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I am working on a method based on moving least square approximation where shape functions do not satisfy Kroneker Delta property. So Dirichlet boundary condition should be enforced. I usually used penalty method for that. In the error estimation based on residual I used the following for Poisson equation:

\begin{align} \eta^2=h_T^2\Vert \Delta u+f \Vert^2+h_T^{-1} \Vert u-g_D\Vert^2_{\partial \Omega}. \end{align} where \begin{align} -\Delta u=f, \quad u_{|\partial \Omega}=g_D. \end{align} Also, it should be noted that, because of the shape functions there is no jump between the internal elements. The problem is that penalty parameter $\gamma$ should be bigger than $1e3$ to get good result. But for large $\gamma$ the effect of adaptive technique is not really good. All the time results by adaptive refinement is better than uniform. But when $\gamma$ is smaller for example $\gamma< 1e2$ the difference is remarkable. By these results I guess the term relates to Dirichlet boundary should be change by $\gamma$. But how?

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It is quite well known in finite element research community that penalty methods easily lead to under/over refinement when used with adaptive methods, see for example the numerical examples in the end of Juntunen, Stenberg: Nitsche's method for general boundary conditions (page 1373). If you want to enforce the boundary conditions weakly, I would advise you to use Nitsche's method instead.

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