# Fictitious Domain Method (FDM)

General description

The Fictitious Domain Methods are generally used to numerically solve a problem, that is initially defined in a geometrically complex domain, in a bigger and geometrically simpler domain. A FDM often involves the imposition of a boundary condition (bc) no longer on an external boundary of the domain but on an « immersed boundary ».

The FDM I have used is based on the Lagrange Multiplier Fictitious Domain Method [1] (see also [2]). The « immersed boundary » is discretized by a set of control points on which a given Dirichlet bc is enforced at a cost of a local modification of the discretized equations (see [3] for more details).

An illustrative example with the Poisson’s equation

We consider the following problem :

At least two possibilities exist to numerically solve the problem : use the « standard » Finite Element Method (FEM) or use the Fictitious Domain Method (FDM).

• with FEM :

•  with FDM (1st option) :

• with FDM (2nd option) :

The FDM used to solve the Stokes flow around an immersed body

We consider now the viscous flow around a rotating rigid cylinder placed near a moving wall. This problem has an exact solution in the 2D Stokes regime [4].

The FDM used to couple a solid solver and a Stokes solver

The Fictitious Domain Methods can also be used to couple two solvers using two different spatial discretizations. This approach is sometimes referred to as a coupling via a non-matching interface method.
For the application to the geodynamic modeling, I use the FDM to couple a solid Lagrangian solver to a Stokes Eulerian solver. The method is described here .

[1] Glowinski, R., Pan, T. W., & Periaux, J. (1994). A fictitious domain method for Dirichlet problem and applications. Computer Methods in Applied Mechanics and Engineering, 111(3), 283-303.
[2] Bertrand, F., Tanguy, P. A., & Thibault, F. (1997). A three‐dimensional fictitious domain method for incompressible fluid flow problems. International Journal for Numerical Methods in Fluids, 25(6), 719-736.
[3] Cerpa, N. G., Hassani, R., Gerbault, M., & Prévost, J. H. (2014). A fictitious domain method for lithosphere‐asthenosphere interaction: Application to periodic slab folding in the upper mantle. Geochemistry, Geophysics, Geosystems, 15(5), 1852-1877.
[4] Wannier, G. H. (1950). A contribution to the hydrodynamics of lubrication. Quarterly of Applied Mathematics, 8(1), 1-32.

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