A Laplace Operator on Semi-Discrete Surfaces

This paper studies a laplace operator on semi-discrete surfaces. a semi-discrete surface is represented by a mapping into three-dimensional euclidean space possessing one discrete variable and one continuous variable. it can be seen as a limit case of a quadrilateral mesh, or as a semi-discretization of a smooth surface. laplace operators on both smooth and discrete surfaces have been an object of interest for a long time, also from the viewpoint of applications. there are a wealth of geometric objects available immediately once a laplacian is defined, e.g., the mean curvature normal. we define our semi-discrete laplace operator to be the limit of a discrete laplacian on a quadrilateral mesh, which converges to the semi-discrete surface. the main result of this paper is that this limit exists under very mild regularity assumptions. moreover, we show that the semi-discrete laplace operator inherits several important properties from its discrete counterpart, like symmetry, positive semi-definiteness, and linear precision. we also prove consistency of the semi-discrete laplacian, meaning that it converges pointwise to the laplace–beltrami operator, when the semi-discrete surface converges to a smooth one. this result particularly implies consistency of the corresponding discrete scheme.