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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:

  1. 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.
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  • 2
    $\begingroup$ You do not need $h$ to define the dual problem (similarly $f$ was not needed). The important step is to be able to take $u - u_h$ as a test function in the dual problem. The function space for the dual problem remains $\{ v \in H^1(\Omega); \; v_{\Gamma} = 0 \}$ (when $g$ and its nodal interpolant are identical --- so $u - u_h = 0$ on $\Gamma$). For arbitrary non-homogeneous Dirichlet conditions, additional work is needed. $\endgroup$ – user7440 Oct 15 '18 at 20:21
  • $\begingroup$ Thank you very much for your answer. I thought before that $h$ was not needed for the dual problem. But I still have some confusion. First, If you don't consider $g$ and $h$, what about elliptic global regularity estimates because they need the inhomogeneous boundary conditions(Dirichlet, Neumann or mixed)? Second, do you mean the variational formulation of the dual problem in the case of inhomogeneity the same as homogeneity? Can you give a concrete variational form? The last, Can you tell me some books about the proof of the $H^1$ and $L^2$ norm error estimates with inhomogeneous condition? $\endgroup$ – David Oct 17 '18 at 3:19
  • $\begingroup$ Here is one reference: Bartels, S. and Carstensen, C. and Dolzmann, G., "Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis", Numer. Math., vol. 99, pp. 1-24, (2004). $\endgroup$ – user7440 Oct 17 '18 at 3:57

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