Abstract Theory

35. Abstract Theory#

We have seen a couple of examples leading to variational problems of the form:

Mixed variational problem: Find $u \in V$ and $p \in Q$ such that

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

We introduce operators

\begin{array}{r} \begin{aligned} A : & V \to V^{*} : & u \mapsto a (u, \cdot), \\ B : & V \to Q^{*} : & u \mapsto b (u, \cdot), \\ B^{*} : & Q \to V^{*} : & p \mapsto b (\cdot, p) \end{aligned} \end{array}

to rewrite the variational problem as operator equation

\begin{array}{r} \begin{array}{ccccl} A u & + & B^{*} p & = & f \\ B u & = & g \end{array} \end{array}

To be solvable, the $B$ operator must be onto (surjective). If we think of matrices, it cannot be a high rectangular matrix, i.e. the space $Q$ must not be too rich.

Surjectivity of $B$ can be rephrased by the LBB-condition:

sup_{u \in V} \frac{b (u, p)}{‖ u ‖_{V}} \geq β ‖ p ‖_{Q}

Typically, the $B$ operator has a non-trivial null-space:

V_{0} = {v \in V : B v = 0}

We split the space $V$ orthogonally as $V = V_{0} \times V_{0}^{⊥}$ , and decompose vectors and operators accordingly:

\begin{array}{r} (\begin{array}{ccc} A_{⊥ ⊥} & A_{⊥ 0} & B_{⊥}^{*} \\ A_{0 ⊥} & A_{00} & 0 \\ B_{⊥} & 0 & 0 \end{array}) (\begin{array}{c} u_{⊥} \\ u_{0} \\ p \end{array}) = (\begin{array}{c} f_{⊥} \\ f_{0} \\ g \end{array}) \end{array}

This is a block-triangular system. First we use the third equation to solve for $u_{⊥}$ , then we use the second equation to solve for $u_{0}$ , and finally we use the first equation to solve for $p$ . For this construction, we need regularity of $B_{⊥}$ (and thus also $B_{⊥}^{*}$ ), and $A_{00}$ . Regularity of $B_{⊥}$ follow from the LBB condition. We require that the bilinear-form $a (., .)$ is coercive on the null-space $V_{0}$ to guarantee regularity of $A_{00}$ .

By forming the product space $X = V \times Q$ , we can consider one problem: find $(u, p) \in X$ such that

B ((u, p), (v, q)) = F ((v, q)) \forall (v, q) \in X

with the big bilinear-form

B ((u, p), (v, q)) = a (u, v) + b (u, q) + b (v, p)

and the big linear-from

F ((v, q)) = f (v) + g (q)

We collect the finding to the following

Brezzi’s theorem:

V and Q Hilbert spaces,

$a (., .)$ and $b (., .)$ continuous bilinear-forms,

$f (.)$ and $g (.)$ continuous linear-forms.

Assume there holds the LBB condition

$sup_{u \in V} \frac{b (u, p)}{‖ u ‖_{V}} \geq β ‖ p ‖_{Q}$

and the kernel coercivity

$A (v, v) \geq α ‖ v ‖_{V}^{2} \forall v \in V_{0} .$

Then the mixed variational problem is uniquely solvable with

$‖ u ‖_{V} + ‖ p ‖_{Q} \leq c (‖ f ‖_{V^{*}} + ‖ g ‖_{Q^{*}})$

Proof: The proof follows solving the triangular system above.

The big bilinear-form $B (., .)$ is continuous

B ((u, p), (v, q)) ⪯ (‖ u ‖ + ‖ p ‖) (‖ v ‖ + ‖ q ‖) .

We prove that it fulfills the $inf - sup$ condition

inf_{v, q} sup_{u, p} \frac{B ((u, p), (v, q))}{(‖ v ‖_{V} + ‖ q ‖_{Q}) (‖ u ‖_{V} + ‖ p ‖_{Q})} ⪰ β .

Then, we use Theorem by Babuska-Aziz to conclude continuous solvability.

To prove the $inf - sup$ -condition, we choose arbitrary $v \in V$ and $q \in Q$ . We will construct $u \in V$ and $p \in Q$ such that

‖ u ‖_{V} + ‖ p ‖_{Q} ⪯ ‖ v ‖_{V} + ‖ q ‖_{Q}

and

B ((u, p), (v, q)) = ‖ v ‖_{V}^{2} + ‖ q ‖_{Q}^{2} .

First, we use the LBB-condition to choose $u_{1} \in V$ such that

b (u_{1}, q) = ‖ q ‖_{Q}^{2} and ‖ u_{1} ‖_{V} \leq 2 β_{1}^{- 1} ‖ q ‖_{Q} .

Next, we solve a problem on the kernel:

Find u_{0} \in V_{0} : a (u_{0}, w_{0}) = (v, w_{0})_{V} - a (u_{1}, w_{0}) \forall w_{0} \in V_{0}

Due to kernel ellipticity, the left hand side is a coercive bilinear-form on $V_{0}$ . The right hand side is a continuous linear-form. By Lax-Milgram, the problem has a unique solution fulfilling

‖ u_{0} ‖_{V} ⪯ ‖ v ‖_{V} + ‖ u_{1} ‖_{V}

We set

u = u_{0} + u_{1} .

By the Riesz-isomorphism, we define a $z \in V$ such that

(z, w)_{V} = (v, w)_{V} - a (u, w) \forall w \in V

By construction, it fulfills $z ⊥_{V} V_{0}$ . The LBB condition implies

‖ p ‖_{Q} \leq β_{1}^{- 1} sup_{v} \frac{b (v, p)}{‖ v ‖_{V}} = β_{1}^{- 1} sup_{v} \frac{(v, B^{*} p)_{V}}{‖ v ‖_{V}} = β_{1}^{- 1} ‖ B^{*} p ‖_{V},

and thus $B^{*} Q$ is closed, and $z \in B^{*} Q$ . Take the $p \in Q$ such that

z = B^{*} p .

It fulfills

‖ p ‖_{Q} \leq β_{1}^{- 1} ‖ z ‖_{V} ⪯ ‖ v ‖_{V} + ‖ q ‖_{Q}

Concluding, we have constructed $u$ and $p$ such that

‖ u ‖_{V} + ‖ p ‖_{Q} ⪯ ‖ v ‖_{V} + ‖ q ‖_{Q},

and

\begin{aligned} B ((u, p), (v, q)) & = a (u, v) + b (v, p) + b (u, q) \\ = a (u, v) + (z, v)_{V} + b (u, q) \\ = a (u, v) + (v, v)_{V} - a (u, v) + b (u, q) \\ = ‖ v ‖_{V}^{2} + b (u_{1}, q) \\ = ‖ v ‖_{V}^{2} + ‖ q ‖_{Q}^{2} . \end{aligned}

$◻$

35.1. Constrained minimization problem#

Now assume that $a (., .)$ is symmetric and coercive. Consider the constrained minimization problem

min_{B v = g} \frac{1}{2} a (v, v) - f (v)

The Lagrangian is

\begin{array}{rcl} L (v, q) & = & \frac{1}{2} a (v, v) - f (v) + ⟨ B v - g, q ⟩ \\ = & \frac{1}{2} a (v, v) - f (v) + b (v, q) - g (q) \end{array}

Zeroing the directional derivatives with respect to the $V$ -variable we get

\partial_{v} L (u, p) = a (u, v) + b (v, p) - f (v) = 0

and for the $Q$ -variable we obtain

\partial_{q} L (u, p) = b (u, q) - g (q) = 0

The mixed variational problem is recovered from the Karush Kuhn Tucker (KKT) conditions.

35.2. Stokes equation within the abstract theory#

The Hilbert spaces are $V := [H_{0}^{1} (Ω)]^{d}$ and $Q = L_{2}^{0}$ , and the forms

\begin{array}{rcl} a (u, v) & = & \int \nabla u \nabla v \\ b (u, q) & = & \int div u q \\ f (v) & = & \int f v \\ g (q) & = & 0 \end{array}

are continuous. The LBB condition

sup_{u \in V} \frac{\int_{Ω} div u q}{‖ u ‖_{H^{1}}} \geq β ‖ q ‖_{L_{2}}

for $q \in L_{2}$ with $\int q = 0$ is true, but non-trivial to prove.

The kernel space is

V_{0} = {v \in [H_{0}^{1}]^{d} : div v = 0},

the bilinear-form $a (., .)$ is coercive on the whole space $V$ , and so also on the sub-space $V_{0}$ .

35.3. Dirichlet boundary conditions as mixed system#

$V = H^{1} (Ω)$ , $Q = H^{- 1 / 2} (\partial Ω)$

\begin{array}{rcl} a (u, v) & = & \int \nabla u \nabla v \\ b (u, μ) & = & {⟨ u_{| \partial Ω}, μ ⟩}_{H^{1 / 2} \times H^{- 1 / 2}} \\ f (v) & = & \int f v \\ g (μ) & = & {⟨ u_{D}, μ ⟩}_{H^{1 / 2} \times H^{- 1 / 2}} \end{array}

Continuity of $a (., .)$ and $f (.)$ are clear. Continuity of $b (., .)$ and $g (.)$ require the trace theorem, and ${⟨ v, μ ⟩}_{H^{1 / 2} \times H^{- 1 / 2}} \leq ‖ v ‖_{H^{1 / 2}} ‖ μ ‖_{H^{- 1 / 2}}$ .

To show the LBB-condition, we use that we can continuously extend a boundary function $v \in H^{1 / 2}$ to a domain function $w \in H^{1}$ :

‖ μ ‖_{H^{- 1 / 2}} = sup_{v \in H^{1 / 2} (\partial Ω)} \frac{< v, μ >_{\partial Ω}}{‖ v ‖_{H^{1 / 2}}} \approx sup_{w \in H^{1} (Ω)} \frac{< w, μ >_{\partial Ω}}{‖ w ‖_{H^{1} (Ω)}}

The kernel space is

V_{0} = {v \in H^{1} (Ω) : v_{| \partial Ω} = 0} = H_{0}^{1}

On $H_{0}^{1}$ , the bilinear-form $a (., .)$ is continuous by Friedrichs’ inequality.

35.4. Mixed method for second order equation#

Our Hilbert-spaces are $Σ \times V = H (div) \times L_{2}$ , the forms are

\begin{array}{rcl} a (σ, τ) & = & \int σ \cdot τ \\ b (σ, v) & = & \int div σ v \\ f (τ) & = & 0 \\ g (v) & = & - \int f v \end{array}

On $H (div)$ we have the norm

‖ σ ‖_{H (div}^{2} = ‖ σ ‖_{L_{2}}^{2} + ‖ div σ ‖_{L_{2}}^{2}

Thus, all forms are continuous. The LBB condition

sup_{σ \in Σ} \frac{\int div σ v}{‖ σ ‖_{H (div)}} ≻ ‖ v ‖_{L_{2}}

is shown as follows: Given a $v \in L_{2}$ , we are going to find a candidate $σ$ . We solve the Poisson equation

Δ w = v w = 0 on \partial Ω

By Friedrichs’ inequality we get $‖ \nabla w ‖_{L_{2}} ≺ ‖ v ‖_{L_{2}}$ . We set $σ = \nabla w$ , and see that $div σ = v$ . Thus

\frac{\int div σ v}{‖ σ ‖_{H (div)}} = \frac{‖ v ‖_{L_{2}}^{2}}{(‖ σ ‖_{L_{2}}^{2} + ‖ v ‖_{L_{2}}^{2})^{1 / 2}} ≻ ‖ v ‖_{L_{2}}

The kernel space is

V_{0} = {τ \in H (div) : div τ = 0}

The bilinear-form $a (., .)$ is not elliptic in general, but on the kernel:

a (τ, τ) = ‖ τ ‖_{L_{2}}^{2} = ‖ τ ‖_{L_{2}}^{2} + ‖ div τ ‖_{L_{2}}^{2} = ‖ τ ‖_{H (div)}^{2} \forall τ \in V_{0}

Abstract Theory

Contents

35. Abstract Theory#

35.1. Constrained minimization problem#

35.2. Stokes equation within the abstract theory#

35.3. Dirichlet boundary conditions as mixed system#

35.4. Mixed method for second order equation#