Error estimation for immersed interface solutions

Previous Article
Accurate two and three dimensional interpolation for particle mesh calculations
DCDS-B Home
This Issue
Next Article
An augmented immersed interface method for moving structures with mass

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
Related Papers

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 (): 425. 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.

show all references

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 (): 425. 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.

[1]	Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 807-823. doi: 10.3934/dcdsb.2007.7.807
[2]	Mario Ahues, Filomena D. d'Almeida, Alain Largillier, Paulo B. Vasconcelos. Defect correction for spectral computations for a singular integral operator. Communications on Pure and Applied Analysis, 2006, 5 (2) : 241-250. doi: 10.3934/cpaa.2006.5.241
[3]	Champike Attanayake, So-Hsiang Chou. An immersed interface method for Pennes bioheat transfer equation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 323-337. doi: 10.3934/dcdsb.2015.20.323
[4]	Jian Hao, Zhilin Li, Sharon R. Lubkin. An augmented immersed interface method for moving structures with mass. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1175-1184. doi: 10.3934/dcdsb.2012.17.1175
[5]	Qiang Du, Manlin Li. On the stochastic immersed boundary method with an implicit interface formulation. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 373-389. doi: 10.3934/dcdsb.2011.15.373
[6]	So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343
[7]	Shyam Sundar Ghoshal. BV regularity near the interface for nonuniform convex discontinuous flux. Networks and Heterogeneous Media, 2016, 11 (2) : 331-348. doi: 10.3934/nhm.2016.11.331
[8]	Sheng Xu. Derivation of principal jump conditions for the immersed interface method in two-fluid flow simulation. Conference Publications, 2009, 2009 (Special) : 838-845. doi: 10.3934/proc.2009.2009.838
[9]	Youngmok Jeon, Dongwook Shin. Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions. Electronic Research Archive, 2021, 29 (5) : 3361-3382. doi: 10.3934/era.2021043
[10]	Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377
[11]	Patrick W. Dondl, Michael Scheutzow. Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients. Networks and Heterogeneous Media, 2012, 7 (1) : 137-150. doi: 10.3934/nhm.2012.7.137
[12]	Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781
[13]	Pierpaolo Soravia. Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Communications on Pure and Applied Analysis, 2006, 5 (1) : 213-240. doi: 10.3934/cpaa.2006.5.213
[14]	Sofia Giuffrè, Giovanna Idone. On linear and nonlinear elliptic boundary value problems in the plane with discontinuous coefficients. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1347-1363. doi: 10.3934/dcds.2011.31.1347
[15]	Franco Flandoli, Enrico Priola, Giovanni Zanco. A mean-field model with discontinuous coefficients for neurons with spatial interaction. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3037-3067. doi: 10.3934/dcds.2019126
[16]	Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177
[17]	Feng Zhou, Zhenqiu Zhang. Pointwise gradient estimates for subquadratic elliptic systems with discontinuous coefficients. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3137-3160. doi: 10.3934/cpaa.2019141
[18]	Konstantinos Chrysafinos, Efthimios N. Karatzas. Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473
[19]	Kim S. Bey, Peter Z. Daffer, Hideaki Kaneko, Puntip Toghaw. Error analysis of the p-version discontinuous Galerkin method for heat transfer in built-up structures. Communications on Pure and Applied Analysis, 2007, 6 (3) : 719-740. doi: 10.3934/cpaa.2007.6.719
[20]	Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Ben A. Vanderlei Matthew M. Hopkins Lisa J. Fauci
on Google Scholar
Ben A. Vanderlei Matthew M. Hopkins Lisa J. Fauci

[Back to Top]