# Solution to Laplace equation with zero average

Suppose that I am looking to solve the following:

\begin{cases} -\Delta u = f \quad &\text{on } \Omega,\\ \nabla u \cdot \textbf{n} = 0 \quad &\text{on } \partial \Omega, \end{cases}

with the additional assumption that \int_\Omega u=0. The following question on Math Stack Exchange suggests the following: Define f \in \mathbb{R}^N be the load vector of the original problem with the components f_j = \int_\Omega f \varphi_i \,\mathrm{d}x, let g \in \mathbb{R}^N with the components g_j = \int_\Omega \varphi_j\,\mathrm{d}x, and let A be the mass matrix . Then the mean value zero is imposed by solving the modified matrix system

\begin{pmatrix} A & g \\ g^T & 0 \end{pmatrix} \begin{pmatrix} u \\ \lambda \end{pmatrix} = \begin{pmatrix} f \\ 0 \end{pmatrix},

where \lambda\in\mathbb{R} and u\in\mathbb{R}^N are the unknowns.

Is there a way to do this in FreeFemm++? If not this way, maybe some other method?

Thank you very much!