unfitted

Unfitted refers to a concept in computational mathematics, particularly within the field of numerical methods for solving partial differential equations (PDEs). It describes a technique where the numerical mesh used to approximate the solution is not required to conform to the geometric boundaries of the domain. This contrasts with traditional finite element methods (FEM) where the mesh elements typically align perfectly with the domain's edges and surfaces.

The unfitted approach offers advantages when dealing with complex or moving geometries. For instance, if a domain's boundary changes over time, regenerating a conforming mesh at each time step can be computationally expensive and challenging. Unfitted methods can utilize a fixed background mesh and incorporate the geometry information implicitly or explicitly. This can involve techniques like immersed boundary methods, cut finite element methods, or extended finite element methods.

In immersed boundary methods, the physical boundary is represented as a curve or surface embedded within a fixed, often Cartesian, background mesh. Special treatment is then applied at the interface between the immersed boundary and the background mesh to enforce boundary conditions. Cut finite element methods, on the other hand, allow the mesh to be cut by the domain's boundary, with special formulations developed to handle the resulting fractional elements. Extended finite element methods enrich the standard finite element basis functions with special functions that can capture discontinuities or singularities, often associated with the geometry.

The development of unfitted methods is an active area of research, aiming to provide efficient and robust solutions for a wide range of problems involving complex geometries, such as fluid-structure interaction, crack propagation, and multiphase flows.