Variational Laplacians for semidiscrete surfaces

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• 作者：Wolfgang Carl ; Johannes Wallner
• 关键词：Semidiscrete surface ; Variational Laplacian ; Mean curvature normal ; Consistency
• 刊名：Advances in Computational Mathematics
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：42
• 期：6
• 页码：1491-1509
• 全文大小：311 KB
• 刊物类别：Computer Science
• 刊物主题：Numeric Computing
  Calculus of Variations and Optimal Control
  Mathematics
  Algebra
  Theory of Computation
  
• 出版者：Springer U.S.
• ISSN：1572-9044
• 卷排序：42

文摘

We study a Laplace operator on semidiscrete surfaces that is defined by variation of the Dirichlet energy functional. We show existence and its relation to the mean curvature normal, which is itself defined via variation of area. We establish several core properties like linear precision (closely related to the mean curvature of flat surfaces), and pointwise convergence. It is interesting to observe how a certain freedom in choosing area measures yields different kinds of Laplacians: it turns out that using as a measure a simple numerical integration rule yields a Laplacian previously studied as the pointwise limit of geometrically meaningful Laplacians on polygonal meshes.