High-Order AFEM for the Laplace–Beltrami Operator: Convergence Rates

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• 作者：Andrea Bonito ; J. Manuel Cascón…
• 关键词：Laplace–Beltrami operator ; Parametric surfaces ; Adaptive finite element method ; Convergence rates ; A posteriori error estimates ; Higher order
• 刊名：Foundations of Computational Mathematics
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：16
• 期：6
• 页码：1473-1539
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Numerical Analysis; Economics, general; Applications of Mathematics; Linear and Multilinear Algebras, Matrix Theory; Math Applications in Computer Science; Computer Science, general;
• 出版者：Springer US
• ISSN：1615-3383
• 卷排序：16

文摘

We present a new AFEM for the Laplace–Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally \(W^1_\infty \) and piecewise in a suitable Besov class embedded in \(C^{1,\alpha }\) with \(\alpha \in (0,1]\). The idea is to have the surface sufficiently well resolved in \(W^1_\infty \) relative to the current resolution of the PDE in \(H^1\). This gives rise to a conditional contraction property of the PDE module. We present a suitable approximation class and discuss its relation to Besov regularity of the surface, solution, and forcing. We prove optimal convergence rates for AFEM which are dictated by the worst decay rate of the surface error in \(W^1_\infty \) and PDE error in \(H^1\).