---
tag: 87VMQ892VW
kind: concept
name: Mixed finite element method
msc: [65N30, 65N12]
specializes: [98GZ9DVHTA]
sources:
  - url: https://en.wikipedia.org/wiki/Mixed_finite_element_method
same_as:
  wikidata: Q16965966
  wikipedia: Mixed finite element method
---

## Summary

A mixed finite element method approximates two or more unknown fields of a boundary value problem at once, each in its own finite element space.
It starts from a saddle-point formulation, such as velocity and pressure for Stokes flow, or flux and potential for the Poisson equation.
The spaces cannot be chosen independently: stability requires the [[Ladyzhenskaya–Babuška–Brezzi condition|SC5FV3GHV5]].
Mixed methods give accurate fluxes that satisfy conservation element by element, which matters for flow in porous media.
See the [[definition|DBVE636BXN]].

## Explanations

### L1 human

In many physical problems one quantity is of main interest, but another one comes with it, such as the flow of heat together with the temperature.
The standard finite element method computes the temperature and gets the heat flow by differentiating, which loses accuracy.
A mixed method computes both at the same time, as separate unknowns linked by the equations.
For the Poisson equation $-\Delta u = f$, one introduces the flux $\sigma = \nabla u$ and solves for the pair $(\sigma, u)$ with $\nabla \cdot \sigma = -f$.