×

Error analysis of the immersed interface method for Stokes equations with an interface. (English) Zbl 1457.65161

Summary: The immersed interface method using the three Poisson equation approach has been successfully developed to solve incompressible Stokes equations with interfaces [Z. Li, Int. J. Numer. Methods Eng. 106, No. 7, 556–575 (2016; Zbl 1352.76083); Z. Li and K. Ito, The immersed interface method. Numerical solutions of PDE involving interfaces and irregular domains. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (2006; Zbl 1122.65096)]. While the numerical results show second order convergence for both velocity and pressure, rigorous error analysis is still missing. Based on recent theoretical development, particularly the error analysis by J. T. Beale and A. T. Layton [Commun. Appl. Math. Comput. Sci. 1, 91–119 (2006; Zbl 1153.35319)], second order convergence has been shown in this paper for both pressure and velocity under some assumptions.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows
76M20 Finite difference methods applied to problems in fluid mechanics
35J25 Boundary value problems for second-order elliptic equations
35Q35 PDEs in connection with fluid mechanics

Software:

IIMPACK
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Peskin, C. S., The immersed boundary method, Acta Numer., 1-39, (2002)
[2] Li, Z.; Ito, K., Maximum principle preserving schemes for interface problems with discontinuous coefficients, SIAM J. Sci. Comput., 23, 339-361, (2001) · Zbl 1001.65115
[3] LeVeque, R. J.; Li, Z., The immersed interface method for elliptic euqations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31, 4, 1019-1044, (1994) · Zbl 0811.65083
[4] Li, Z., An overview of the immersed interface method and its applications, J. Math., 7, 1-49, (2003) · Zbl 1028.65108
[5] Li, Z.; Ito, K., The immersed interface method-numerical solutions of PDEs involving interfaces and irregular domains, SIAM Front. Ser. Appl. Math., FR33, (2006)
[6] Ito, K.; Li, Z.; Wan, X., Pressure jump conditions for Stokes equations with discontinuous viscosity in 2D and 3D, Methods Appl. Anal., 13, 199-214, (2006) · Zbl 1142.76022
[7] Johnston, H.; Liu, J., Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term, J. Comput. Phys., 199, 221-259, (2004) · Zbl 1127.76343
[8] J.T. Beale, A.T. Layton, On the accuracy of finite difference methods for elliptic problems with interfaces, 2006.; J.T. Beale, A.T. Layton, On the accuracy of finite difference methods for elliptic problems with interfaces, 2006. · Zbl 1153.35319
[9] Li, Z., An augmented Cartesian grid method for Stokes-Darcy fluid-structure interactions, Internat. J. Numer. Methods Engrg., 106, 556-575, (2015) · Zbl 1352.76083
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.