The approximation of Sobolev homeomorphisms by smooth diffeomorphisms is well understood in first-order spaces \(W^{1,p}\) , but remains largely open in the second-order space \(W^{2,1}\) due to a fundamental tension between curvature control and injectivity. In this paper we isolate and resolve the local analytical component of this problem. We construct explicit local regularisations both across flat interfaces and near multi-cell vertices, and prove convergence in \(W^{2,1}\) together with quantitative preservation of the Jacobian. We prove that any piecewise quadratic \(C^1\) -compatible planar homeomorphism on a finite conforming rectangular partition, satisfying a quantitative lower bi-Lipschitz bound and the uniform nondegeneracy condition \(\det Dg \ge \lambda >0\) , can be approximated in \(W^{2,1}\) by injective \(C^1\) maps which are smooth outside arbitrarily small neighborhoods of the endpoints of the interior edges. Under the additional completion assumption stated in Section 6, the localized result formally yields globally smooth injective approximants. Thus the paper separates the localized analytic smoothing established here from the additional global completion property postulated in Section 6.
