Fictitious domain method

In numerical analysis, the fictitious domain method (FD) is a numerical technique designed to solve partial differential equations on complex geometries by embedding the physical domain into a larger and simpler computational domain. The method consists of extending the equations beyond the physical boundaries and enforcing the interface conditions through a distributed Lagrange multiplier in order to recover the correct solution within the original domain of interest.

This method belongs to the more general family of unfitted methods (also known as embedded or immersed), which allow solving interface problems on complex or evolving domains without generating a mesh that conforms to the domain’s boundaries. For this reason, the construction of two independent meshes is considered in the fictitious domain method: one for the unfitted background domain and another overlapping mesh that conforms to the interface, but which can move freely during the simulation. As with the immersed boundary method and the level-set method, the fictitious domain method is particularly effective for problems with evolving geometries since there is no need to rebuild the mesh but just to move the overlapping one.

The Dirichlet boundary conditions on immersed closed interfaces are imposed weakly by the introduction of the distributed Lagrange multiplier which enforces the boundary condition over the whole overlapping region.

A particular application of the fictitious domain method is in fluid–structure interaction (FSI) problems, where a fluid and an immersed solid interact one each other, exchanging forces through the interface. In this context, the solid is considered to be embedded within the computational fluid dynamics domain and the coupling conditions at the interface are enforced weakly over the whole fictitious fluid region.