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Navier boundary conditions

Here we consider Navier boundary conditions:

\left. u\right|_{\partial{\Omega}}=g, \left. \Delta{u}\right|_{\partial{\Omega}}=f, \left. \phi\right|_{\partial{\Omega}}=p,

where g,f,p\in{H^{1/2}}(\partial{\Omega}). As the second of the above boundary conditions is a natural boundary condition (rather than an essential boundary condition) we have to modify the energy functional

\mathcal{F}(u,\phi) := \frac{1}{2} \kappa\int_\Omega |\Delta u |^2 +\frac{1}{2} \sigma\int_\Omega |\nabla u |^2 +\frac{1}{2} a\int_\Omega \phi^2 +\frac{1}{2} b\int_\Omega |\nabla \phi |^2 + c\int_\Omega \phi \Delta u

by adding the respective boundary terms

 -\int_{\partial\Omega }(\kappa{f}+cp) \frac{\partial{u}}{\partial\nu} dS(x) .

These terms correspond to the energy required to impose the boundary conditions on the solution. Hence we will consider the energy functional of the form

\overline{\mathcal{F}}(u,\phi):=\mathcal{F}(u,\phi)-\int_{\partial\Omega}(\kappa{f}+cp) \frac{\partial{u}}{\partial\nu} dS(x) .

We will consider the function set


which is a convex, closed subset of the Hilbert space H^2(\Omega)\times{H^1}(\Omega). The respective energy minimisation problem is to find  (u, \phi)\in{W} such that

\overline{\mathcal{F}}(u,\phi)=\inf_{(v,\psi)\in{W}}\overline{\mathcal{F}}(v,\psi). \qquad (1)

Our aim is to find a solution to the above minimisation problem. We have, similarly as in the case of Dirichlet boundary conditions, that if c<\sqrt{\kappa{a}}+\sqrt{\sigma{b}} then the functional \overline{\mathcal{F}} is \alpha-convex W and the minimisation problem has a unique solution (see Lemma 3 of the RSG report).

The Euler-Lagrange equations

The respective Euler-Lagrange equations are

\overline{a} ((u,\phi),(v,\psi))=0 \qquad \forall (v,\psi)\in{W_0}, \qquad (2)

where (u,\phi)\in{W} is the minimiser, W_0:=H^1_0(\Omega)\cap{H^2}(\Omega)\times{H^1_0}(\Omega) and

+c\int_\Omega\phi\Delta{v}+c\int_\Omega\psi\Delta{u}-\int_{\partial\Omega}(\kappa{f}+cp)\frac{\partial{v}}{\partial\nu}dS(x) \qquad (3)

We note that if (u,\phi)\in{W} is a solution to (2) then it is also a solution to (1) (because \overline{\mathcal{F}} is convex on W).

We shall now focus on finding the solution to (1). We will find the solution by formal integration by parts some of the terms appearing in (3) and considering the substitution w=\Delta{u}.

Precisely, let w\in{H^1_f}(\Omega), \phi\in{H^1_p}(\Omega) be the solution of

0=\kappa\int_\Omega\nabla{w}\cdot\nabla{v}\,dx+\sigma\int_\Omega{w}\,v \,dx+a\int_\Omega\phi\,\psi\,dx+b\int_\Omega\nabla\phi\cdot\nabla\psi\,dx\nonumber
+c\int_\Omega\nabla\phi\cdot\nabla{v}\,dx+c\int_\Omega\psi\,w\,dx,\qquad\forall v,\psi\in{H^1_0}(\Omega).

We note that this equation comes from integrating by parts the Euler-Lagrange equations and formally substituting w:=\Delta{u}.
From here we define u\in{H^1_g} to be the solution of


One can show that if c^2<2\min\{\kappa{b}},\sigma{a}\}, then there exists a unique solution (w,\phi)\in{H^1_f}(\Omega)\times{H^1_p}(\Omega) and that, given w, there exists a unique solution u\in{H^1_g}. In particular, this gives us the pair of functions u\in{H^1_g}(\Omega), \phi\in{H^1_p}(\Omega). One can then show that this pair of functions is the solution to the Euler-Lagrange equations. For the full proof we refer to Section 4.2.2 of RSG report.

The discretised Euler-Lagrange equations and the convergence rate of the approximation