Many physical phenomena in science and engineering can be modelled by partial di erential equations (PDEs) and solved using the nite element method (FEM). Such a method uses as computational spatial support a mesh of the domain where the equations are formulated. The mesh quality i s a k ey-point for the accuracy of the numerical solution. This paper describes a methodology to construct a quality mesh of the domain from a given discretization of its boundary. W e show that the size map related to such a mesh constitutes a minimal variational surface supported by a given contour. This surface can be constructed, from its boundary, using the nite element method or by the resolution of a simple discrete optimization problem. The quality mesh of the domain is then a mesh conforming to the size map given by this surface. A numerical example is given to demonstrate the method.