Minimal variational surfaces and quality meshes

Many physical phenomena in science and engineering can be modeled by partial differential equations (PDEs) and solved using Finite Element Method (FEM). Such a method uses as computational spatial support a mesh of the domain where the equations are formulated. The mesh quality is a key-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. We 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 Finite 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. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.