Fictitious domain method

In mathematics, the Fictitious domain method is a method to find the solution of a partial differential equations on a complicated domain [math]\displaystyle{ D }[/math], by substituting a given problem posed on a domain [math]\displaystyle{ D }[/math], with a new problem posed on a simple domain [math]\displaystyle{ \Omega }[/math] containing [math]\displaystyle{ D }[/math].

## Contents

* 1 General formulation
* 2 Simple example, 1-dimensional problem
* 2.1 Prolongation by leading coefficients
* 2.2 Prolongation by lower-order coefficients
* 3 Literature

## General formulation

Assume in some area [math]\displaystyle{ D \subset \mathbb{R}^n }[/math] we want to find solution [math]\displaystyle{ u(x) }[/math] of the equation:

[math]\displaystyle{ Lu = - \phi(x), x = (x_1, x_2, \dots , x_n) \in D }[/math]

with boundary conditions:

[math]\displaystyle{ lu = g(x), x \in \partial D }[/math]

The basic idea of fictitious domains method is to substitute a given problem posed on a domain [math]\displaystyle{ D }[/math], with a new problem posed on a simple shaped domain [math]\displaystyle{ \Omega }[/math] containing [math]\displaystyle{ D }[/math] ([math]\displaystyle{ D \subset \Omega }[/math]). For example, we can choose n-dimensional parallelotope as [math]\displaystyle{ \Omega }[/math].

Problem in the extended domain [math]\displaystyle{ \Omega }[/math] for the new solution [math]\displaystyle{ u_{\epsilon}(x) }[/math]:

[math]\displaystyle{ L_\epsilon u_\epsilon = - \phi^\epsilon(x), x = (x_1, x_2, \dots , x_n) \in \Omega }[/math]

[math]\displaystyle{ l_\epsilon u_\epsilon = g^\epsilon(x), x \in \partial \Omega }[/math]

It is necessary to pose the problem in the extended area so that the following condition is fulfilled:

[math]\displaystyle{ u_\epsilon (x) \xrightarrow[\epsilon \rightarrow 0]{ } u(x), x \in D }[/math]

## Simple example, 1-dimensional problem

[math]\displaystyle{ \frac{d^2u}{dx^2} = -2, \quad 0 \lt x \lt 1 \quad (1) }[/math]

[math]\displaystyle{ u(0) = 0, u(1) = 0 }[/math]

### Prolongation by leading coefficients

[math]\displaystyle{ u_\epsilon(x) }[/math] solution of problem:

[math]\displaystyle{ \frac{d}{dx}k^\epsilon(x)\frac{du_\epsilon}{dx} = - \phi^{\epsilon}(x), 0 \lt x \lt 2 \quad (2) }[/math]

Discontinuous coefficient [math]\displaystyle{ k^{\epsilon}(x) }[/math] and right part of equation previous equation we obtain from expressions:

[math]\displaystyle{ k^\epsilon (x)=\begin{cases} 1, & 0 \lt x \lt 1 \\\ \frac{1}{\epsilon^2}, & 1 \lt x \lt 2 \end{cases} }[/math]
[math]\displaystyle{ \phi^\epsilon (x)=\begin{cases} 2, & 0 \lt x \lt 1 \\\ 2c_0, & 1 \lt x \lt 2 \end{cases}\quad (3) }[/math]

Boundary conditions:

[math]\displaystyle{ u_\epsilon(0) = 0, u_\epsilon(2) = 0 }[/math]

Connection conditions in the point [math]\displaystyle{ x = 1 }[/math]:

[math]\displaystyle{ [u_\epsilon] = 0,\ \left[k^\epsilon(x)\frac{du_\epsilon}{dx}\right] = 0 }[/math]

where [math]\displaystyle{ [ \cdot ] }[/math] means:

[math]\displaystyle{ [p(x)] = p(x + 0) - p(x - 0) }[/math]

Equation (1) has analytical solution therefore we can easily obtain error:

[math]\displaystyle{ u(x) - u_\epsilon(x) = O(\epsilon^2), \quad 0 \lt x \lt 1 }[/math]

### Prolongation by lower-order coefficients

[math]\displaystyle{ u_\epsilon(x) }[/math] solution of problem:

[math]\displaystyle{ \frac{d^2u_\epsilon}{dx^2} - c^\epsilon(x)u_\epsilon = - \phi^\epsilon(x), \quad 0 \lt x \lt 2 \quad (4) }[/math]

Where [math]\displaystyle{ \phi^{\epsilon}(x) }[/math] we take the same as in (3), and expression for [math]\displaystyle{ c^{\epsilon}(x) }[/math]

[math]\displaystyle{ c^\epsilon(x)=\begin{cases} 0, & 0 \lt x \lt 1 \\\ \frac{1}{\epsilon^2}, & 1 \lt x \lt 2. \end{cases} }[/math]

Boundary conditions for equation (4) same as for (2).

Connection conditions in the point [math]\displaystyle{ x = 1 }[/math]:

[math]\displaystyle{ [u_\epsilon(0)] = 0,\ \left[\frac{du_\epsilon}{dx}\right] = 0 }[/math]

Error:

[math]\displaystyle{ u(x) - u_\epsilon(x) = O(\epsilon), \quad 0 \lt x \lt 1 }[/math]

## Literature

* P.N. Vabishchevich, The Method of Fictitious Domains in Problems of Mathematical Physics, Izdatelstvo Moskovskogo Universiteta, Moskva, 1991.
* Smagulov S. Fictitious Domain Method for Navier–Stokes equation, Preprint CC SA USSR, 68, 1979.
* Bugrov A.N., Smagulov S. Fictitious Domain Method for Navier–Stokes equation, Mathematical model of fluid flow, Novosibirsk, 1978, p. 79–90

* v
* t
* e

Numerical partial differential equations by method

Finite difference|

| Parabolic|

* Forward-time central-space (FTCS)
* Crank–Nicolson

|

Hyperbolic|

* Lax–Friedrichs
* Lax–Wendroff
* MacCormack
* Upwind
* Method of characteristics

Others|

* Alternating direction-implicit (ADI)
* Finite-difference time-domain (FDTD)

Finite volume|

* Godunov
* High-resolution
* Monotonic upstream-centered (MUSCL)
* Advection upstream-splitting (AUSM)
* Riemann solver
* essentially non-oscillatory (ENO)
* weighted essentially non-oscillatory (WENO)

Finite element|

* hp-FEM
* Extended (XFEM)
* Discontinuous Galerkin (DG)
* Spectral element (SEM)
* Mortar
* Gradient discretisation (GDM)
* Loubignac iteration
* Smoothed (S-FEM)

Meshless/Meshfree|

* Smoothed-particle hydrodynamics (SPH)
* Moving particle semi-implicit method (MPS)
* Material point method (MPM)
* Particle-in-cell (PIC)

Domain decomposition|

* Schur complement
* Fictitious domain
* Schwarz alternating
* additive
* abstract additive
* Neumann–Dirichlet
* Neumann–Neumann
* Poincaré–Steklov operator
* Balancing (BDD)
* Balancing by constraints (BDDC)
* Tearing and interconnect (FETI)
* FETI-DP

Others|

* Spectral
* Pseudospectral (DVR)
* Method of lines
* Multigrid
* Collocation
* Level-set
* Boundary element
* Immersed boundary
* Analytic element
* Particle-in-cell
* Isogeometric analysis
* Infinite difference method
* Infinite element method
* Galerkin method
* Petrov–Galerkin method
* Validated numerics
* Computer-assisted proof

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