L2 and pointwise a posteriori error estimates for FEM for elliptic PDEs on surfaces
L2 and pointwise a posteriori error estimates for FEM for elliptic PDEs on surfaces
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曲面上椭圆 PDE 的 FEM 的 L2 和逐点后验误差估计
DOI：
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发表时间：
2015
期刊：
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通讯作者：
A. Demlow
中科院分区：
文献类型：
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作者：
F. Camacho;A. Demlow
表面有限元法（SFEM）被广泛应用于求解晶体生长、流体力学和计算机图形学等领域中的表面偏微分方程。后验误差估计是误差的可计算度量，用于实现自适应网格细化。以往对SFEM后验误差估计的研究主要集中在约束能量范数误差上。在这项工作中，我们推导出一个后验L2和逐点误差估计分段线性SFEM的Laplace-Beltrami方程隐式定义的表面上。在SFEM中有两个主要的误差源，一个是有限元方法中常见的“伽辽金误差”，另一个是在编写有限元方程时用离散近似代替连续表面所产生的“几何误差”。我们的工作包括数值估计的误差界的依赖表面的几何性质。我们还提供了数值实验中的估计已被用来实现一个自适应有限元表面具有不同的曲率。
Surface Finite Element Methods (SFEM) are widely used to solve surface partial differential equations arising in applications including crystal growth, fluid mechanics and computer graphics. A posteriori error estimators are computable measures of the error and are used to implement adaptive mesh refinement. Previous studies of a posteriori error estimation in SFEM have mainly focused on bounding energy norm errors. In this work we derive a posteriori L2 and pointwise error estimates for piecewise linear SFEM for the Laplace-Beltrami equation on implicitly defined surfaces. There are two main error sources in SFEM, a “Galerkin error” arising in the usual way for finite element methods, and a “geometric error” arising from replacing the continuous surface by a discrete approximation when writing the finite element equations. Our work includes numerical estimation of the dependence of the error bounds on the geometric properties of the surface. We provide also numerical experiments where the estimators have been used to implement an adaptive FEM over surfaces with different curvatures.