水平集曲率计算的Tikhonov方法
A Tikhonov approach to level set curvature computation
摘要：
在两相流的数值模拟中，界面曲率的计算是一个关键的组成部分。使用有限元和水平集离散化，离散界面通常是低阶多项式的水平集，这通常导致界面曲率的差的近似。我们提出了一种方法来计算曲率使用的近似反演的$L^2$投影算子从索伯列夫空间$H^2$或$H^3$。对于近似逆的有限元计算，所得的高阶方程被重新表示为二阶方程组。由于Tikhonov正则化，该方法被证明是稳定的离散不规则性。数值例子显示内部接口以及接口相交的域的边界。
英文摘要：
In numerical simulations of two-phase flows, the computation of the curvature of the interface is a crucial ingredient. Using a finite element and level set discretization, the discrete interface is typically the level set of a low order polynomial, which often results in a poor approximation of the interface curvature. We present an approach to curvature computation using an approximate inversion of the $L^2$ projection operator from the Sobolev space $H^2$ or $H^3$. For finite element computation of the approximate inverse, the resulting higher order equation is reformulated as a system of second order equations. Due to the Tikhonov regularization, the method is demonstrated to be stable against discretization irregularities. Numerical examples are shown for interior interfaces as well as interfaces intersecting the boundary of the domain.
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