# A priori error estimates - finite element method - mixed boundary conditions

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 $$H^2$$)