial derivative of the fundamental solution of the negative Laplace operator.

pdy_laplace_kernel()

HEADER_PREFIX real pdy_laplace_kernel	(	const field *	x,
		const field *	y,
		const uint	dim,
		const uint	i
	)

Calculates the partial derivative of the negative Laplace operator \(- \Delta\) in the i th component of the second operand in 2D and 3D.

Calculates the partial derivative of the fundamental solution of the negative Laplace operatorin in the i th component of the second operand:

\[ \frac{\delta g}{\delta y_{i}}(x, y) := \begin{cases} \frac{x_i - y_i}{\pi \left \Vert x - y \right \Vert_{2}^{2}} & \text{if } x \neq y, \ dim = 2 \\ \\ \frac{x_i - y_i}{4 \pi \left \Vert x - y \right \Vert_{2}^{3}} & \text{if } x \neq y, \ dim = 3 \\ \\ 0 & \text{otherwise} \end{cases} \]

Parameters

[in]	x	First operand as an array of dim coefficients.
[in]	y	Second operand as an array of dim coefficients. Parital derivating the i th component of this operand.
[in]	dim	Dimension in which to evaluate the partial derivative of the negative Laplace operator
[in]	i	Index of the component of y in which to parital derive.

Returns

The result of the dim dimensional partial derivative of the fundamental solution of the negative Laplace operator.

Variable Documentation

r_four_pi

const real r_four_pi = 0.0795774715459476679

static

\(\frac{1}{4 \pi}\)

r_minus_pi

const real r_minus_pi = -0.318309886183790671

static

\(- \frac{1}{\pi}\)

r_minus_two_pi

const real r_minus_two_pi = -0.159154943091895336

static

\(- \frac{1}{2 \pi}\)