Mixed Finite Elements for Variational Surface Modeling

Many problems in geometric modeling can be described using variational formulations that define the smoothness of the shape and its behavior w.r.t. the posed modeling constraints. For example, high-quality C2 surfaces that obey boundary conditions on positions, tangents and curvatures can be conveniently defined as solutions of high-order geometric PDEs; the advantage of such a formulation is its conceptual representation-independence. In practice, solving high-order problems efficiently and accurately for surfaces approximated by meshes is notoriously difficult. Classical FEM approaches require high-order elements which are complex to construct and expensive to compute. Recent discrete geometric schemes are more efficient, but their convergence properties are hard to analyze, and they often lack a systematic way to impose boundary conditions. In this paper, we present an approach to discretizing common PDEs on meshes using mixed finite elements, where additional variables for the derivatives in the problem are introduced. Such formulations use first-order derivatives only, allowing a discretization with simple linear elements. Various boundary conditions can be naturally discretized in this setting. We formalize continuous region constraints, and show that these seamlessly fit into the mixed framework. We demonstrate mixed FEM in the context of diverse modeling tasks and analyze its effectiveness and convergence behavior.