Mixed finite element method

In numerical analysis, the mixed finite element method, is a type of finite element method in which extra independent variables are introduced during the discretization of a partial differential equation problem. Somewhat related is the hybrid finite element method. The extra independent variables are constrained by using Lagrange multipliers. To be distinguished from the mixed finite element method, usual finite element methods that do not introduce such extra independent variables are also called irreducible or primal finite element methods. The mixed finite element method is efficient for some problems that would be numerically ill-posed if discretized by using the irreducible finite element method; one example of such problems is to compute the stress and strain fields in an almost incompressible elastic body.

In mixed methods, the Lagrange multipliers represents fields inside the elements, usually enforcing the applicable partial differential equations. In hybrid methods, the Lagrange multipliers are for jumps of fields between elements, living on the boundary of the elements, weakly enforcing continuity; continuity from fields in the elements does not need to be enforced through shared degrees of freedom between elements anymore.