一类界面问题的一些新分析结果。
Some new analysis results for a class of interface problems.
作者信息
Li Zhilin, Wang Li, Aspinwall Eric, Cooper Racheal, Kuberry Paul, Sanders Ashley, Zeng Ke
机构信息
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA, and School of Mathematical Sciences, Nanjing Normal University, Nanjing, China.
School of Mathematical Sciences and Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing, China.
出版信息
Math Methods Appl Sci. 2015 Dec 1;38(18):4530-4539. doi: 10.1002/mma.2865. Epub 2013 Jun 20.
Interface problems modeled by differential equations have many applications in mathematical biology, fluid mechanics, material sciences, and many other areas. Typically, interface problems are characterized by discontinuities in the coefficients and/or the Dirac delta function singularities in the source term. Due to these irregularities, solutions to the differential equations are not smooth or discontinuous. In this paper, some new results on the jump conditions of the solution across the interface are derived using the distribution theory and the theory of weak solutions. Some theoretical results on the boundary singularity in which the singular delta function is at the boundary are obtained. Finally, the proof of the convergency of the Immersed Boundary method is presented. The IB method is shown to be first order convergent in norm.
由微分方程建模的界面问题在数学生物学、流体力学、材料科学以及许多其他领域有诸多应用。通常，界面问题的特征在于系数的不连续性和/或源项中的狄拉克δ函数奇点。由于这些不规则性，微分方程的解不光滑或不连续。在本文中，利用分布理论和弱解理论推导出了关于解在界面处的跳跃条件的一些新结果。得到了奇异δ函数位于边界处的边界奇点的一些理论结果。最后，给出了浸入边界法收敛性的证明。结果表明，浸入边界法在 范数下是一阶收敛的。
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