Mixed finite element method

Remarks

Example: the mixed Poisson problem. For $-\Delta u = s$ with $s \in L^2(\Omega)$ in $\Omega$ with $u = 0$ on $\partial \Omega$, set $\sigma = \nabla u$. Take $V = H(\operatorname{div}, \Omega)$, the vector fields in $L^2$ whose divergence is in $L^2$, and $Q = L^2(\Omega)$. Multiplying $\sigma = \nabla u$ by $\tau \in V$ and integrating by parts with $u = 0$ on the boundary gives $\int_\Omega \sigma \cdot \tau + \int_\Omega u \, \nabla \cdot \tau = 0$, and the equation gives $\int_\Omega (\nabla \cdot \sigma) q = -\int_\Omega s q$. This has the form above with $a(\sigma, \tau) = \int_\Omega \sigma \cdot \tau$, $b(\tau, q) = \int_\Omega q \, \nabla \cdot \tau$, $f = 0$ and $g(q) = -\int_\Omega s q$. A stable choice is the lowest-order Raviart–Thomas space for $V_h$ with piecewise constants for $Q_h$.

Non-example: the standard method seeks $u_h$ alone, in continuous piecewise linear functions in the Sobolev space $H^1_0(\Omega)$. It has one unknown field and no form $b$, so it is not a mixed method.

The matrix of the discrete problem has the block form $\begin{pmatrix} A & B^T \\ B & 0 \end{pmatrix}$, which is symmetric when $a$ is but never positive definite.