We consider a Partial Differential Equation model combining second and fourth-order operators for solving the geometry inpainting and denoising problems. The model allows the accurate recovery of curvatures and the singular set of the reconstructed image (edges, corners). The approach proposed permits a dynamical modelling by constructing a family of simple discrete energies that admit as a $Γ$-limit Mumford-Shah-Euler like functional. The approximation functionals are build within an adaptive strategy, based on two ingredients: a fine location of the singular set using mesh refinement, and second, a local choice of the diffusion coefficients which modify the reconstruction operator. Unlike the usual methods, mostly based on prior guess on the continuous solution and leading to complex and nonlinear systems of PDEs, our method consists in solving linear problems and updating the diffusion coefficients. The high order of the operator allows us to perform simultaneously efficient filtering of the data and the interpolation in the damaged regions. The method turns out to be superior to any second-order model in restoring large gap connections and curvy features. In order to validate this approach, we compare the results of our method with those of some existing one in the fields of geometry-oriented inpainting and we present several numerical examples.