Pollution free discretization of the wave equation

References

• Adams R. A. (1975) Sobolev Spaces Academic Press New York
   Google Scholar
• Babuška , I. M. , Ihlenburg , F. , Paik , E. T. and Sauter , S. A. (1995) . A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution . Comput. Methods Appl. Mech. Engrg , 128 : 325 – 359 .
   Web of Science ®Google Scholar
• Babuška , I. M. and Melenk , J. M. (1997) . The partition of unity finite element method . Internat. J. Numer. Methods Engrg , 40 : 727 – 758 .
   Web of Science ®Google Scholar
• Babuška , I. M. and Sauter , S. A. (2000) . Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? . SIAM Rev. , 42 ( 3 ) : 451 – 484 .
   Google Scholar
• Bochev , P. and Gunzburger , M. D. (1994) . Analysis of least squares finite element methods for the Stokes equations . Math. Comp. , 63 : 479 – 506 .
   Web of Science ®Google Scholar
• Bochev P. Cai Z. Manteuffel T. A. McCormick S. F. (1995) First-order system least-squares for the Navier-Stokes equations Technical Report University of Colorado at Boulder, Department of Mathematics
   Google Scholar
• Bramble J. H. Lazarov R. D. Pasciak J. E. (1994) A least-squares approach based on a discrete minus one inner product for first order systems Technical Report Brookhaven National Laboratory
   Google Scholar
• Bramble , J. H. and Pasciak , J. E. (1996) . Least-squares methods for Stokes equations based on a discrete minus one inner product . J. Comp. Appl. Math. , 74 : 155 – 173 .
   Web of Science ®Google Scholar
• Cai , Z. , Manteuffel , T. A. and McCormick , S. F. (1994) . First-order system least-squares for second-order partial differential equations: Part I . SIAM J. Numer. Anal. , 31 : 1785 – 1799 .
   Web of Science ®Google Scholar
• Cai Z. Manteuffel T. A. McCormick S. F. (1995) First-order system least squares for the Stokes equations, with applications to linear elasticity Technical Report University of Colorado at Boulder, Department of Mathematics
   Google Scholar
• Cai , Z. , Manteuffel , T. A. and McCormick , S. F. (1997) . First-order system least squares for the second-order partial differential equations: Part II . SIAM J. Numer. Anal. , 34 : 425 – 454 .
   Web of Science ®Google Scholar
• Carey , G. F. , Pehlivanov , A. I. and Vassilevski , P. S. (1995) . Least-squares mixed finite elements for non-selfadjoint elliptic problems: II, performance of block-ILU factorization methods . SIAM J. Sci. Comp. , 16 : 1126 – 1136 .
   Google Scholar
• Chang , C. L. (1992) . Finite element approximation for grad-dic type systems in the plane . SIAM J. Numer. Anal. , 29 : 452 – 461 .
   Web of Science ®Google Scholar
• Colton D. Kress R. (1992) Inverse Acoustic Electromagnetic Scattering Theory Springer-Verlag Berlin
   Google Scholar
• Elschner , J. and Schmidt , G. (1998) . Numerical solution of optimal design problems for binary gratings . J. Comp. Phys. , 146 : 603 – 626 .
   Web of Science ®Google Scholar
• Franca , L. , Farhat , C. and Macedo , A. (1997) . A residual-free bubbles for the Helmholtz equation . Internat. J. Numer. Methods Engrg , 40 : 4003 – 4009 .
   Web of Science ®Google Scholar
• Franca , L. and Macedo , A. (1998) . A two-level finite element method and its application to the Helmholtz equation . Internat. J. Numer. Methods Engrg , 43 : 23 – 32 .
   Google Scholar
• Hackbusch W. (1995) Integral Equations. Internat. Ser. Numer. Math. Birkhäuser Basel
   Google Scholar
• Harari , I. and Hughes , T. J. R. (1991) . Finite element methods for Helmholtz equation in an exterior domain: Model problems . Comput. Methods Appl. Mech. Engrg , 87 : 59 – 96 .
   Web of Science ®Google Scholar
• Harari , I. and Huges , T. J. R. (1992) . Galerkin/least-squares finite element methods for the reduced wave equation with non-reflecting boundary conditions in unbounded domains . Comput. Methods Appl. Mech. Engrg , 98 : 411 – 454 .
   Google Scholar
• Jackson J. D. (1975) Classical Electrodynamics John Wiley & Sons New York
   Google Scholar
• Ihlenburg F. (1998) Finite Element Analysis of Acoustic Scattering Springer-Verlag New York
   Google Scholar
• Keller , J. B. and Givoli , D. (1989) . Exact non-reflecting boundary conditions . J. Comp. Phys. , 82 : 172 – 192 .
   Web of Science ®Google Scholar
• Melenk J. M. (1995) On generalized finite element methods PhD thesis University of Maryland College Park MD
   Google Scholar
• Nédélec J. C. (2001) Acoustic Electromagnetic Equations Springer New York
   Google Scholar
• Oberai , A. A. and Pinsky , P. M. (1998) . A multiscale finite element method for the Helmholtz equation . Comput. Methods Appl. Mech. Engrg , 154 : 281 – 297 .
   Google Scholar
• Pehlivanov , A. I. , Carey , G. F. and Lazarov , R. D. (1994) . Least-squares mixed finite elements for second-order elliptic problems . SIAM J. Numer. Anal. , 31 : 1368 – 1377 .
   Web of Science ®Google Scholar
• Suleau , S. and Bouillard , P. (2000) . One dimensional dispersion analysis for the element-free Galerkin method for the Helmholtz equation . Internat. J. Numer. Methods Engrg , 47 : 1169 – 1188 .
   Google Scholar
• Thompson , L. L. and Pinsky , P. M. (1995) . A Galerkin least squares finite element method for the two-dimensional Helmholtz equation . Internat. J. Numer. Methods Engrg , 38 : 371 – 398 .
   Google Scholar