Abstract
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.
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Acknowledgments
The author would like to thank J. Wallner for fruitful discussions and comments. This research was supported by the DFG-Collaborative Research Center, TRR 109, “Discretization in Geometry and Dynamics” and the Doctoral Program “Discrete Mathematics” through grants I 706-N26 and W1230 of the Austrian Science Fund (FWF). We further acknowledge support from NAWI Graz and are grateful to the anonymous reviewers for their remarks and suggestions.
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Communicated by Wolfgang Dahmen.
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Carl, W. A Laplace Operator on Semi-Discrete Surfaces. Found Comput Math 16, 1115–1150 (2016). https://doi.org/10.1007/s10208-015-9271-y
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DOI: https://doi.org/10.1007/s10208-015-9271-y