On the accuracy of finite element approximations to a class of interface problems
On the accuracy of finite element approximations to a class of interface problems
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DOI：
10.1090/mcom3051
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发表时间：
2015-11
期刊：
影响因子：
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通讯作者：
J. Guzmán;M. Sánchez;M. Sarkis
中科院分区：
文献类型：
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作者：
J. Guzmán;M. Sánchez;M. Sarkis
定义跳跃为[∇u·n] =∇u·n +∇u·n，其中u = u|Ω±，n为向外垂直于Ω的单位（见图1）。同时，我们表示[u] = u - u。对于问题（1.1）已经开发了许多数值方法。其中最值得注意的可能是Peskin[11]的有限差分法（即浸入边界法）和LeVeque和Li[11]的方法（即浸入界面法；另见Mayo的方法[14,15,16]）。LeVeque和Li[11]方法用于更一般的具有不连续扩散系数的问题，而Peskin[18]方法用于具有浸入边界的流体流动问题。虽然Peskin[18]的方法是用包含浸入边界弹性力Γ的力函数F来表示的，但[19]表明，它可以重新表述为一个界面问题（α = 0），其中β编码弹性力。自这两篇重要论文[18,11]以来，有许多文章扩展或改进了这些方法。特别是，这些方法的有限元版本已经出现；参见[3,9,6,2]。对于上述问题（α = 0），众所周知，Peskin[18]方法只有一阶精度，而LeVeque和Li[11]方法是二阶精度
The jump is defined as [∇u · n] = ∇u · n + ∇u ·n where u = u|Ω± and n is the unit outward pointing normal to Ω (see figure 1). Also, we denote [u] = u − u. Many numerical methods have been developed for problem (1.1). Perhaps the most notable ones are the finite difference method of Peskin [18] (i.e., immersed boundary method) and the method of LeVeque and Li [11] (i.e., the immersed interface method ; see also the method of Mayo [14, 15, 16]) .The method of LeVeque and Li [11] was developed for the more general problem with discontinuous diffusion coefficients, while the method of Peskin [18] was developed for fluid flow problems with an immersed boundary. Although the method of Peskin [18] is formulated with a force function F that incorporates the elastic force of the immersed boundary Γ, it was shown in [19] that it can be re-formulated as an interface problem (with α = 0) where β encodes the elastic force. Since the two important papers [18, 11] there have been many articles extending or improving these methods. In particular, finite element versions of these methods have appeared; see for example [3, 9, 6, 2]. For the above problem (α = 0), it is well known that the method of Peskin [18] is only first-order accurate whereas the method of LeVeque and Li [11] is second-order