Universal image systems for non-periodic and periodic Stokes flows above a no-slip wall
Center for Computational Biology, Flatiron Institute（Flatiron研究所计算生物学中心）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
众所周知，通过在平壁上方精心选择的图像点力和双极来放置斯托克斯let，可以方便地在壁上施加无滑动边界条件[Blake, J. R. Math. Proc. Camb. Philos. Soc. 70(2), 1971: 303.]。然而，为了在平行于壁的方向上进一步施加周期性边界条件，通常需要繁琐的推导，因为斯托克斯流中的单或双周期性可能要求周期单元没有净力，而众所周知的图像系统并不满足这一条件。在本文中，我们提出了一种力中性的图像系统。这种中性允许我们将斯托克斯图像系统表示为适用于非周期、单周期和双周期几何的通用公式。这种公式使快速核求和方法能够以黑盒方式使用。我们通过在非周期和双周期几何中使用周期核独立快速多极方法，展示了这种新图像方法的效率和准确性。然后，我们将这种新图像系统扩展到其他广泛使用的斯托克斯基本解，包括斯托克斯let的拉普拉斯和罗内-帕格-山马张量。
英文摘要：
It is well-known that by placing judiciously chosen image point forces and doublets to the Stokeslet above a flat wall, the no-slip boundary condition can be conveniently imposed on the wall [Blake, J. R. Math. Proc. Camb. Philos. Soc. 70(2), 1971: 303.]. However, to further impose periodic boundary conditions on directions parallel to the wall usually involves tedious derivations because single or double periodicity in Stokes flow may require the periodic unit to have no net force, which is not satisfied by the well-known image system. In this work we present a force-neutral image system. This neutrality allows us to represent the Stokes image system in a universal formulation for non-periodic, singly periodic and doubly periodic geometries. This formulation enables the black-box style usage of fast kernel summation methods. We demonstrate the efficiency and accuracy of this new image method with the periodic kernel independent fast multipole method in both non-periodic and doubly periodic geometries. We then extend this new image system to other widely used Stokes fundamental solutions, including the Laplacian of the Stokeslet and the Rotne-Prager-Yamakawa tensor.
We present an isoparametric unfitted finite element approach of the CutFEM or Nitsche-XFEM family for the simulation of two-phase Stokes problems with slip between phases. For the unfitted generalized Taylor--Hood finite element pair $\mathbf{P}_{k+1}-P_k$, $k\ge1$, we show an inf-sup stability property with a stability constant that is independent of the viscosity ratio, slip coefficient, position of the interface with respect to the background mesh and, of course, mesh size. In addition, we prove stability and optimal error estimates that follow from this inf-sup property. We provide numeric…
H(curl)-based approximation of the Stokes problem with slip boundary conditions
Faculty of Mathematics, University of Duisburg-Essen（杜伊斯堡-埃森大学数学学院）；SAM, ETH Zürich（苏黎世联邦理工学院应用数学系）；Department of Mathematics, University of Vienna（维也纳大学数学系）
On the application of the SCD semismooth* Newton method to solving Stokes problem with stick-slip boundary conditions
VSB-Technical University of Ostrava（俄斯特拉发理工大学）；Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences（奥地利科学院约翰·拉德纳计算与应用数学研究所）；Institute of Information Theory and Automation, Czech Academy of Sciences（捷克科学院信息论与自动化研究所）
Equal order stabilized finite elements with Nitsche for stationary Navier-Stokes problem with slip boundary conditions : a priori and a posteriori error analysis
RBFs methods for null control problems of the Stokes system with Dirichlet and Navier slip boundary conditions
Instituto de Matemáticas, Universidad Nacional Autónoma de México（墨西哥国立自治大学数学研究所）；Departamento de Matemática, Universidad Técnica Federico Santa María（智利弗雷德里科·圣玛丽亚理工大学数学系）
Nonlinear entropy stability analysis is used to derive entropy stable no-slip wall boundary conditions for the Eulerian model proposed by Svärd (Physica A: Statistical Mechanics and its Applications, 2018). and its spatial discretization based on entropy stable collocated discontinuous Galerkin operators with the summation-by-parts property for unstructured grids. A set of viscous test cases of increasing complexity are simulated using both the Eulerian and the classic compressible Navier-Stokes models. The numerical results obtained with the two models are compared, and differences and simila…
In this paper, we introduce a nonlocal model for linear steady Stokes system with physical no-slip boundary condition. We use the idea of volume constraint to enforce the no-slip boundary condition and prove that the nonlocal model is well-posed. We also show that and the solution of the nonlocal system converges to the solution of the original Stokes system as the nonlocality vanishes.
Peskin's Immersed Boundary (IB) model and method are among one of the most important modeling tools and numerical methods. The IB method has been known to be first order accurate in the velocity. However, almost no rigorous theoretical proof can be found in the literature for Stokes equations with a prescribed velocity boundary condition. In this paper, it has been shown that the pressure of the Stokes equation has a convergence order $O(\sqrt{h} |\log h| )$ in the $L^2$ norm while the velocity has an $O(h |\log h| )$ convergence order in the infinity norm in two-space dimensions. The proofs a…
The simulation of long, nonlinear dispersive waves in bounded domains usually requires the use of slip-wall boundary conditions. Boussinesq systems appearing in the literature are generally not well-posed when such boundary conditions are imposed, or if they are well-posed it is very cumbersome to implement the boundary conditions in numerical approximations. In the present paper a new Boussinesq system is proposed for the study of long waves of small amplitude in a basin when slip-wall boundary conditions are required. The new system is derived using asymptotic techniques under the assumption…
