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