## Developer Reference

• 2020.2
• 07/15/2020
• Public Content
Contents

# Poisson Solver Implementation

Poisson Solver routines enable approximate solving of certain two-dimensional and three-dimensional problems. Figure
"Structure of the Poisson Solver"
shows the general structure of the Poisson Solver.
Although in the Cartesian case, both periodic and non-periodic solvers are also supported, they use the same interfaces.
Sections below provide details of the problems that can be solved using
Intel® MKL
Poisson Solver.

## Two-Dimensional Problems

Notational Conventions
The Poisson Solver interface description uses the following notation for boundaries of a rectangular domain
a
x
<
x
<
b
x
,
a
y
<
y
<
b
y
on a Cartesian plane:
bd_a
x
= {
x
=
a
x
,
a
y
y
b
y
},
bd_b
x
= {
x
=
b
x
,
a
y
y
b
y
}
bd_a
y
= {
a
x
x
b
x
,
y
=
a
y
},
bd_b
y
= {
a
x
x
b
x
,
y
=
b
y
}.
The following figure shows these boundaries: The wildcard "+" may stand for any of the symbols
a
x
,
b
x
,
a
y
,
b
y
, so
bd_+
denotes any of the above boundaries.
The Poisson Solver interface description uses the following notation for boundaries of a rectangular domain
a
φ
< φ <
b
φ
,
a
θ
< θ <
b
θ
on a sphere 0 ≤ φ ≤ 2
π
, 0 ≤ θ ≤
π
:
bd_a
φ
= {φ =
a
φ
,
a
θ
≤ θ ≤
b
θ
},
bd_b
φ
= {φ =
b
φ
,
a
θ
≤ θ ≤
b
θ
},
bd_a
θ
= {
a
φ
≤ φ ≤
b
φ
, θ =
a
θ
},
bd_b
θ
= {
a
φ
≤ φ ≤
b
φ
, θ =
b
θ
}.
The wildcard "~" may stand for any of the symbols
a
φ
,
b
φ
,
a
θ
,
b
θ
, so
bd_~
denotes any of the above boundaries.
Two-dimensional Helmholtz problem on a Cartesian plane
The two-dimensional (2D) Helmholtz problem is to find an approximate solution of the Helmholtz equation in a rectangle, that is, a rectangular domain
a
x
<
x
<
b
x
,
a
y
<
y
<
b
y
, with one of the following boundary conditions on each boundary
bd_+
:
• The Dirichlet boundary condition