Skip to main content
Log in

Summary

The Dirichlet-to-Neumann (DtN) Finite Element Method is a general technique for the solution of problems in unbounded domains, which arise in many fields of application. Its name comes from the fact that it involves the nonlocal Dirichlet-to-Neumann (DtN) map on an artificial boundary which encloses the computational domain. Originally the method has been developed for the solution of linear elliptic problems, such as wave scattering problems governed by the Helmholtz equation or by the equations of time-harmonic elasticity. Recently, the method has been extended in a number of directions, and further analyzed and improved, by the author's group and by others. This article is a state-of-the-art review of the method. In particular, it concentrates on two major recent advances: (a) the extension of the DtN finite element method tononlinear elliptic and hyperbolic problems; (b) procedures forlocalizing the nonlocal DtN map, which lead to a family of finite element schemes with local artificial boundary conditions. Possible future research directions and additional extensions are also discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Givoli (1992),Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam.

    MATH  Google Scholar 

  2. Exterior Problems of Wave Propagation, special issue, D. Givoli and I. Harari, eds. (1998),Comput. Meth. Appl. Mech. Engng., Vol.164, Nos. 1–2.

  3. D. Givoli (1991), “Non-Reflecting Boundary Conditions”,J. of Computational Physics,94, pp. 1–29.

    Article  MATH  MathSciNet  Google Scholar 

  4. B. Gustafsson and H.-O. Kreiss (1979), “Boundary Conditions for Time Dependent Problems With an Artificial Boundary”,J. Comp. Phys.,30, pp. 333–351.

    Article  MATH  MathSciNet  Google Scholar 

  5. L. Ferm and B. Gustafsson (1982), “A Down-Stream Boundary Procedure for the Euler Equations”,Comp. Fluids,10, pp. 261–276.

    Article  MATH  Google Scholar 

  6. T. Hagstrom and H.B. Keller (1986), “Exact Boundary Conditions at an Artificial Boundary for Partial Differential Equations in Cylinders”,SIAM J. Math. Anal.,17, pp. 322–341.

    Article  MATH  MathSciNet  Google Scholar 

  7. T. Hagstrom and H.B. Keller (1987), “Asymptotic Boundary Conditions and Numerical Methods for Nonlinear Elliptic Problems on Unbounded Domains”,Math. Comput.,48, pp. 449–470.

    Article  MATH  MathSciNet  Google Scholar 

  8. L. Ting and M.J. Miksis (1986), “Exact Boundary Conditions for Scattering Problems”,J. Acoust. Soc. Am.,80, pp. 1825–1827.

    Article  Google Scholar 

  9. D. Givoli and D. Cohen (1995), “Non-reflecting Boundary Conditions Based on Kirchhoff-type Formulac”,J. of Computational Physics,117, pp. 102–113.

    Article  MATH  MathSciNet  Google Scholar 

  10. M.J. Grote and J.B. Keller (1995), “Exact Nonreflecting Boundary Conditions for the Time Dependent Wave Equation”,SIAM J. of Appl. Math.,55, pp. 280–297.

    Article  MATH  MathSciNet  Google Scholar 

  11. M.J. Grote and J.B. Keller (1996), “Nonreflecting Boundary Conditions for Time Dependent Scattering”,J. of Comput. Phys.,127, pp. 52–65.

    Article  MATH  MathSciNet  Google Scholar 

  12. S.V. Tsynkov (1995), “An Application of Nonlocal External Conditions to Viscous Flow Computations”,J. Comput. Phys.,116, pp. 212–225.

    Article  MATH  MathSciNet  Google Scholar 

  13. S.V. Tsynkov, E. Turkel and S. Abarbanel (1996), “External Flow Computations Using Global Boundary Conditions”,AIAA J.,34, pp. 700–706.

    Google Scholar 

  14. H. Han and X. Wu (1992), “The Approximation of the Exact Boundary Conditions at an Artificial Boundary for Linear Elastic Equations and its Application”,Math. of Comput.,59, 21–37.

    Article  MATH  MathSciNet  Google Scholar 

  15. J.B. Keller and D. Givoli (1989), “Exact Non-Reflecting Boundary Conditions”,J. Comput. Phys.,82, pp. 172–192.

    Article  MATH  MathSciNet  Google Scholar 

  16. D. Givoli and J.B. Keller (1989), “A Finite Element Method for Large Domains”,Comput. Meth. Appl. Mech. Engng.,76, pp. 41–66.

    Article  MATH  MathSciNet  Google Scholar 

  17. G.J. Fix and S.P. Marin (1978), “Variational Methods for Underwater Acoustic Problems”,J. Comp. Phys.,28, pp. 253–270.

    Article  MATH  MathSciNet  Google Scholar 

  18. R.C. MacCamy and S.P. Marin (1980), “A Finite Element Method for Exterior Interface Problems”,Int. J. Math. Math. Sci.,3, pp. 311–350.

    Article  MATH  MathSciNet  Google Scholar 

  19. S.P. Marin (1982), “Computing Scattering Amplitudes for Arbitrary Cylinders Under Incident Plane Waves”,IEEE Trans. Antennas and Propagat.,AP-30, pp. 1045–1049.

    Article  Google Scholar 

  20. C.I. Goldstein (1982), “A Finite Element Method for Solving Helmholtz Type Equations in Wave Guides and Other Unbounded Domains”,Math. Comp.,39, pp. 309–324.

    Article  MATH  MathSciNet  Google Scholar 

  21. A. Bayliss, C.I. Goldstein and E. Turkel (1985), “The Numerical Solution of the Helmholtz Equation for Wave Propagation Problems in Underwater Acoustics”,Comp. and Math. with Appl.,11, pp. 655–665.

    Article  MATH  MathSciNet  Google Scholar 

  22. C. Canuto, S.I. Hariharan and L. Lustman (1985), “Spectral Methods for Exterior Elliptic Problems”,Numer. Math.,46, pp. 505–520.

    Article  MATH  MathSciNet  Google Scholar 

  23. Feng Kang (1983), “Finite Element Method and Natural Boundary Reduction”,Proc. of the International Congress of Mathematicians, Warsaw, pp. 1439–1453.

  24. Feng Kang (1984), “Asymptotic Radiation Conditions for Reduced Wave Equation”,J. of Computational Math. (published in China),2, 130–138.

    MATH  Google Scholar 

  25. M. Masmoudi (1987), “Numerical Solution of Exterior Problems”,Numer. Math.,51, pp. 87–101.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Lenoir and A. Tounsi (1988), “The Localized Finite Element Method and Its Application to the 2D Sea-Keeping Problem”,SIAM J. Num. Anal.,25, pp. 729–752.

    Article  MATH  MathSciNet  Google Scholar 

  27. S.W. Marcus (1991), “A Generalized Impedance Method for Application of the Parabolic Approximation to Underwater Acoustics”,J. Acoust. Soc. Am.,90, pp. 391–398.

    Article  Google Scholar 

  28. S.W. Marcus (1992), “A Hybrid (Finite Difference-Surface Green's Function) Method for Computing Transmission Losses in an Inhomogeneous Atmosphere Over Irregular Terrain”,IEEE Trans. Antennas and Propag.,40, pp. 1451–1458.

    Article  Google Scholar 

  29. M. Verriere and M. Lenoir (1992), “Computation of Waves Generated by Submarine Landslides”,Int. J. Num. Meth. Fluids,14, pp. 403–421.

    Article  MATH  Google Scholar 

  30. G.N. Gatica and G.C. Hsiao (1992), “The Coupling of Boundary Integral and Finite Element Methods for Nonmonotone Nonlinear Problems”,Numer. Func. Anal. Optim.,13, pp. 431–447.

    Article  MATH  MathSciNet  Google Scholar 

  31. G.N. Gatica and G.C. Hsiao (1992), “On The Coupled BEM and FEM for a Nonlinear Exterior Dirichlet Problem in R2Numer. Math.,61, pp. 171–214.

    Article  MATH  MathSciNet  Google Scholar 

  32. Araya, Rodolfo A.; Gatica, Gabriel N.; Mennickent, Hubert (1996), “Boundary-field equation method for a nonlinear exterior elasticity problem in the plane”,Journal of Elasticity,43, pp. 45–68.

    MATH  MathSciNet  Google Scholar 

  33. D. Givoli (1990), “Finite Element Analysis of Long Cylindrical Shells”,AIAA Journal,28, pp. 1331–1333.

    Google Scholar 

  34. D. Givoli (1990), “A Combined Analytic-Finite Element Method for Elastic Shells”,Int. J. of Solids and Structures,26, pp. 185–198.

    Article  MATH  Google Scholar 

  35. D. Givoli and J.B. Keller (1990), “Non-Reflecting Boundary Conditions for Elastic Waves”,Wave Motion,12, pp. 261–279.

    Article  MATH  MathSciNet  Google Scholar 

  36. D. Givoli and S. Vigdergauz (1993), “Artificial Boundary Conditions for 2D Problems in Geophysics”,Comput. Meth. Appl. Mech. Engng.,110, pp. 87–101.

    Article  MATH  MathSciNet  Google Scholar 

  37. I. Patlashenko and D. Givoli (1997), “Non-Reflecting Finite Element Schemes for Three-Dimensional Acoustic Waves”,J. Computational Acoustics,5, pp. 95–115.

    Article  Google Scholar 

  38. I. Harari, I. Patlashenko and D. Givoli (1998), “DtN Maps for Unbounded Wave Guides”,J. Computational Physics,143, pp. 200–223.

    Article  MATH  MathSciNet  Google Scholar 

  39. G. Ben-Porat and D. Givoli (1995), “Solution of Unbounded Domain Problems Using Elliptic Artificial Boundaries”,Commun. Num. Meth. Engng.,11, pp. 735–741.

    Article  MATH  MathSciNet  Google Scholar 

  40. M.J. Grote and J.B. Keller (1995), “On Nonreflecting Boundary Conditions”,J. Comput. Phys.,122, pp. 231–243.

    Article  MATH  MathSciNet  Google Scholar 

  41. K.G. Ayappa, H.T. Davis, E.A. Davis and J. Gordon (1992), “Two-Dimensional Finite Element Analysis of Microwave Heating”,AIChE J.,38, pp. 1577–1590.

    Article  Google Scholar 

  42. G.N. Gatica and M.E. Mellado (1998), “On the Numerical Solution of Linear Exterior Problems via the Uncoupling Method”,Int. J. Numer. Meth. Engng.,41, pp. 233–251.

    Article  MATH  MathSciNet  Google Scholar 

  43. M. Malhotra and P. Pinsky (1996), “Matrix-free interpretation of the non-local Dirichlet-to-Neumann radiation boundary condition”,Int. J. Numer. Meth. Engng.,39, pp. 3705–3713.

    Article  MATH  Google Scholar 

  44. I. Harari and T.J.R. Hughes (1992), “A Cost Comparison of Boundary Element and Finite Element Methods for Problems of Time-Harmonic Acoustics”,Comput. Meth. Appl. Mech. Engng.,97, pp. 77–102.

    Article  MATH  MathSciNet  Google Scholar 

  45. D. Givoli, I. Patlashenko and J.B. Keller (1997), “High order Boundary Conditions and Finite Elements for Infinite Domains”,Comput. Meth. Appl. Mech. Engng.,143, pp. 13–39.

    Article  MATH  MathSciNet  Google Scholar 

  46. I. Harari and T.J.R. Hughes (1992), “Galerkin/Least-squares Finite Element Methods for the Reduced Wave Equation with Non-reflecting Boundary Conditions in Unbounded Domains”,Comput. Meth. Appl. Mech. Engng.,98, pp. 411–454.

    Article  MATH  MathSciNet  Google Scholar 

  47. D. Givoli (1989), “Finite Element Analysis of Heat Problems in Unbounded Domains”, inNumerical Methods In Thermal Problems, Volume VI, R.W. Lewis and K. Morgan, eds., pp. 1094–1104, Pineridge Press, Swansca, U.K.

    Google Scholar 

  48. D. Goldman and P.E. Barbone (1996), “Dirichlet to Neumann maps for the representation of equipment with weak nonlinearities”, in Proc. 1996 ASME Winter Annual Meeting, Atlanta, GA, USA, Nov. 96, ASME Noise Control and Acoustics Division, 22, 1996, New York, NY, pp. 71–76.

  49. P.E. Barbone (1995), “Equipment Representations for Shock Calculations: Time Domain Dirichlet to Neumann Maps”, Tech. Report No. AM-95-012, Boston University, Boston, MA, April.

    Google Scholar 

  50. D. Givoli (1992), “A Numerical Solution Procedure for Exterior Wave Problems”,Computers and Structures,43, pp. 77–84.

    Article  MATH  Google Scholar 

  51. D. Givoli (1992), “A Spatially Exact Non-Reflecting Boundary Condition for Time Dependent Problems”,Comput. Meth. Appl. Mech. Engng.,95, pp. 97–113.

    Article  MATH  MathSciNet  Google Scholar 

  52. L. Sierevogel (1997),Time-Domain Calculations of Ship Motions, PhD Thesis, Delft Technical University, Delft, The Netherlands.

    Google Scholar 

  53. L.L. Thompson and P.M. Pinsky (1993), “New space-time finite element methods for fluid-structure interaction in exterior domains”, in Proc. of the 1993 ASME Winter Annual Meeting, New Orleans, LA, USA Nov. 1993, Computational Methods for Fluid/Structure Interaction, American Society of Mechanical Engineers, Applied Mechanics Division, AMD 178, ASME, New York, NY, 1993, pp. 101–120.

    Google Scholar 

  54. L.L. Thompson and P.M. Pinsky (1996), “Space-time finite element method for structural acoustics in infinite domains. Part 2: exact time-dependent non-reflecting boundary conditions”,Comput. Meth. Appl. Mech. Engng.,132, pp. 229–258.

    Article  MathSciNet  Google Scholar 

  55. R.E. Kleinman and G.F. Roach (1974), “Boundary Integral Equations for the Three Dimensional Helmholtz Equation”,SIAM Review,16, pp. 214–236.

    Article  MATH  MathSciNet  Google Scholar 

  56. I. Harari and T.J.R. Hughes (1992), “Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains”,Comput. Meth. Appl. Mech. Engng.,97, pp. 103–124.

    Article  MATH  MathSciNet  Google Scholar 

  57. I. Harari and T.J.R. Hughes (1994), “Studies of domain-based formulations for computing exterior problems of acoustics”,Int. J. Num. Meth. Engng.,37, pp. 2935–2950.

    Article  MATH  MathSciNet  Google Scholar 

  58. I. Harari, K. Grosh, T.J.R. Hughes, M. Malhotra, P.M. Pinsky, J.R. Stewart and L.L. Thompson (1996), “Recent Developments in Finite Element Methods for Structural Acoustics”,Archives Comput. Meth. Engng.,3, pp. 131–309.

    Article  MathSciNet  Google Scholar 

  59. I. Harari and Z. Shohet (1998), “On non-reflecting boundary conditions in unbounded elastic solids”,Comput. Meth. Appl. Mech. Engng.,163, pp. 123–139.

    Article  MATH  MathSciNet  Google Scholar 

  60. D.S. Burnett (1994), “A Three-Dimensional Acoustic Infinite Element Based on a Prolate Spheroidal Multipole Expansion”,J. Acoust. Soc. Am.,96, pp. 2798–2816.

    Article  MathSciNet  Google Scholar 

  61. A.S. Deakin and J.R. Dryden (1995), “Numerically Derived Boundary Conditions on Artificial Boundaries”,J. Comput. and. Applied Math.,58, pp. 1–16.

    Article  MATH  MathSciNet  Google Scholar 

  62. D. Givoli, I. Patlashenko and J.B. Keller (1998), “Discrete Dirichlet-to-Neumann Maps for Unbounded Domains”,Comput. Meth. Appl. Mech. Engng.,164, pp. 173–185.

    Article  MATH  MathSciNet  Google Scholar 

  63. R.J. Astley (1996), “FE Mode-Matching Schemes for the Exterior Helmholtz Problem and Their Relationship to the FE-DtN Approach”,Common. Num. Meth. Engng.,12, pp. 257–267.

    Article  MATH  Google Scholar 

  64. A.S. Deakin and H. Rasmussen (1996), “Sparse Boundary Conditions on Artificial Boundaries for Three-Dimensional Potential Problems”,J. Comput. Phys.,129, pp. 111–120.

    Article  MATH  Google Scholar 

  65. A.A. Oberai, M. Malhotra and P.M. Pinsky (1998), “On the Implementation of the DtN Radiation Condition for Iterative Solution of the Helmholtz Equation”,Appl. Num. Math.,27, pp. 443–464.

    Article  MATH  MathSciNet  Google Scholar 

  66. M. Malhotra and P.M. Pinsky, “Parallel Preconditioning Based onh-Hierarchical Finite Elements with Application to Acoustics”,Comput. Meth. Appl. Mech. Engng., submitted.

  67. R.F. Susan-Resiga and H.M. Atassi, “A Domain Decomposition Method for the Exterior Helmholtz Problem”, J. Comput. Phys., to appear.

  68. I. Patlashenko and D. Givoli (1998), “A Numerical Method for Problems in Infinite Strips with Irregularities Extending to Infinity”,Numerical Methods for PDEs,14, pp. 233–249.

    MATH  MathSciNet  Google Scholar 

  69. Hughes, Thomas J.R. (1995), “Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods”,Comput. Meth. Appl. Mech. Engng.,127, pp. 387–401.

    Article  MATH  Google Scholar 

  70. J.R. Stewart and T.J.R. Hughes (1997), “h-Adaptive Finite Element Computation of Time-Harmonic Exterior Acoustics Problems in Two Dimensions”,Comput. Meth. Appl. Mech. Engng.,146, pp. 65–89.

    Article  MATH  MathSciNet  Google Scholar 

  71. I. Harari, P.E. Barbone and J.M. Montgomery (1997), “Finite element formulations for exterior problems: Application to hybrid methods, non-reflecting boundary conditions, and infinite elements”,Int. J. Num. Meth. Engng.,40, pp. 2791–2805.

    Article  MATH  MathSciNet  Google Scholar 

  72. R. A. Uras, C. T. Chang, Y. Chen and W. K. Liu (1997), “Multiresolution Reproducing Kernel Particle Methods in Acoustics”,J. Computational Acoustics,5, pp. 71–94.

    Article  Google Scholar 

  73. W. K. Liu, W. Hao, Y. Chen, S. Jun and J. Gosz (1997), “Multiresolution Reproducing Kernel Particle Methods”,Computational Mechanics,20, pp. 295–309.

    Article  MATH  MathSciNet  Google Scholar 

  74. W. K. Liu, Y. Chen, C. T. Chang and T. Belytschko (1996), “Advances in Multiple Scale Kernel Particle Methods”,Computational Mechanics,18, pp. 73–111.

    Article  MATH  MathSciNet  Google Scholar 

  75. W. K. Liu, Y. Chen, R. A. Uras and C. T. Chang (1996), “Generalized Multiple Scale Reproducing Kernel Particle Methods”,Comput. Meth. Appl. Mech. Engng.,139, pp. 91–158.

    Article  MATH  MathSciNet  Google Scholar 

  76. W. K. Liu, Y. Chen, S. Jun, J. S. Chen, T. Belytschko, C. Pan, R. A. Uras and C. T. Chang (1996), “Overview and Applications of the Reproducing Kernel Particle Methods”,Archives Comput. Meth. Engng.,3, pp. 3–80.

    MathSciNet  Google Scholar 

  77. B. Engquist and A. Majda (1981), “Numerical Radiation Boundary Conditions for Unsteady Transonic Flow”,J. Comp. Phys.,40, pp. 91–103.

    Article  MATH  MathSciNet  Google Scholar 

  78. Thomas and M.D. Salas (1986), “Far Field Boundary Conditions for Transonic Lifting Solutions to the Euler Equations”,AIAA J.,24, pp. 1074–1080.

    Article  Google Scholar 

  79. T. Hagstrom and S.I. Hariharan (1988), “Accurate Boundary Conditions for Exterior Problems in Gas Dynamics”,Math. Comput.,51, pp. 581–97.

    Article  MATH  MathSciNet  Google Scholar 

  80. W.-S. Don and D. Gottlieb (1990), “Spectral Simulations of an Unsteady Compressible Flow Past a Circular Cylinder”,Comp. Meth. Appl. Mech. Engng.,80, pp. 39–58.

    Article  MATH  MathSciNet  Google Scholar 

  81. S. Abarbanel, W.-S. Don, D. Gottlieb, D.H. Rudy and J.C. Townsend (1991), “Secondary Frequencies in the Wake of a Circular Cylinder with Vortex Shedding”,J. Fluid Mech.,225, pp. 557–574.

    Article  MATH  Google Scholar 

  82. A. Verhoff and D. Stookesberry (1992), “Second-Order Far-Field Computational Boundary Conditions for Inviscid Duct Flow Problems”,AIAA J.,30, pp. 1268–1276.

    MATH  Google Scholar 

  83. T. Hagstrom (1991), “Conditions at the Downstream Boundary for Simulations of Viscous, Incompressible Flow”,SIAM J. Sci. Stat. Comp.,12, pp. 843–858.

    Article  MATH  MathSciNet  Google Scholar 

  84. R.A. Pearson (1974), “Consistent Boundary Conditions for the Numerical Models of Systems That Admit Dispersive Waves”,J. Atmos. Sci.,31, pp. 1418–1489.

    Article  Google Scholar 

  85. L. Halpern and M. Schatzman (1989), “Artificial Boundary Conditions for Viscous Incompressible Flows”,SIAM J. Math. Anal.,20, pp. 308–353.

    Article  MATH  MathSciNet  Google Scholar 

  86. G. Jin and M. Braza (1993), “A Nonreflecting Outlet Boundary Condition for Incompressible Unsteady Navier-Stokes Calculations”,J. Comput. Phys.,107, pp. 239–253.

    Article  MATH  Google Scholar 

  87. J.S. Danowitz, S.A. Abarbanel and E. Turkel (1995), “A Far-Field Non-Reflecting Boundary Condition for Two-Dimensional Wake Flows”, ICASE Rep. 95-63, NASA Langley, Hampton, VA

    Google Scholar 

  88. G.W. Hedstrom (1979), “Nonreflecting Boundary Conditions for Nonlinear Hyperbolic Systems”,J. Comp. Phys.,30, pp. 222–237.

    Article  MATH  MathSciNet  Google Scholar 

  89. K.W. Thompson (1987), “Time Dependent Boundary Conditions for Hyperbolic Systems”,J. Comp. Phys.,68, pp. 1–24.

    Article  MATH  Google Scholar 

  90. B. Gustafsson (1988), “Far-field Boundary Conditions for Time-Dependent Hyperbolic Systems”,SIAM J. Sci. Stat. Comput.,9, pp. 812–828.

    Article  MATH  MathSciNet  Google Scholar 

  91. K.W. Thompson (1990), “Time Dependent Boundary Conditions for Hyperbolic Systems, II”,J. Comp. Phys.,89, pp. 439–461.

    Article  MATH  Google Scholar 

  92. S. Karmi (1996), “Far-field Filtering Operators for Suppression of Reflections from Artificial Boundaries”,SIAM J. Numer. Anal.,33, pp. 1014–1047.

    Article  MathSciNet  Google Scholar 

  93. D. Givoli and I. Patlashenko (1998), “Finite Element Schemes for Nonlinear Problems in Infinite Domains”,Int. J. Numer. Meth. Engng.,42, pp. 341–360.

    Article  MATH  MathSciNet  Google Scholar 

  94. P.J. Davis and P. Rabinowitz (1984),Methods of Numerical Integration, 2nd ed., Academic Press, New York.

    MATH  Google Scholar 

  95. M. Van Dyke (1975),Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford CA.

    MATH  Google Scholar 

  96. T. Hagstrom (1987), “Boundary Conditions at Outflow for a Problem with Transport and Diffusion”,J. Comp. Phys.,69, pp. 69–80.

    Article  MATH  MathSciNet  Google Scholar 

  97. S. Buonincontri and T. Hagstrom (1989), “Multidimensional Traveling Wave Solutions to Reaction-Diffusion Equations”,IMA J. Appl. Math.,43, pp. 261–271.

    Article  MATH  MathSciNet  Google Scholar 

  98. T. Hagstrom (1991), “Asymptotic Boundary Conditions for Dissipative Waves: General Theory”,Math. Comput.,56, pp. 589–606.

    Article  MATH  MathSciNet  Google Scholar 

  99. C.M. Bender and S.A. Orszag (1978),Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York.

    MATH  Google Scholar 

  100. G.B. Whitham (1974),Linear and Nonlinear Waves, Wiley, New York.

    MATH  Google Scholar 

  101. T.W. Wright, ed. (1986),Nonlinear Wave Propagation in Mechanics, ASME Publications, New York.

    Google Scholar 

  102. L.A. Segel (1987),Mathematics Applied to Continuum Mechanics, Dover, New York.

    Google Scholar 

  103. D. Givoli and I. Patlashenko (1998), “Finite Element Solution of Nonlinear Time-Dependent Exterior Wave Problems”,J. Comput. Phys.,143, pp. 241–258.

    Article  MATH  MathSciNet  Google Scholar 

  104. W.A. Strauss (1989),Nonlinear Wave Equations, American Mathematical Society, Providence, RI.

    MATH  Google Scholar 

  105. R. Courant and D. Hilbert (1962),Methods of Mathematical Physics, Volume II, Wiley, New York.

    MATH  Google Scholar 

  106. D. Givoli and I. Patlashenko (1998), “Optimal Local Non-Reflecting Boundary Conditions”,Applied Numerical Mathematics,27, pp. 367–384.

    Article  MATH  MathSciNet  Google Scholar 

  107. I. Patlashenko and D. Givoli (1996), “Nonlocal and Local Artificial Boundary Conditions for Two-dimensional Flow in an Infinite Channel”,Int. J. Numerical Methods for Heat and Fluid Flow,6, pp. 47–62.

    Article  MATH  Google Scholar 

  108. L. Demkowicz and F. Ihlenburg (1996), “Analysis of Coupled Finite-Infinite Element Method for Exterior Helmholtz Problems”, TICAM Report 96-52, University of Texas, Austin, Nov.

    Google Scholar 

  109. D. Givoli (1997), “A Family of Matrix Problems”, Problems and Solutions,SIAM Review,39, p. 514.

    Google Scholar 

  110. A. Sidi (1998), “A Family of Matrix Problems: Solution”, Problems and Solutions,SIAM Review,40, pp. 719–723.

    Google Scholar 

  111. D. Givoli, I. Patlashenko and A. Sidi (1998), “A Hierarchy of Non-Reflecting Boundary Conditions and Finite Elements”, Proc. 4th Int. Conf. on Spectral and High Order Methods (ICOSA-HOM 98), Tel Aviv, Israel, June.

  112. A. Sidi and D. Givoli, “Stability and Accuracy of Optimal Local Non-Reflecting Boundary Conditions”,Appl. Num. Math., submitted.

  113. G. Strang and G. J. Fix (1973),An Analysis of the Finite Element Method, Prentice-Hall, New Jersey.

    MATH  Google Scholar 

  114. D. Givoli and J.B. Keller (1994), “Special Finite Elements for use with High-order Boundary Conditions”,Comput. Meth. Appl. Mech. Engng.,119, pp. 199–213.

    Article  MATH  MathSciNet  Google Scholar 

  115. T.L. Geers (1998), “Computational Methods for Unbounded Domains: Benchmark Problems”, in Proc. IUTAM Symposium on Computational Methods for Unbounded Domains, Boulder, CO, USA, 1997, Kluwer Academic Publishers, to appear.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Givoli, D. Recent advances in the DtN FE Method. ARCO 6, 71–116 (1999). https://doi.org/10.1007/BF02736182

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02736182

Keywords

Navigation