Consider the problem $$ \left\{\begin{array} {rcl} -\Delta u & = 0 & \text{ in } \Omega \\ u & = 0 & \text{ on } \Gamma_D \\ \frac{\partial u}{\partial n} &= g &\text{ on } \Gamma_R \\ \frac{\partial u}{\partial n} &= 0 &\text{ on } \Gamma_N \end{array}\right. $$ Where $g$ is a trace of a $H^{1+\varepsilon}(\Omega)$ function. An example of such a configuration in is shown in the Figure below.

I am aware that depending on the function $g$ the solution to this problem might be less regular than $H^2(\Omega)$. I am interested in reading about the a priori error estimates that can be obtained when solving this problem with triangular piecewise linear finite elements (P1, Lagrange).

My questions are:

  • Is it possible to indicate a reference which treats in detail this classical case?
  • What is the best convergence result that can be expected in this case in terms of the regularity properties of $u$? (i.e. what is the best exponent one can get in the estimate $\|u-u_h\|_{H^1(\Omega)}\leq Ch^\gamma$ when $u$ is not necessarily in $H^2$)

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