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On the Solution of the Stokes Equation on Regions with Corners
Communications on Pure and Applied Mathematics ( IF 3 ) Pub Date : 2020-09-20 , DOI: 10.1002/cpa.21937 Manas Rachh 1 , Kirill Serkh 2
Communications on Pure and Applied Mathematics ( IF 3 ) Pub Date : 2020-09-20 , DOI: 10.1002/cpa.21937 Manas Rachh 1 , Kirill Serkh 2
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In Stokes flow, the stream function associated with the velocity of the fluid satisfies the biharmonic equation. The detailed behavior of solutions to the biharmonic equation on regions with corners has been historically difficult to characterize. The problem was first examined by Lord Rayleigh in 1920; in 1973, the existence of infinite oscillations in the domain Green's function was proven in the case of the right angle by S.~Osher. In this paper, we observe that, when the biharmonic equation is formulated as a boundary integral equation, the solutions are representable by rapidly convergent series of the form $\sum_{j} ( c_{j} t^{\mu_{j}} \sin{(\beta_{j} \log{(t)})} + d_{j} t^{\mu_{j}} \cos{(\beta_{j} \log{(t)})} )$, where $t$ is the distance from the corner and the parameters $\mu_{j},\beta_{j}$ are real, and are determined via an explicit formula depending on the angle at the corner. In addition to being analytically perspicuous, these representations lend themselves to the construction of highly accurate and efficient numerical discretizations, significantly reducing the number of degrees of freedom required for the solution of the corresponding integral equations. The results are illustrated by several numerical examples.
中文翻译:
关于带角区域的斯托克斯方程的解
在斯托克斯流中,与流体速度相关的流函数满足双调和方程。双调和方程在带角区域上的解的详细行为在历史上一直难以表征。瑞利勋爵于 1920 年首次研究了这个问题。1973年,S.~Osher在直角的情况下证明了域格林函数的无限振荡的存在。在本文中,我们观察到,当双调和方程被表述为边界积分方程时,解可以用 $\sum_{j} ( c_{j} t^{\mu_{j} } \sin{(\beta_{j} \log{(t)})} + d_{j} t^{\mu_{j}} \cos{(\beta_{j} \log{(t)}) } )$,其中$t$是到角点的距离,参数$\mu_{j},\beta_{j}$是实数,并通过一个明确的公式确定,具体取决于拐角处的角度。除了在分析上清晰可见之外,这些表示还有助于构建高精度和高效的数值离散化,从而显着减少求解相应积分方程所需的自由度数。结果由几个数值例子说明。
更新日期:2020-09-20
中文翻译:
关于带角区域的斯托克斯方程的解
在斯托克斯流中,与流体速度相关的流函数满足双调和方程。双调和方程在带角区域上的解的详细行为在历史上一直难以表征。瑞利勋爵于 1920 年首次研究了这个问题。1973年,S.~Osher在直角的情况下证明了域格林函数的无限振荡的存在。在本文中,我们观察到,当双调和方程被表述为边界积分方程时,解可以用 $\sum_{j} ( c_{j} t^{\mu_{j} } \sin{(\beta_{j} \log{(t)})} + d_{j} t^{\mu_{j}} \cos{(\beta_{j} \log{(t)}) } )$,其中$t$是到角点的距离,参数$\mu_{j},\beta_{j}$是实数,并通过一个明确的公式确定,具体取决于拐角处的角度。除了在分析上清晰可见之外,这些表示还有助于构建高精度和高效的数值离散化,从而显着减少求解相应积分方程所需的自由度数。结果由几个数值例子说明。