# American Institute of Mathematical Sciences

2011, 2011(Special): 1091-1100. doi: 10.3934/proc.2011.2011.1091

## Constructing approximate solutions for three dimensional stationary Stokes flow

 1 Berufsakademie Nordhessen, University of Cooperative Education, Eichlerstr. 25, 34537 Bad Wildungen, Germany 2 Fachbereich Mathematik, Universität Kassel, Heinrich Plett Str. 40 (AVZ), D-34132 Kassel

Received  July 2010 Revised  February 2011 Published  October 2011

In this article we develop a method for the numerical solution of a boundary value problem for the system of Stokes equations in three dimensions. Simultaneously we develop the method for linear splines and for quasi interpolants. For this we start with a representation of the uniquely determined velocity field $v$ by the double layer potential. In a fi rst step we approximate the kernel an the source density of the double layer potential. In a second step we use a collocation method for the determination of the unknown values of the source density in the collocation points. The convergence analysis is carried out and error estimates are given.
Citation: Frank Müller, Werner Varnhorn. Constructing approximate solutions for three dimensional stationary Stokes flow. Conference Publications, 2011, 2011 (Special) : 1091-1100. doi: 10.3934/proc.2011.2011.1091
 [1] César Nieto, Mauricio Giraldo, Henry Power. Boundary integral equation approach for stokes slip flow in rotating mixers. Discrete and Continuous Dynamical Systems - B, 2011, 15 (4) : 1019-1044. doi: 10.3934/dcdsb.2011.15.1019 [2] André Nachbin, Roberto Ribeiro-Junior. A boundary integral formulation for particle trajectories in Stokes waves. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3135-3153. doi: 10.3934/dcds.2014.34.3135 [3] Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137 [4] Sergey E. Mikhailov, Carlos F. Portillo. Boundary-Domain Integral Equations equivalent to an exterior mixed BVP for the variable-viscosity compressible Stokes PDEs. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1103-1133. doi: 10.3934/cpaa.2021009 [5] Carlos Fresneda-Portillo. A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5097-5114. doi: 10.3934/cpaa.2020228 [6] Lucio Boccardo, Daniela Giachetti. A nonlinear interpolation result with application to the summability of minima of some integral functionals. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 31-42. doi: 10.3934/dcdsb.2009.11.31 [7] Antonella Falini, Francesca Mazzia, Cristiano Tamborrino. Spline based Hermite quasi-interpolation for univariate time series. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022039 [8] Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011 [9] Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006 [10] Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277 [11] Abdelkader Boucherif. Positive Solutions of second order differential equations with integral boundary conditions. Conference Publications, 2007, 2007 (Special) : 155-159. doi: 10.3934/proc.2007.2007.155 [12] Oleksandr Boichuk, Victor Feruk. Boundary-value problems for weakly singular integral equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1379-1395. doi: 10.3934/dcdsb.2021094 [13] Tsukasa Iwabuchi. On analyticity up to the boundary for critical quasi-geostrophic equation in the half space. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1209-1224. doi: 10.3934/cpaa.2022016 [14] Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems and Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159 [15] C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403 [16] Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks and Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343 [17] Hongjie Dong, Kunrui Wang. Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5289-5323. doi: 10.3934/dcds.2020228 [18] Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595 [19] Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113 [20] Laurence Cherfils, Madalina Petcu. On the viscous Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1419-1449. doi: 10.3934/cpaa.2016.15.1419

Impact Factor:

## Metrics

• PDF downloads (52)
• HTML views (0)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]