In this paper we introduce a mesh approximation method that uses a volume-based metric. After a geometric simplification, we minimize the volume between the simplified mesh and the original mesh using a gradient-based optimization algorithm and a finite-element interpolation model implicitly defined on meshes. The notable contribution of this paper is the theoretical framework which permits the construction of a volume minimization process between two triangular meshes. We chose this volume-based metric because of its good perceptual properties, as it naturally and accurately fits the geometric singularities on 3D meshes. Furthermore, this metric corresponds well to a sort of intuitive error between two 3D surfaces and the resulting optimization algorithm only requires a few parameters. We show that this approach permits geometric compression leading to multi-resolution meshes with minimal visual losses.