Error estimation for immersed interface solutions

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June 2012, 17(4): 1185-1203. doi: 10.3934/dcdsb.2012.17.1185

Error estimation for immersed interface solutions

Ben A. Vanderlei ^1, , Matthew M. Hopkins ^2, and Lisa J. Fauci ^3,

1.	Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada
2.	Nanoscale and Reactive Processes, Sandia National Laboratories, Albuquerque, NM 87185-0836, United States
3.	Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, LA 70118, United States

Received September 2010 Revised August 2011 Published February 2012

Abstract
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We present an error estimation method for immersed interface solutions of elliptic boundary value problems. As opposed to an asymptotic rate that indicates how the errors in the numerical method converge to zero, we seek a posteriori estimates of the errors, and their spatial distribution, for a given solution. Our estimate is based upon the classical idea of defect corrections, which requires the application of a higher-order discretization operator to a solution achieved with a lower-order discretization. Our model problem will be an elliptic boundary value problem in which the coefficients are discontinuous across an internal boundary.

Keywords: Immersed interface, error estimation, defect correction, discontinuous coefficients..

Mathematics Subject Classification: Primary: 65N22, 65N06; Secondary: 82B2.

Citation: Ben A. Vanderlei, Matthew M. Hopkins, Lisa J. Fauci. Error estimation for immersed interface solutions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1185-1203. doi: 10.3934/dcdsb.2012.17.1185

References:

[1]	W. Auzinger, Defect correction for nonlinear elliptic difference equations, Numerische Mathematik, 51 (1987), 199-208. doi: 10.1007/BF01396749.
[2]	R. LeVeque and Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM Journal on Numerical Analysis, 31 (1994), 1019-1044. doi: 10.1137/0731054.
[3]	R. LeVeque, "Finite Difference Methods for Ordinary and Partial Differential Equations. Steady-State and Time-Dependent Problems," SIAM, Philadelphia, PA, 2007.
[4]	Z. Li, A fast iterative algorithm for elliptic interface problems, SIAM Journal on Numerical Analysis, 35 (1998), 230-254. doi: 10.1137/S0036142995291329.
[5]	Z. Li and M.-C. Lai, The immersed interface method for Navier-Stokes equations with singular forces, Journal of Computational Physics, 171 (2001), 822-842. doi: 10.1006/jcph.2001.6813.
[6]	Z. Li and K. Ito, Maximum principle preserving schemes for interface problems with discontinuous coefficients, SIAM Journal on Scientific Computing, 23 (2001), 339-361. doi: 10.1137/S1064827500370160.
[7]	Z. Li and K. Ito, "The Immersed Interface Method. Numerical Solutions of PDE's Involving Interfaces and Irregular Domains," Frontiers in Applied Mathematics, 33, SIAM, Philadelphia, PA, 2006.
[8]	B. Lindberg, Error estimation and iterative improvement for discretization algorithms, BIT, 20 (1980), 486-500. doi: 10.1007/BF01933642.
[9]	W. Oberkampf and C. Roy, "Verification and Validation in Scientific Computing," Cambridge University Press, Cambridge, 2010.
[10]	C. Peskin, The immersed boundary method, Acta Numerica, 11 (2002), 479-517. doi: 10.1017/S0962492902000077.
[11]	C. Pozrikidis, "Boundary Integral and Singularity Methods for Linearized Viscous Flow," Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511624124.
[12]	P. Roache, "Verification and Validation in Computational Science and Engineering," Hermosa Publishers, Socorro, New Mexico, 1998.
[13]	C. Roy, A. Raju and M. Hopkins, Estimation of discretization errors using the method of nearby problems, AIAA Journal, 45 (2007), 1232-1243. doi: 10.2514/1.24282.
[14]	C. Roy and A. Sinclair, On the generation of exact solutions for evaluating numerical schemes and estimating discretization error, Journal of Computational Physics, 228 (2009), 1790-1802. doi: 10.1016/j.jcp.2008.11.008.
[15]	H. Stetter, The defect correction principle and discretization methods, Numerische Mathematik, 29 (1977/78), 425-443. doi: 10.1007/BF01432879.
[16]	S. Xu and Z. Wang, An immersed interface method for simulating the interaction of a fluid with moving boundaries, Journal of Computational Physics, 216 (2006), 454-493. doi: 10.1016/j.jcp.2005.12.016.
[17]	X. Zhong, A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity, Journal of Computational Physics, 225 (2007), 1066-1099. doi: 10.1016/j.jcp.2007.01.017.
[18]	Y. Zhou, S. Zhou, M. Feig and G. Wei, High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources, Journal of Computational Physics, 213 (2006), 1-30. doi: 10.1016/j.jcp.2005.07.022.

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References:

[1]	W. Auzinger, Defect correction for nonlinear elliptic difference equations, Numerische Mathematik, 51 (1987), 199-208. doi: 10.1007/BF01396749.
[2]	R. LeVeque and Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM Journal on Numerical Analysis, 31 (1994), 1019-1044. doi: 10.1137/0731054.
[3]	R. LeVeque, "Finite Difference Methods for Ordinary and Partial Differential Equations. Steady-State and Time-Dependent Problems," SIAM, Philadelphia, PA, 2007.
[4]	Z. Li, A fast iterative algorithm for elliptic interface problems, SIAM Journal on Numerical Analysis, 35 (1998), 230-254. doi: 10.1137/S0036142995291329.
[5]	Z. Li and M.-C. Lai, The immersed interface method for Navier-Stokes equations with singular forces, Journal of Computational Physics, 171 (2001), 822-842. doi: 10.1006/jcph.2001.6813.
[6]	Z. Li and K. Ito, Maximum principle preserving schemes for interface problems with discontinuous coefficients, SIAM Journal on Scientific Computing, 23 (2001), 339-361. doi: 10.1137/S1064827500370160.
[7]	Z. Li and K. Ito, "The Immersed Interface Method. Numerical Solutions of PDE's Involving Interfaces and Irregular Domains," Frontiers in Applied Mathematics, 33, SIAM, Philadelphia, PA, 2006.
[8]	B. Lindberg, Error estimation and iterative improvement for discretization algorithms, BIT, 20 (1980), 486-500. doi: 10.1007/BF01933642.
[9]	W. Oberkampf and C. Roy, "Verification and Validation in Scientific Computing," Cambridge University Press, Cambridge, 2010.
[10]	C. Peskin, The immersed boundary method, Acta Numerica, 11 (2002), 479-517. doi: 10.1017/S0962492902000077.
[11]	C. Pozrikidis, "Boundary Integral and Singularity Methods for Linearized Viscous Flow," Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511624124.
[12]	P. Roache, "Verification and Validation in Computational Science and Engineering," Hermosa Publishers, Socorro, New Mexico, 1998.
[13]	C. Roy, A. Raju and M. Hopkins, Estimation of discretization errors using the method of nearby problems, AIAA Journal, 45 (2007), 1232-1243. doi: 10.2514/1.24282.
[14]	C. Roy and A. Sinclair, On the generation of exact solutions for evaluating numerical schemes and estimating discretization error, Journal of Computational Physics, 228 (2009), 1790-1802. doi: 10.1016/j.jcp.2008.11.008.
[15]	H. Stetter, The defect correction principle and discretization methods, Numerische Mathematik, 29 (1977/78), 425-443. doi: 10.1007/BF01432879.
[16]	S. Xu and Z. Wang, An immersed interface method for simulating the interaction of a fluid with moving boundaries, Journal of Computational Physics, 216 (2006), 454-493. doi: 10.1016/j.jcp.2005.12.016.
[17]	X. Zhong, A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity, Journal of Computational Physics, 225 (2007), 1066-1099. doi: 10.1016/j.jcp.2007.01.017.
[18]	Y. Zhou, S. Zhou, M. Feig and G. Wei, High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources, Journal of Computational Physics, 213 (2006), 1-30. doi: 10.1016/j.jcp.2005.07.022.

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