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Hermite finite element methods, or Hermite FEM, use a set of basis functions that parameterise derivative values as well as function values at various points over a computational domain. This allows the use of function spaces that are continuous in derivatives as well as value over the computational domain. Doing this is shown to have computational benefits, such as allowing a larger stable time-step in time-dependent problems, and solving equations with higher spatial derivatives. This is particularly useful for modifying equations with higher derivative terms to increase the order of accuracy of a time-stepping scheme.

A limitation of these methods is that basis functions corresponding to derivatives in 2D or higher are more dependent on the shape of grid elements than basis functions corresponding to value. By using cut finite element methods (cutFEM), problems can be solved on domains without the exact shape of the domain being represented in the grid. This allows the benefits of Hermite FEM to be gained, without placing restrictions on the computational domain or deriving a new basis for every grid.