Finite elements on evolving surfaces
Finite elements on evolving surfaces
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DOI：
10.1093/imanum/drl023
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发表时间：
2007-04-01
影响因子：
2.1
通讯作者：
Elliott, C. M.
中科院分区：
文献类型：
--
作者：
Dziuk, G.;Elliott, C. M.
本文定义了一种新的演化曲面有限元方法，用于数值逼近随时间演化的超曲面Gamma(T)上的偏微分方程组。其核心思想是基于用一个演化的内插多面体(如果n=1是多边形)曲面Gamma(H)(T)来逼近Gamma(T)，该曲面由顶点位于Gamma(T)上的单形(对于n=2的三角形)的并组成。然后通过取Gamma(H)(T)上的所有连续函数的集合来定义函数的有限元空间，这些函数在每个单纯形上都是线性仿射的。有限元节点基函数具有输运性质，简化了计算。我们建立了Gamma(T)上标量的守恒定律，并在扩散通量的情况下，导出了考虑切向速度和表面局部伸展的输运和扩散方程。使用曲面梯度来定义椭圆算子的弱形式，自然地生成关于Gamma(T)的椭圆和抛物型方程的弱公式。将我们的有限元方法应用于守恒方程的弱形式。质量矩阵和单元刚度矩阵的计算简单明了。给出了空间半离散情况下的误差界。数值实验表明了该方法的收敛阶和算法的有效性。我们描述了如何在应用程序中使用该框架。
In this article, we define a new evolving surface finite-element method for numerically approximating partial differential equations on hypersurfaces Gamma(t) in < Ropf >(n+1) which evolve with time. The key idea is based on approximating Gamma(t) by an evolving interpolated polyhedral (polygonal if n = 1) surface Gamma(h)(t) consisting of a union of simplices (triangles for n = 2) whose vertices lie on Gamma(t). A finite-element space of functions is then defined by taking the set of all continuous functions on Gamma(h)(t) which are linear affine on each simplex. The finite-element nodal basis functions enjoy a transport property which simplifies the computation. We formulate a conservation law for a scalar quantity on Gamma(t) and, in the case of a diffusive flux, derive a transport and diffusion equation which takes into account the tangential velocity and the local stretching of the surface. Using surface gradients to define weak forms of elliptic operators naturally generates weak formulations of elliptic and parabolic equations on Gamma(t). Our finite-element method is applied to the weak form of the conservation equation. The computations of the mass and element stiffness matrices are simple and straightforward. Error bounds are derived in the case of semi-discretization in space. Numerical experiments are described which indicate the order of convergence and also the power of the method. We describe how this framework may be employed in applications.