11. Inf-sup stable variational problems

The coercivity condition is by no means a necessary condition for a stable solvable system. A simple, stable problem with non-coercive bilinear form is to choose \(V = {\mathbb R}^2\), and the bilinear form \(B(u,v) = u_1 v_1 - u_2 v_2\). The solution of \(B(u,v) = f^T v\) is \(u_1 = f_1\) and \(u_2 = -f_2\). We will follow the convention to call coercive bilinear forms \(A(\cdot,\cdot)\), and the more general ones \(B(\cdot,\cdot)\).

Let \(V\) and \(W\) be Hilbert spaces, and \(B(\cdot,\cdot) : V \times W \rightarrow {\mathbb R}\) be a continuous bilinear form with bound

\[ B(u,v) \leq \beta_2 \| u \|_V \| v \|_W \qquad \forall \, u \in V, \; \forall \, v \in W. \]

The general condition is the inf-sup condition

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

Define the linear operator \(B : V \rightarrow W^\ast\) by \(\left< B u, v \right>_{W^\ast \times W} = B(u,v)\). The inf-sup condition can be reformulated as

\[ \sup_{v \in W} \frac{ \left< B u, v \right> } { \| v \|_W } \geq \beta_1 \| u \|_V, \qquad \forall \, u \in V \]

and, using the definition of the dual norm,

\[ \| B u \|_{W^\ast} \geq \beta_1 \| u \|_V. \]

We immediately obtain that \(B\) is one to one (aka injective), since

\[ B u = 0 \Rightarrow u = 0 \]

Lemma: Assume that the continuous bilinear form \(B(\cdot,\cdot)\) fulfills the inf-sup condition. Then the according operator \(B\) has closed range.

Proof: Let \(B u^n\) be a Cauchy sequence in \(W^\ast\). From \(\| Bu \| \geq \beta_1 \| u \|\) we conclude that also \(u^n\) is Cauchy in \(V\). Since \(V\) is complete, \(u_n\) converges to some \(u \in V\). By continuity of \(B\), the sequence \(B u^n\) converges to \(B u \in W^\ast\). \(\Box\)

The inf-sup condition does not imply that \(B\) is onto \(W^\ast\). To insure that, we can pose an inf-sup condition the other way around (aka surjective):

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

It will be sufficient to state the weaker condition

\[ \sup_{u \in V \atop u \neq 0} \frac{ B(u,v) } { \| u \|_V } > 0 \qquad \forall \, v \neq 0 \in W. \]

Theorem by Babuška and Aziz:

Assume that the continuous bilinear form \(B(\cdot,\cdot)\) fulfills the inf-sup condition

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

and the condition

\[ \sup_{u \in V \atop u \neq 0} \frac{ B(u,v) } { \| u \|_V } > 0 \qquad \forall \, v \neq 0 \in W. \]

Then, the variational problem: find \(u \in V\) such that

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

has a unique solution. The solution depends continuously on the right hand side:

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

Proof: We have to show that the range \(R(B) = W^\ast\). The Hilbert space \(W^\ast\) can be split into the orthogonal, closed subspaces

\[ W^\ast = R(B) \oplus R(B)^\bot. \]

Assume that there exists some \(0 \neq g \in R(B)^\bot\). This means that

\[ (B u, g)_{W^\ast} = 0 \qquad \forall \, u \in V. \]

Let \(v_g \in W\) be the Riesz representation of \(g\), i.e., \((v_g, w)_W = g(w)\) for all \(w \in W\). This \(v_g\) is in contradiction to the second assumption, namely

\[ \sup_{u \in V} \frac{B(u,v_g)}{ \| u \|_V} = \sup_{u \in V} \frac{(Bu,g)_{W^\ast}}{ \| u \|_V} = 0. \]

Thus, \(R(B)^\bot = \{ 0 \}\) and \(R(B) = W^\ast\). \(\Box\)

Example: A coercive bilinear form is inf-sup stable.

Example: A complex-valued symmetric variational problem: Consider the complex valued PDE

\[ -\Delta u + i u = f, \]

with Dirichlet boundary conditions, \(f \in L_2\), and \(i = \sqrt{-1}\). The weak form for the real system \(u = (u_{r}, u_i) \in V^2\) is

\[\begin{align*} (\nabla u_{r}, \nabla v_{r})_{L_2} &+ (u_i, v_{r})_{L_2} &= (f,v_{r}) \quad & \forall \, v_r \in V \\ (u_{r}, v_i)_{L_2} &- (\nabla u_i, \nabla v_i)_{L_2} &= -(f,v_i) \quad & \forall \, v_i \in V \end{align*}\]

We can add up both lines, and define the large bilinear form \(B(\cdot,\cdot) : V^2 \times V^2 \rightarrow {\mathbb R}\) by

\[ B ((u_r,u_i), (v_r, v_i)) = (\nabla u_{r}, \nabla v_{r}) + (u_i, v_{r}) + (u_{r}, v_i) -(\nabla u_i, \nabla v_i) \]

With respect to the norm \(\|v\|_V = ( \| v \|_{L_2}^2 + \| \nabla v \|_{L_2}^2)^{1/2}\), the bilinear form is continuous, and fulfills the inf-sup conditions (exercises !) Thus, the variational formulation: find \(u \in V^2\) such that

\[ B(u,v) = (f,v_r) - (f,v_i) \qquad \forall \, v \in V^2 \]

is stable solvable.

11.1. Approximation of inf-sup stable variational problems

Again, to approximate the variational problem \(B(u,v) = f(v)\), we pick finite dimensional subspaces \(V_h \subset V\) and \(W_h \subset W\) with \(\operatorname{dim} V_h = \operatorname{dim} W_h\) , and pose the finite dimensional variational problem: find \(u_h \in V_h\) such that

\[ B(u_h, v_h) = f(v_h) \qquad \forall \, v_h \in W_h. \]

But now, in contrast to the coercive case, the solvability of the finite dimensional equation does not follow from the solvability conditions of the original problem on \(V \times W\). E.g., take the example in \({\mathbb R}^2\) above, and choose the subspaces \(V_h = W_h = \mbox{span} \{ (1,1) \}\).

We have to pose an extra inf-sup condition for the discrete problem:

\[ \inf_{u_h \in V_h \atop u_h \neq 0} \sup_{v_h \in W_h \atop v_h \neq 0} \frac{ B(u_h,v_h) } { \| u_h \|_V \, \| v_h \|_W } \geq \beta_{1h}. \]

On a finite dimensional space, one to one is equivalent to onto, and we can skip the second condition.

Theorem: Assume that \(B(\cdot,\cdot)\) is continuous with bound \(\beta_2\), and fulfills the conditions of the Babuška-Aziz theorem. Furthermore, \(B(\cdot,\cdot)\) fulfills the discrete inf-sup condition with bound \(\beta_{1h}\). Then there holds the quasi-optimal error estimate

(11.1)\[\begin{equation} \| u - u_h \| \leq (1 + \beta_2 / \beta_{1h}) \inf_{v_h \in V_h} \| u - v_h \| \end{equation}\]

Proof: Again, there holds the Galerkin orthogonality \(B(u,w_h) = B(u_h,w_h)\) for all \(w_h \in V_h\). Again, we choose an arbitrary \(v_h \in V_h\):

\[\begin{align*} \| u - u_h \|_V & \leq \| u - v_h \|_V + \| v_h - u_h \|_V \\ & \leq \| u - v_h \|_V + \beta_{1h}^{-1} \sup_{w_h \in W_h} \frac{ B(v_h-u_h, w_h) } { \| w_h \|_V } \\ & = \| u - v_h \|_V + \beta_{1h}^{-1} \sup_{w_h \in W_h} \frac{ B(v_h-u, w_h) } { \| w_h \|_V } \\ & \leq \| u - v_h \|_V + \beta_{1h}^{-1} \sup_{w_h \in W_h} \frac{ \beta_2 \| v_h-u \|_V \, \| w_h \|_W } { \| w_h \|_W } \\ & = (1 + \beta_2 / \beta_{1h}) \| u - v_h\|_V. \end{align*}\]

\(\Box\)

Note: Actually, the factor \((1 + \beta_2 / \beta_{1h})\) can be improved to \(\beta_2 / \beta_{1h}\), see Some observations on Babuška and Brezzi theories, J. Xu and L. Zikatanov.