Consider Poisson’s equation
$$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$
with following mixed boundary cconditions
$$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$ $$\frac{{\partial u}}{{\partial n}} = h{\qquad\rm{ on }}\;\partial \Omega \backslash \Gamma$$
where $\frac{{\partial u}}{{\partial n}}$ denotes the derivative of $u$ in the direction normal to the boundary, $\partial \Omega$. When $g=0$ and $h=0$, the problem becomes homogeneous boundary condition.
I already know the Aubin-Nitsche trick to get error estimates for $u - {u_h}$ in the $L^2$ norm with homogeneous boundary condition. I learned the duality argument from Brenner and Scott's book The mathematical theory of finite element methods. But when the boundary condition is inhomogeneous, it is difficult for me to get error estimates for $u - {u_h}$ in the $L^2$ norm using duality argument. Although the book says,
...inhomogeneous boundary conditions are easily treated' and the proof is analogous
I encountered a problem arising from the Neumann boundary condition. When the boundary condition is homogeneous, I can get the dual problem
$$a\left( {v,w} \right) = \left( {u - {u_h},v} \right)\qquad\forall v \in H_0^1\left( \Omega \right)$$
and the elliptic regularity estimates
$${\left| w \right|_{{H^2}}} \le C{\left\| {u - {u_h}} \right\|_{{L^2}}}$$
which is crucial to the proof. In the case of inhomogeneous boundary condition, however, I get the dual problem
$$a\left( {v,w} \right) = \left( {r - {r_h},v} \right) + {\left( {h,v} \right)_{\partial \Omega \backslash \Gamma }}\qquad\forall v \in H_0^1\left( \Omega \right)$$
and the elliptic regularity estimates
$${\left| w \right|_{{H^2}}} \le C\left( {{{\left\| {r - {r_h}} \right\|}_{{L^2}}} + {{\left\| h \right\|}_{{H^{\frac{1}{2}}}}}} \right)$$
which $r = u - \tilde u$, $\tilde u \in {H^1}\left( \Omega\right)$ and ${\left.{\tilde u} \right|_\Gamma } = g$. In the following steps of proof, I try to estimate ${\left( {h,v} \right)_{\partial \Omega \backslash \Gamma }}$ and ${{{\left\| h \right\|}_{{H^{\frac{1}{2}}}}}}$, but I didn't get the result.
Questions:
- Is my idea about the proof of the inhomogeneous boundary condition, such as the dual problem and the elliptic regularity estimate which I get, correct?
- If my idea about the proof is correct, how can I continue to prove the error estimates for $u - {u_h}$ in the $L^2$ norm(my idea to estimate ${\left( {h,v} \right)_{\partial \Omega \backslash \Gamma }}$ and ${{{\left\| h \right\|}_{{H^{\frac{1}{2}}}}}}$)?
- If my idea about the proof is wrong, how to obtain the error estimates for $u - {u_h}$ in the $L^2$ norm(the case of inhomogeneous boundary condition)?Please tell me the proof or the main idea of the proof.
- I already know that the proof of error estimates about model problems which are homogeneous boundary conditions. Who can tell me some references about the proof of the $H^1$ and $L^2$ norm error estimates with inhomogeneous boundary conditions?
Note:Many thanks for Anton Menshov's reedit.
You can set up the dual problem with homogeneous boundary conditions, $$ -\Delta w = u-u_h, \ w =0 \text{ on } \Gamma, \frac{\partial w}{\partial n}=0 \text{ on } \partial \Omega\setminus \Gamma. $$ Then you have to make sure that you have these two identities $$ \|u-u_h\|_{L^2(\Omega)}^2 = a(u-u_h,w)= a(u-u_h, w-I_hw). $$ That is, you can take $u-u_h$ as test function in the dual problem, and you have some Galerkin orthogonality. This depends in the weak formulation of the problem with inhomogeneous boundary conditions.
If you prescribe a Neumann boundary condition $$ \frac{\partial w}{\partial n}=u-u_h \text{ on } \partial \Omega\setminus \Gamma $$ then you can get error estimates in the $L^2(\partial \Omega\setminus \Gamma)$-norm.