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In mathematics, The Difference Potentials Method (DPM) is a numerical technique to find an approximate solutions of boundary value and/or interface problems in difference and differential formulations. It can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics, and fracture mechanics.

Difference potentials and DPM play the same role in the theory of solutions of linear systems of difference equations on multi-dimensional non-regular meshes as the classical Cauchy integral and the method of singular integral equations do in the theory of analytical functions (solutions Cauchy-Riemann system).

DPM provides stable and robust algorithm for high order approximation of the problems with discontinuities in coefficients of underlying equations describing the physical problem. DPM handles arbitrary shaped boundaries and interfaces using (but not limited to) regularly structured grids, e.g. Polar\Cartesian. The boundaries and interfaces not conforming with the numerical grid are handled without reducing the accuracy or hampering stability.

History

Difference Potentials Method was originated by Viktor Solomonovich Ryabenk'ii in 1969 and were generalized later by his students and their followers.

The phylosophy

DPM use advantages of several algorithmic approaches and engineering principles, among those the reduction, divide and conquer, modularity and reusability. That is, DPM reduces a boundary (or an interface) problem to a problem of finding coefficients of the expansion of the solution into some basis, e.g. Fourier or Chebyshev along the boundary(interface). The reduction described reminds the spectral approach, however DPM is not a spectral method. DPM divides the original domain/problem is into several numerically simple auxiliary domains (AD) and define numerically efficient auxiliary problems (AP). The solution to the original problem is assembled from the solutions to the AP's. DPM creates a basis to a functional space of solution to AP's, which make it modular and reusable. In addition, this makes DPM a very efficient algorithm for a multiple data problem, since different input data leads to the different set of the coefficients to the same basis expansion, so the basis to a functional space of solution to AP's needs to be computed only once.

Difference potentials and Calderon's Operator

Let $L$ be a linear differential operator and let $G$ be a Green's function of $L$. Consider a differential problem in domain $\Omega \subset \mathbb {R} ^{n}$

$Lu=0$

subject to some boundary condition providing existence and uniqueness.

Definition: The Calderon's operator, a potential with density $v_{\Gamma }=(v_{1},v_{2})$ is defined as

$\mathrm {P} _{\Omega }v_{\Gamma }=\int _{\Gamma }v_{1}(\mathbf {y} )G(\mathbf {x} -\mathbf {y} )+v_{2}(\mathbf {y} ){\frac {\partial G(\mathbf {x} -\mathbf {y} )}{\partial \mathbf {n} }}d\mathbf {s} _{\mathbf {y} }$

where ${\frac {\partial }{\partial \mathbf {n} }}$ denote a derivative in direction of a vector normal to the boundary shape of $\Gamma =\partial \Omega$.

Definition: A trace operator that maps a function $u(\mathbf {x} )$ defined for $\mathbf {x} \in \Omega$ to a vector on $\Gamma$ is defined as.

$\mathrm {Tr} \;u=\left(u,{\frac {\partial u}{\partial \mathbf {n} }}\right)_{\Gamma }$

Thus, the Green's solution to $Lu=0$ is given by

$u(\mathbf {x} )=\mathrm {P} _{\Omega }\mathrm {Tr} \;u=\mathrm {P} _{\Omega }\left(u,{\frac {\partial u}{\partial \mathbf {n} }}\right)=\int _{\Gamma }{\frac {\partial u(\mathbf {y} )}{\partial \mathbf {n} }}G(\mathbf {x} -\mathbf {y} )+u(\mathbf {y} ){\frac {\partial G(\mathbf {x} -\mathbf {y} )}{\partial \mathbf {n} }}d\mathbf {s} _{\mathbf {y} }$

Definition: Let $v_{\Gamma }=\mathrm {Tr} \;u$. Then a projection operator is defined as

$\mathrm {P} _{\Gamma }v_{\Gamma }=\mathrm {Tr} \mathrm {P} _{\Omega }u$

Theorem: A given boundary function $v_{\Gamma }$ is a trace of $u(\mathbf {x} )$ the solution to the homogeneous equation $Lu=0$ on $\Omega$: $v_{\Gamma }=\mathrm {Tr} \;u$, if and only if $v_{\Gamma }$ satisfies the Boundary Equation with Projection (BVP):

$\mathrm {P} _{\Gamma }v_{\Gamma }=v_{\Gamma }$

Assume that $v_{\Gamma }$ is given and take an arbitrary sufficiently smooth and compactly supported function $w(\mathbf {x} )\in R^{n}$, that satisfies $v_{\Gamma }=\mathrm {Tr} \;w$. Since $Lw\neq 0,(\mathbf {x} )\in \Omega$ in general, the Green's solution for $w$ becomes

$w(\mathbf {x} )=\int _{\Omega }G(Lw)d\mathbf {y} +\int _{\Gamma }{\frac {\partial w(\mathbf {y} )}{\partial \mathbf {n} }}G(\mathbf {x} -\mathbf {y} )+w(\mathbf {y} ){\frac {\partial G(\mathbf {x} -\mathbf {y} )}{\partial \mathbf {n} }}d\mathbf {s} _{\mathbf {y} }=\int _{\Omega }G(Lw)d\mathbf {y} +\mathrm {P} _{\Omega }\mathrm {Tr} \;w|_{\Gamma }=\int _{\Omega }G(Lw)d\mathbf {y} +\mathrm {P} _{\Omega }v_{\Gamma }$

Therefore we arrive at new definition of Calderon's potential

$\mathrm {P} _{\Omega }v_{\Gamma }=w-\int _{\Omega }G(Lw)d\mathbf {y}$

1

which is insensitive of choice of $w$. Furthermore, now the Calderon Potential does not contain surface integrals and allows us to define Calderon’s operators for the case of variable coefficients, when there is no known fundamental solution.

Numerical Algorithm

Let $L,B$ be linear differential operators. Let $S$ be a solver, a program that solves $Lu(\mathbf {x} )=f$ in a rectangular domain (using a Cartesian grid) subject to given boundary condition providing uniqueness.

Consider the following problem

${\begin{cases}Lu(\mathbf {x} )=0&\mathbf {x} \in \Omega \\Bu(\mathbf {x} )=0&\mathbf {x} \in \partial \Omega \end{cases}}$

2

where $\Omega$ is a given smooth curvilinear domain.

We describe below an algorithm that uses $S$ on Cartesian grid $\mathbb {N} \supset \Omega$ to solve (2) which domain, $\Omega$, is not necessarily conform the grid $\mathbb {N}$. However first we need several definitions.

The Grid Representation of $\Gamma \equiv \partial \Omega$

Let $\mathbb {M}$ be a set of all grid points of the Cartesian domain except all the boundary points. In order to define $\gamma$ define the following sets

• $\mathbb {M} ^{+}=\mathbb {M} \cap \Omega$
• $\mathbb {M} ^{-}=\mathbb {M} \setminus \Omega$
• $\mathbb {N} _{m}$ a stencil as used in $S$ centered at the grid point $m$
• $\mathbb {N} ^{+}=\bigcup _{m\in \mathbb {M} ^{+}}\mathbb {N} _{m}$
• $\mathbb {N} ^{-}=\bigcup _{m\in \mathbb {M} ^{-}}\mathbb {N} _{m}$

Finally

$\gamma =\mathbb {N} ^{+}\cap \mathbb {N} ^{-}$

The Algorithm

Assume that along the curvilinear boundary $\Gamma$, the solution to (2) and its normal derivative has an expansion in some basis, e.g. Fourier. Denote it as $u|_{\Gamma }$ and consider that it can be approximated by a finite subspace of the basis, e.g. $\{b_{n}(s)\}_{n\in I\subset \mathbb {Z} }$, where $s$ is an arc-length parameter, that is

$u|_{\Gamma }\approx u(s)=\sum \limits _{n\in I}c_{n}^{1}b_{n}(s)$

and

${\frac {\partial u}{\partial \mathbf {n} }}{\Big |}_{\Gamma }\approx u_{\mathbf {n} }(s)=\sum \limits _{n\in I}c_{n}^{2}b_{n}(s)$

1. For each $n\in I$
   1. Define an auxiliary function $w_{n}$ as following:

      $w_{n}(\mathbf {x} )={\begin{cases}\mathrm {T} b_{n}&\mathbf {x} \in \gamma \\0&\mathbf {x} \in \mathbb {N} \setminus \gamma \end{cases}}$

      where the operator $\mathrm {T}$ defines a Whitney extension, e.g. Taylor.
   2. Use the (direct) operator $L$ to define

      $f_{n}={\begin{cases}Lw_{n}&\in \mathbb {M} ^{+}\\0&\in \mathbb {M} ^{+}\end{cases}}$

   3. Solve $Lv_{n}(\mathbf {x} )=f_{n}$ using $S$
   , Note, the numerical counterpart of the Calderon Potential $P_{\Omega }\left(v_{n},{\frac {\partial v_{n}}{\partial n}}\right){\Bigg |}_{\gamma }$ is given by $w_{n}-v_{n}$, see(1), and the projection is its restriction to the $\gamma$, that is $P_{n}\equiv (w_{n}-v_{n}){\Big |}_{\gamma }$
   4. Define $Q_{n}=P_{n}-w_{n}{\Big |}_{\gamma }=v_{n}{\Big |}_{\gamma }$
3. Set a matrix $Q$ from the column vectors $Q_{n}$ numerical counterpart of BVP $Q=\mathrm {P} _{\Gamma }v_{\Gamma }-v_{\Gamma }$

Given a solver to $Lu=f$ for a rectangular domain. in Cartesian Domain

Note that numerically $\int _{\Omega }G(Lw)d\mathbf {y}$ can be any solution $Lw=f,(\mathbf {x} )\in \Omega$.

Mmedvin (talk) 03:59, 22 February 2015 (UTC)