Imposing total pressure over surface in FEM

You must add an additional equation to your system. It is: $$ int_{Gamma_p}{p,dsigma}=0implies sum_{p_iin Gamma_p}p_iint_{Gamma_{pi}}{phi(t),dt}=0$$

Where $phi(t)$ is the basis function of $p$ along the parameterised boundary described with the parameter $t$ in the element $Gamma_{pi}$. Remember that $Gamma_p=cup{ Gamma_{pi}}$

For Finite Element formulation for fluid flow, it is sufficient to impose zero-pressure on one of the nodes.

Usually, this node is chosen on the outflow boundary. If the outflow boundary does not exist, then pick a node far away from the region(s) of steep pressure gradients.