- Representation by a finite set of simple base functions (piecewise polynomials) with compact (``finite'') support
- Usually used on an unstructured grid to model the flow around complex bodies
- More often used in engineering than in astrophysics

- Representation by a finite set of harmonical functions (e.g. sine waves)
- Spatial derivatives become multiplications with wavevector
- Good for flows with small non-linear interactions
- Typically used for flows in the stellar interior (with low Mach numbers)

Smoothed Particle Hydrodynamics (SPH):

- Representation by a finite set of ``large particles'' that move freely
- Grid is replaced by particle positions
- Particle density translates into fluid density