American Institute of Mathematical Sciences

Advanced
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

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 & 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.  doi: 10.1007/BF01396749.  Google Scholar

[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.  doi: 10.1137/0731054.  Google Scholar

[3]

R. LeVeque, "Finite Difference Methods for Ordinary and Partial Differential Equations. Steady-State and Time-Dependent Problems,", SIAM, (2007).   Google Scholar

[4]

Z. Li, A fast iterative algorithm for elliptic interface problems,, SIAM Journal on Numerical Analysis, 35 (1998), 230.  doi: 10.1137/S0036142995291329.  Google Scholar

[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.  doi: 10.1006/jcph.2001.6813.  Google Scholar

[6]

Z. Li and K. Ito, Maximum principle preserving schemes for interface problems with discontinuous coefficients,, SIAM Journal on Scientific Computing, 23 (2001), 339.  doi: 10.1137/S1064827500370160.  Google Scholar

[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 (2006).   Google Scholar

[8]

B. Lindberg, Error estimation and iterative improvement for discretization algorithms,, BIT, 20 (1980), 486.  doi: 10.1007/BF01933642.  Google Scholar

[9]

W. Oberkampf and C. Roy, "Verification and Validation in Scientific Computing,", Cambridge University Press, (2010).   Google Scholar

[10]

C. Peskin, The immersed boundary method,, Acta Numerica, 11 (2002), 479.  doi: 10.1017/S0962492902000077.  Google Scholar

[11]

C. Pozrikidis, "Boundary Integral and Singularity Methods for Linearized Viscous Flow,", Cambridge Texts in Applied Mathematics, (1992).  doi: 10.1017/CBO9780511624124.  Google Scholar

[12]

P. Roache, "Verification and Validation in Computational Science and Engineering,", Hermosa Publishers, (1998).   Google Scholar

[13]

C. Roy, A. Raju and M. Hopkins, Estimation of discretization errors using the method of nearby problems,, AIAA Journal, 45 (2007), 1232.  doi: 10.2514/1.24282.  Google Scholar

[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.  doi: 10.1016/j.jcp.2008.11.008.  Google Scholar

[15]

H. Stetter, The defect correction principle and discretization methods,, Numerische Mathematik, 29 (): 425.  doi: 10.1007/BF01432879.  Google Scholar

[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.  doi: 10.1016/j.jcp.2005.12.016.  Google Scholar

[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.  doi: 10.1016/j.jcp.2007.01.017.  Google Scholar

[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.  doi: 10.1016/j.jcp.2005.07.022.  Google Scholar

show all references

References:
[1]

W. Auzinger, Defect correction for nonlinear elliptic difference equations,, Numerische Mathematik, 51 (1987), 199.  doi: 10.1007/BF01396749.  Google Scholar

[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.  doi: 10.1137/0731054.  Google Scholar

[3]

R. LeVeque, "Finite Difference Methods for Ordinary and Partial Differential Equations. Steady-State and Time-Dependent Problems,", SIAM, (2007).   Google Scholar

[4]

Z. Li, A fast iterative algorithm for elliptic interface problems,, SIAM Journal on Numerical Analysis, 35 (1998), 230.  doi: 10.1137/S0036142995291329.  Google Scholar

[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.  doi: 10.1006/jcph.2001.6813.  Google Scholar

[6]

Z. Li and K. Ito, Maximum principle preserving schemes for interface problems with discontinuous coefficients,, SIAM Journal on Scientific Computing, 23 (2001), 339.  doi: 10.1137/S1064827500370160.  Google Scholar

[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 (2006).   Google Scholar

[8]

B. Lindberg, Error estimation and iterative improvement for discretization algorithms,, BIT, 20 (1980), 486.  doi: 10.1007/BF01933642.  Google Scholar

[9]

W. Oberkampf and C. Roy, "Verification and Validation in Scientific Computing,", Cambridge University Press, (2010).   Google Scholar

[10]

C. Peskin, The immersed boundary method,, Acta Numerica, 11 (2002), 479.  doi: 10.1017/S0962492902000077.  Google Scholar

[11]

C. Pozrikidis, "Boundary Integral and Singularity Methods for Linearized Viscous Flow,", Cambridge Texts in Applied Mathematics, (1992).  doi: 10.1017/CBO9780511624124.  Google Scholar

[12]

P. Roache, "Verification and Validation in Computational Science and Engineering,", Hermosa Publishers, (1998).   Google Scholar

[13]

C. Roy, A. Raju and M. Hopkins, Estimation of discretization errors using the method of nearby problems,, AIAA Journal, 45 (2007), 1232.  doi: 10.2514/1.24282.  Google Scholar

[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.  doi: 10.1016/j.jcp.2008.11.008.  Google Scholar

[15]

H. Stetter, The defect correction principle and discretization methods,, Numerische Mathematik, 29 (): 425.  doi: 10.1007/BF01432879.  Google Scholar

[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.  doi: 10.1016/j.jcp.2005.12.016.  Google Scholar

[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.  doi: 10.1016/j.jcp.2007.01.017.  Google Scholar

[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.  doi: 10.1016/j.jcp.2005.07.022.  Google Scholar

[1]

Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343
[7]

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
[8]

Shyam Sundar Ghoshal. BV regularity near the interface for nonuniform convex discontinuous flux. Networks & Heterogeneous Media, 2016, 11 (2) : 331-348. doi: 10.3934/nhm.2016.11.331
[9]

Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377
[10]

Patrick W. Dondl, Michael Scheutzow. Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients. Networks & Heterogeneous Media, 2012, 7 (1) : 137-150. doi: 10.3934/nhm.2012.7.137
[11]

Pierpaolo Soravia. Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Communications on Pure & Applied Analysis, 2006, 5 (1) : 213-240. doi: 10.3934/cpaa.2006.5.213
[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]

Sofia Giuffrè, Giovanna Idone. On linear and nonlinear elliptic boundary value problems in the plane with discontinuous coefficients. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1347-1363. doi: 10.3934/dcds.2011.31.1347
[14]

Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177
[15]

Franco Flandoli, Enrico Priola, Giovanni Zanco. A mean-field model with discontinuous coefficients for neurons with spatial interaction. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3037-3067. doi: 10.3934/dcds.2019126
[16]

Feng Zhou, Zhenqiu Zhang. Pointwise gradient estimates for subquadratic elliptic systems with discontinuous coefficients. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3137-3160. doi: 10.3934/cpaa.2019141
[17]

Konstantinos Chrysafinos, Efthimios N. Karatzas. Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473
[18]

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 & Applied Analysis, 2007, 6 (3) : 719-740. doi: 10.3934/cpaa.2007.6.719
[19]

Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185
[20]

Serena Dipierro, Aram Karakhanyan, Enrico Valdinoci. Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6073-6090. doi: 10.3934/dcds.2018262

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences