We present a numerical framework to approximate the $μ$-domain in the planar Skorokhod embedding problem (PSEP), recently appeared in \cite{gross2019}. Our approach investigates the continuity and convergence properties of the solutions with respect to the underlying distribution $μ$. We establish that, under weak convergence of a sequence of probability measures $(μ_n)$ with bounded support, the corresponding sequence of $μ_n$-domains converges to the domain associated with $μ$, limit of $(μ_n)$. We derive explicit convergence results in the $L^1$ norm, supported by a generalization using the concept of $α_p$-convergence. Furthermore, we provide practical implementation techniques, convergence rate estimates, and numerical simulations using various distributions. The method proves robust and adaptable, offering a concrete computational pathway for approximating $μ$-domains in the PSEP.