A Laplace Operator on Semi-Discrete Surfaces

详细信息    查看全文

• 作者：Wolfgang Carl
• 关键词：Laplace operator ; Semi ; discrete surfaces ; Quadrilateral meshes ; Consistency
• 刊名：Foundations of Computational Mathematics
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：16
• 期：5
• 页码：1115-1150
• 全文大小：1,241 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Computer Science, general
  Math Applications in Computer Science
  Linear and Multilinear Algebras and Matrix Theory
  Applications of Mathematics
  
• 出版者：Springer New York
• ISSN：1615-3383
• 卷排序：16

文摘

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.