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Numerical solution of 2D and 3D elliptic-type interface models with regular interfaces

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Abstract

In the current paper, two numerical methods are proposed for the numerical solution of two- and three-dimensional elliptic partial differential equations (PDEs) with regular interfaces. The proposed methods are based on meshless collocation and Haar wavelet collocation. Numerical tests are performed to check accuracy and robustness of the proposed methods. Numerical results of the proposed methods are measured in terms of \(L_{\infty }\) error norm to show their better accuracy than the existing methods.

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References

  1. Li Z, Ito K (2001) Maximum principle preserving schemes for interface problems with discontinuous coefficients. SIAM J Sci Comput 23:339–361

    Article  MathSciNet  MATH  Google Scholar 

  2. Baruch G, Fibich G, Tsynkov S, Turkel E (2009) Fourth order schemes for time-harmonic wave equations with discontinuous coefficients. Commun Comput Phys 5:442–455

    MathSciNet  MATH  Google Scholar 

  3. Li Z, McTigue D, Heine J (1997) A numerical method for diffusive transport with moving boundaries and discontinuous material properties. Int J Numer Anal Methods Geomech 21:653–662

    Article  Google Scholar 

  4. Li Z, Wang D, Zou J (1998) Theoretical and numerical analysis on a thermo-elastic system with discontinuities. Comput Appl Math 92:37–58

    Article  MathSciNet  MATH  Google Scholar 

  5. Linnick MN, Fasel HF (2005) A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains. J Comput Phys 204:157–192

    Article  MathSciNet  MATH  Google Scholar 

  6. Liu X, Sideris TC (2003) Convergence of the ghost fluid method for elliptic equations with interfaces. Math Comput 72:1731–1746

    Article  MathSciNet  MATH  Google Scholar 

  7. Rutka V, Li Z (2008) An explicit jump immersed interface method for two-phase Navier–Stokes equations with interfaces. Comput Methods Appl Mech Eng 197:2317–2328

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhou YC, Zhao S, Feig M, Wei GW (2006) High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J Comput Phys 213:1–30

    Article  MathSciNet  MATH  Google Scholar 

  9. Xu S, Wang ZJ (2006) Systematic derivation of jump conditions for the immersed interface method in three-dimensional flow simulation. SIAM J Sci Comput 27:1948–1980

    Article  MathSciNet  MATH  Google Scholar 

  10. Wiegmann A, Bube K (1998) The immersed interface method for nonlinear differential equations with discontinuous coefficients and singular sources. SIAM J Numer Anal 35:177–200

    Article  MathSciNet  MATH  Google Scholar 

  11. LeVeque RJ, Li Z (1994) The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J Numer Anal 31:1019–1044

    Article  MathSciNet  MATH  Google Scholar 

  12. Li Z, Ito K (2006) The immersed interface method: numerical solutions of PDEs involving interfaces and irregular domains. In: Frontiers in applied mathematics, vol 33. Soc. Ind. Appl. Math. (SIAM), Philadelphia

  13. Calhoun D (2002) A cartesian grid method for solving the streamfunction-vorticity equation in irregular regions. J Comput Phys 176:231–275

    Article  MathSciNet  MATH  Google Scholar 

  14. Li Z, Wang C (2003) A fast finite difference method for solving Navier–Stokes equations on irregular domains. J Commun Math Sci 1:180–196

    Article  MathSciNet  MATH  Google Scholar 

  15. Russell D, Wang ZJ (2003) A cartesian grid method for modeling multiple moving irregular objects in 2D incompressible viscous flow. J Comput Phys 191:177–205

    Article  MathSciNet  MATH  Google Scholar 

  16. Bell JB, Colella P, Glaz HM (1989) A second-order projection method for the incompressible Navier–Stokes equations. J Comput Phys 85:257–283

    Article  MathSciNet  MATH  Google Scholar 

  17. Kan J (1986) A second-order accurate pressure-correction scheme for viscous incompressible flow. SIAM J Sci Comput 7:870–891

    Article  MathSciNet  MATH  Google Scholar 

  18. Kim J, Moin P (1985) Application of a fractional-step method to incompressible Navier–Stokes equations. J Comput Phys 59:308–323

    Article  MathSciNet  MATH  Google Scholar 

  19. Dahmen W, Kurdila A, Oswald P (1997) Multiscale wavelet methods for partial differential equations. Academic Press, New York

    Google Scholar 

  20. Siraj-ul-Islam, Aziz I, Haq F (2010) A comparative study of numerical integration based on Haar wavelets and hybrid functions. Comput Math Appl 59:2026–2036

    Article  MathSciNet  MATH  Google Scholar 

  21. Aziz I, Siraj-ul-Islam, Khan W (2011) Quadrature rules for numerical integration based on Haar wavelets and hybrid functions. Comput Math Appl 61:2770–2781

    Article  MathSciNet  MATH  Google Scholar 

  22. Dehghan M, Lakestani M (2008) Numerical solution of nonlinear system of second-order boundary value problems using cubic B-spline scaling functions. Int J Comput Math 85:1455–1461

    Article  MathSciNet  MATH  Google Scholar 

  23. Siraj-ul-Islam, Aziz I, Šarlar B (2010) The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets. Math Comput Model 52:1577–1590

    Article  MathSciNet  MATH  Google Scholar 

  24. Comincioli V, Naldi G, Scapolla T (2000) A wavelet-based method for numerical solution of nonlinear evolution equations. Appl Numer Math 33:291–297

    Article  MathSciNet  MATH  Google Scholar 

  25. Wu JL (2009) A wavelet operational method for solving fractional partial differential equations numerically. Appl Math Comput 214:31–40

    MathSciNet  MATH  Google Scholar 

  26. Diaz L, Martin M, Vampa V (2009) Daubechies wavelet beam and plate finite elements. Finite Elem Anal Des 45:200–209

    Article  MathSciNet  Google Scholar 

  27. Zhu X, Lei G, Pan G (1997) On application of fast and adaptive Battle–Lemarie wavelets to modeling of multiple lossy transmission lines. J Comput Phys 132:299–311

    Article  MathSciNet  MATH  Google Scholar 

  28. Babolian E, Fattahzdeh F (2007) Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration. Appl Math Comput 188:417–426

    MathSciNet  MATH  Google Scholar 

  29. Banifatemi E, Razzaghi M, Yousefi S (2007) Two-dimensional Legendre wavelets method for the mixed Volterra–Fredholm integral equations. J Vib Control 13:1667–1675

    Article  MathSciNet  MATH  Google Scholar 

  30. Lepik U (2007) Numerical solution of evolution equations by the Haar wavelet method. Appl Math Comput 185:695–704

    MathSciNet  MATH  Google Scholar 

  31. Chen C, Hsiao C (1997) Haar wavelet method for solving lumped and distributed-parameter systems. IEE Proc Control Theory Appl 144:87–94

    Article  MATH  Google Scholar 

  32. Kansa EJ (1990) Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates. Comput Math Appl 19:127–145

    Article  MathSciNet  MATH  Google Scholar 

  33. Franke C, Schaback R (1998) Convergence order estimates of meshless collocation methods using radial basis functions. Adv Comput Math 8:381–399

    Article  MathSciNet  MATH  Google Scholar 

  34. Chinchapatnam P, Djidjeli K, Nair P (2006) Unsymmetric and symmetric meshless schemes for the unsteady convection–diffusion equation. Comput Methods Appl Mech Eng 195:2432–2453

    Article  MathSciNet  MATH  Google Scholar 

  35. Franke C, Schaback R (1998) Solving partial differential equations by collocation with radial basis functions. Appl Math Comput 93:72–82

    MathSciNet  MATH  Google Scholar 

  36. Siraj-ul-Islam, Ahmad I (2016) A comparative analysis of local meshless formulation for multi-asset option models. Eng Anal Bound Elem 65:159–176

    Article  MathSciNet  MATH  Google Scholar 

  37. Siraj-ul-Islam, Singh V, Kumar S (2017) Estimation of dispersion in an open channel from an elevated source using an upwind local meshless method. Int J Comput Methods 14:1750009

    Article  MathSciNet  MATH  Google Scholar 

  38. Bellman RE, Kalaba RE (1965) Quasilinearization and non-linear boundary value problems. American Elsevier, New York

    MATH  Google Scholar 

  39. Feng X, Li Z (2012) Simplified immersed interface methods for elliptic interface problems with straight interfaces. Numer Methods Partial Differ Eqs 28:188–203

    Article  MathSciNet  MATH  Google Scholar 

  40. Majak J, Shvartsman BS, Kirs M, Pohlak M, Herranen H (2015) Convergence theorem for the Haar wavelet based discretization method. Compos Struct 126:227–232

    Article  Google Scholar 

  41. Majak J, Shvartsman BS, Kurjust K, Mikola M, Haavajoe A, Pohlak M (2015) On the accuracy of the Haar wavelet discretization method. Compos Part B Eng 80:321–327

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Peshawar, Peshawar, Pakistan

    Nadeem Haider & Imran Aziz

  2. Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan

    Siraj-ul-Islam

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  1. Nadeem Haider
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  2. Imran Aziz
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  3. Siraj-ul-Islam
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Correspondence to Imran Aziz.

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Haider, N., Aziz, I. & Siraj-ul-Islam Numerical solution of 2D and 3D elliptic-type interface models with regular interfaces. Engineering with Computers 35, 1081–1102 (2019). https://doi.org/10.1007/s00366-018-0652-0

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  • DOI: https://doi.org/10.1007/s00366-018-0652-0

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