User:Daomcg/sandbox

=Mixed Space Saddle Point Problem= Let $$V,W$$ be reflexive Hilbert Spaces with dual spaces denoted $$V^\prime,W^\prime$$ respectively. On these spaces define linear operators $$ \mathcal A\colon V\to V^\prime $$ and $$\mathcal B \colon W\to V^\prime$$. Consider abstract loading data $$ F\in V^\prime, G\in W^\prime$$ Denote the adjoint operator of $$\mathcal B$$ by $$\mathcal B^\prime$$.

We seek solutions to the following variational saddle point problem

$$\begin{align}(u,p) \in V\times W: \begin{cases}\mathcal A u(v) + \mathcal B p(v) =F & \forall v \in V \\ \mathcal B^\prime u (w) = G & \forall q \in W \end{cases}\end{align}$$

This problem is well-posed if the following conditions are satisfied
 * 1) The operators $$\mathcal A,\mathcal B$$ are continuous (i.e. bounded).
 * 2) $$\mathcal A$$ is coercive on the kernel of $$\mathcal B$$; i.e. there exists $$ c >0$$ such that $$ \mathcal A u(u) \geq c \|u\|_V^2, \forall u\in \text{Ker}(\mathcal B).$$
 * 3) The operator $$\mathcal B$$ satisfies the Inf-Sup Condition also called the Ladyzenskaia-Babushka-Brezzi condition $$\inf_{v\in V} \sup_{w\in W} \frac{\mathcal B v (w) }{\|v\|_V\|w\|_W}\geq C$$. This condition is equivalent to the Range of $$\mathcal B$$ being closed, see Closed Range Theorem.

When the operator $$\mathcal A$$ the mixed variational problem is equivalent to a constrained minimization problem.

Example: Poisson Equation
We can pose the Poisson equation, clasically written as

$$ \text{div}K\nabla p = f$$

as a mixed space problems by introducing a flux variable $$\mathbf{u}$$ where

$$ K^{-1}\mathbf{u} = \nabla p.$$

The strong form of the mixed system is then posed as

$$\begin{cases}K^{-1}\mathbf{u} - \nabla p = 0 \\ \text{div}\mathbf u = f\end{cases}.$$

There are two typical ways to pose the mixed Poisson equation in the variational framework. The primal form is called so as the variable $$\mathbf{u}$$ can be eliminated reducing the problem to the classical $$H^1$$ formulation of the Poison equation. We consider our spaces $$V=[L^2(\Omega)]^d$$ and $$W$$ an appropriate subspace of $$H^1$$, i.e.

$$ W = \{v\in H^1(\Omega): \int_\Omega v = 0\} \text{ or } W=\{v\in H^1(\Omega): v|_\Gamma = 0, \Gamma \subset \partial\Omega\}. $$ The operators are defined as follows.

$$ \mathcal{A}: V\to V^\prime, \qquad \mathcal{A}\mathbf{u}(\mathbf{v}) = \int_\Omega K^{-1}\mathbf{u}\cdot\mathbf{v} $$

$$\mathcal{B}:W\to V^\prime, \qquad \mathcal{B}p(\mathbf{v}) = \int_\Omega \nabla p \cdot\mathbf{v}.$$

The dual formulation of the Mixed Poisson equation involves a lower regularity Sobelov space known as $$\mathbf{H}(\text{div},\Omega)$$ defined as

$$ \mathbf{H}(\text{div},\Omega) = \{\mathbf{u}\in[L^2(\Omega)]^d: \text{div} \ \mathbf{u} \in L^2(\Omega)\}. $$

We then choose our spaces as

$$ V = \mathbf{H}(\text{div},\Omega), \quad W = \{v\in L^2(\Omega): \int_\Omega v= 0\}. $$

and define operators as

$$ \mathcal{A}:V\to V^\prime, \qquad \mathcal{A}\mathbf{u}(\mathbf{v}) = \int K^{-1}\mathbf{u}\cdot\mathbf{v} $$

$$ \mathcal{B}:W\to V^\prime, \qquad \mathcal{B}p(\mathbf{v}) = -\int_\Omega p \ \text{div} \ \mathbf v. $$

Note that in this case the variable $$p$$ requires no well-defined weak derivatives.