$L^2$ norm error estimates of conforming FEM about <PERSON>’s equation with mixed boundary conditions
Consider <PERSON>’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 <PERSON>-Nitsche trick to get error estimates for $u - {u_h}$ in the $L^2$ norm with homogeneous boundary condition.I learnt the duality argument from <PERSON> and <PERSON>'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 <PERSON> 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.
Question:
1. 474
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?
2. 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 ${{{\left\| h \right\|}{{H^{\frac{1}{2}}}}}}$)?
3. 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. 875