4. Mixed Methods

Many variational problems with constraints can formulated as saddle point problems:

Let \(V\) and \(Q\) be Hilbert spaces, and

\[ X = V \times Q \]

We consider the variational formulation: Find \((u,p) \in X\) such that

\[\begin{split} \begin{array}{ccccll} a(u,v) & + & b(v,p) & = & f(v) \quad & \forall \, v \in V \\ b(u,q) & & & = & g(q) & \forall \, q \in Q \end{array} \end{split}\]

Where \(a, b, f, g\) are all continuous forms. They define linear operators

\[\begin{eqnarray*} \left< A u, v \right>_{V^\prime \times V} & = & a(u,v) \\ \left< B u, q \right>_{Q^\prime \times Q} & = & b(u,q) \\ \left< B^\prime p, u \right>_{V^\prime \times V} & = & b(u,q) \\ \end{eqnarray*}\]

which allow to write the the variational formulation as operator equation

\[\begin{split} \left( \begin{array}{cc} A & B^\prime \\ B \end{array} \right) \left( \begin{array}{c} u \\ p \end{array} \right) = \left( \begin{array}{c} f \\ g \end{array} \right) \qquad \in X^\prime \end{split}\]

In most of our cases \(a\) is symmetric, and \(a(u,u) \geq 0\). Then the mixed formulation is related to optimization problems with constraints:

Primal minimization problem:

\[ \min_{u \in V \atop B u = g} \tfrac{1}{2} a(u,u) - f(u) \]

The Lagrangian \(L(v,q) := \tfrac{1}{2} a(v,v) - f(v) + b(v,q) - g(q)\) has a saddle point at \((u,p)\):

\[ L(u,q) \leq L(u,p) \leq L(v,p) \qquad \forall \, (v,q) \in X \]

Dual maximization problem:

\[ \max_{p \in Q} -\tfrac{1}{2} \left< B A^{-1} B^\prime p, p \right> - g(q) + f(A^{-1} B^\prime q) \]

4.1. The inf-sup conditions

[Babuška-Aziz, Banach-Nečas-Babuška]

Take two Hilbert spaces \(U\) and \(V\), and a continuous bilinear-form

\[ B(.,.) : U \times V \rightarrow {\mathbb R} \]

defining a continuous linear operator \(B : U \rightarrow V^\prime\).

The operator is \(B\) is surjective iff

\[ \inf_{u \in U} \sup_{v \in V} \frac{B(u,v)}{\| u \|_U \, \| v \|_V} \geq \beta > 0. \]

The operator is \(B\) is injective iff

\[ \inf_{v \in V} \sup_{u \in U} \frac{B(u,v)}{\| u \|_U \, \| v \|_V} \geq \beta > 0. \]

If both conditions are satisfied we get a continuous inverse \(B^{-1}\) with norm bounded by \(\beta^{-1}\).

If one of the conditions is satisfied, the second one can be weakend to

\[ \forall \, 0 \neq u \in U \; \exists v \in V : B(u,v) > 0 \]

on \(U \times V\) surjective, injective, closed range

4.2. Solvability of saddle-point problems

The saddle-point system can be recast as one big variational formulation: find \((u,p) \in X\) such that

\[ B((u,p), (v,q)) = F(v,q) \qquad (v,q) \in X \]

with

\[\begin{eqnarray*} B((u,p),(v,q)) & = & a(u,v) + b(u,q) + b(v,p) \\ F(v,q) & = & f(v) + g(q) \end{eqnarray*}\]

Define kernel of \(b(.,.)\):

\[ V_0 = \{ v \in V : b(v,q) = 0 \; \forall q \in Q \} \]

Theorem [Brezzi]

All forms shall be continuous. Furthermore, we assume

• \(a(.,.)\) is coercive on the kernel \(V_0\):

  \[ a(u,u) \geq \alpha \, \| u \|^2 \qquad \forall \, u \in V_0 \]

• \(b(.,.)\) satisfies the \(\inf-\sup\) condition (aka LBB condition):

  \[ \inf_{q \in Q} \sup_{v \in V} \frac{b(u,q)}{ \|u \| \, \| q \|} \geq \beta > 0 \]

Proof: Let us \(V\)-orthogonally decompose \(V = V_1 + V_0\). Then the operators \(A\) and \(B\) have block-structure

\[\begin{split} A = \left( \begin{array}{cc} A_{11} & A_{10} \\ A_{01} & A_{00} \end{array} \right) \qquad \text{and} \qquad B = \left( \begin{array}{cc} B_{1} & 0 \end{array} \right) \end{split}\]

This allows to rewrite the mixed problem as a triangular system

\[\begin{split} \left( \begin{array}{ccc} A_{11} & A_{10} & B_1^\prime \\ A_{01} & A_{00} & \\ B_1 \end{array} \right) \left( \begin{array}{c} u_1 \\ u_0 \\ p \end{array} \right) = \left( \begin{array}{c} f_1 \\ f_0 \\ g \end{array} \right) \end{split}\]

Brezzi conditions guarantee the invertibility of the operators \(A_{00}\) and \(B_1\), and the triangular system can be solved by backward substitution.

Actually, \(\inf-\sup\) conditions for \(a(.,.)\) on \(V_0\) are enough for invertibility of \(A_{00}\).

4.3. Discretization of saddle-point problems

• Replace \(V\) and \(Q\) by sub-spaces \(V_h\) and \(Q_h\).

• Discrete null-space is \(V_{0h} = \{ v_h \in V_h : b(v_h, q_h) = 0 \; \forall \, q_h \in Q_h \}\)

• verify Brezzi conditions on sub-spaces

Ceá’s Lemma provides error estimates by best approximation of both variables:

\[ \| u - u_h \|_V + \| p - p_h \|_Q \preceq \inf_{v_h, q_h} \| u - v_h \|_V + \| p - q_h \|_Q \]

4.3.1. Kernel inclusion

A very interesting improvement appears if

\[ V_{0h} \subset V_0. \]

Then the error estimates can be improved to

\[ \| u - u_h \|_V + \| \Pi_h p - p_h \|_Q \preceq \inf_{v_h} \| u - v_h \|_V + something(g) \]

Only the approximability of \(u\) enteres the estimates, convergence for the Lagrange parameter \(p\) is proven in a filtered version, where \(\Pi_h : Q \rightarrow Q_h\) is some projection.

Obtaining the kernel inclusion is the focus of FEEC