Exercises

12. Exercises#

12.1. Minimization problem#

Let \(V\) be a Hilbert space, \(A(.,.)\) a symmetric, coercive and continuous bilinear-form on \(V\), and \(f(.)\) a continuous linear from. Let \(u \in V\) solve the variational problem

\[ A(u,v) = f(v) \qquad \forall v \in V. \]

Then \(u\) is the unique solution of the minimization problem

\[ \min_{v \in V} J(v) \quad \text{with} \quad J(v) := \tfrac{1}{2} A(v,v) - f(v) \]

Hint: Show that \(J(v) - J(u) = \tfrac{1}{2} A(u-v, u-v)\)

12.2. inf-sup condition of the first-order derivative bilinear-form#

Define \(V = \{ v \in H^1(0,1) : v(0) = 0 \}\), with \(\| v \|_V = \| v^\prime \|_{L_2}\), and \(W = L_2(0,1)\). Consider the bilinear-form \(B : V \times W \rightarrow {\mathbb R}\) such that

\[ B(u,v) = \int_0^1 u^\prime v \, dx \]

Show that the bilinear-form is continuous. Prove that the bilinear-form satisfies both \(\inf-\sup\) conditions, start with

\[ \inf_{ u \in V} \sup_{v \in W} \frac{B(u,v)}{ \| u \|_V \| v \|_W} \geq \beta_1 > 0 \]

For a given \(u \in V\), find some \(v \in W\) such that the fraction is large (how ?). This proves that the derivative is one-to-one (injective) on \(V\). What if the boundary condition \(v(0)=0\) is skipped ?

Then swap order of the \(\inf-\sup\) condition:

\[ \inf_{ v \in W} \sup_{u \in V} \frac{B(u,v)}{ \| u \|_V \| v \|_W} \geq \beta_1 > 0 \]

For a given \(v\) find a good \(u\), how ? This shows that the derivative operator is onto (surjective). Can we add boundary conditions on both sides ?

12.2.1. Repeat the exercise in 2D.#

Let \(\Omega = (0,1)^2\), set \(V = H_0^1( \Omega )\), and \(W = L_2(\Omega)^2\), and

\[ B(u,v) = \int_\Omega \nabla u \cdot v \; dx \]

Try again proving both inf-sup conditions.

12.3. Building systems from building-blocks#

12.3.1. Coercive examples#

Let \(V\) be a Hilbert-space, and \(a(\cdot,\cdot)\), \(b(\cdot,\cdot)\) and \(c(\cdot,\cdot)\) be continuous and coercive bilinear-forms with continuity bounds \(\alpha_2\), \(\beta_2\), and \(\gamma_2\), and coercivity constants \(\alpha_1\), \(\beta_1\), and \(\gamma_1\) (all positive). Define the big bilinear-form \(B : V^2 \times V^2 \rightarrow {\mathbb R}\) as

\[ B( (u_1,u_2), (v_1, v_2) ) = a(u_1, v_1) + b(u_1, v_2) + c(u_2, v_2) \]

Show that \(B(\cdot, \cdot)\) is continuous. Show that for \(\beta_2 < 2 \sqrt{\alpha_1 \gamma_1}\) the bilinear-form \(B(\cdot, \cdot)\) is coercive. Give expressions for continuity and coercivity constants in terms of \(\alpha_1, \alpha^2, \beta_1, \ldots\).

Hint: express \(\beta_2 = 2 q \sqrt{\alpha_1 \gamma_1}\) with \(q < 1\). use Young’s inequality \(2 x y \leq x^2 + y^2\).

12.3.2. inf-sup condition#

Repeat the example from above, but now assume inf-sup instead of coercive for \(a(.,.)\), \(b(.,.)\), and \(c(.,.)\). Prove that then \(B(.,.)\) also satisfies inf-sup, without further restrictions onto the constants.

12.3.3. complex-valued problem as real system#

Let \(a(\cdot,\cdot)\) and \(b(\cdot,\cdot)\) be two symmetric, continuous, and coercive bilinear-forms on the Hilbert-space \(V\).

Define the form \(B(\cdot, \cdot) : V^2 \times V^2 \mapsto {\mathbb R}\) as

\[ B( (u_1, u_2), (v_1, v_2) ) := a(u_1, v_1) + b(u_2, v_1) + b(u_1, v_2) - a(u_2, v_2) \]

Use the norms (why are they norms?)

\[ \| u \|_V := \sqrt{ a(u,u) + b(u,u) } \qquad \text{for} \; u \in V \]

and

\[ \| (u_1, u_2) \|_{V^2} := \sqrt{ \| u_1 \|_V^2 + \| u_2 \|_V^2 } \quad \text{for } (u_1, u_2) \in V^2. \]

Show that \(B(\cdot, \cdot)\) is continuous and inf-sup stable. Give explicit constants.

Hint: Try first with the simplified forms \(\tilde B((u_1,u_2),(v_1,v_2)) := a(u_1,v_1) - a(u_2,v_2)\), and \(\hat B((u_1,u_2),(v_1,v_2)) := b(u_2,v_1) + b(u_1,v_2)\) and prove continuity and inf-sup. For given \(v = (v_1,v_2)\), come up with some candidates \(u = (u_1,u_2)\) for the supremum. Try a combination of both for the original bilinear-from \(B(.,.)\)

12.4. One sup is enough#

Let \(A(.,.)\) be a continuous and symmetric bilinear-form on the Hilbert-space \(V\). Prove that

\[ \sup_{x \in V} \sup_{y \in V} \frac{A(x,y)}{\| x \| \, \| y \| } = \sup_{x \in V} \frac{|A(x,x)|}{\| x \|^2 } \]

Hints:

  • Show the claim for \(V = {\mathbb R}^2\), and \(A(u,v) = v^T A u\) with a symmetric, positive definite \(2 \times 2\) matrix \(A\).

  • Show the claim for an arbitrary, two-dimensional sub-space of \(V\).

  • Prove it for the general case by reducing it to a two-dimensional sub-space

12.5. Second inf-sup condition means onto#

Let \(V\) and \(W\) be Hilbert spaces, \(B(\cdot, \cdot) : V \times W \rightarrow {\mathbb R}\) a continuous bilinear-form satisfying the second inf-sup condition

\[ \inf_{v \in W} \sup_{u \in V} \frac{B(u,v)}{\|u\|_V \, \|v\|_W} \geq \beta_1 \]

Prove that for every \(f \in W^\ast\) there exists an \(u \in V\) (not necessarily unique) such that

\[ B(u,v) = f(v) \quad \forall \, v \in W \]

with

\[ \| u \|_V \leq \frac{1}{\beta_1} \| f \|_{W^\ast} \]

Hint: Define the operator \(B^\ast : W \rightarrow V^\ast\) via

\[ \left< B v, z \right>_{V^\ast \times V} = B(z,v) \quad \forall \, z \in V \; \forall \, v \in W \]

and the bilinear-form \(A(\cdot, \cdot) : W \times W \rightarrow {\mathbb R}\)

\[ A(z,v) = (B^\ast z, B^\ast v)_{W^\ast} \]

Verify that \(A(\cdot,\cdot)\) satisfies the conditions of the Lax-Milgram theorem, which implies the unique solvability of: find \(z \in W\) such that

\[ A(z,v) = f(v) \quad \forall \, v \in W \]

Construct the searched solution \(u \in V\) from that \(z \in W\).

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