Second-order absorbing boundary conditions for the wave equation in a rectangular domain

Author:
Dongwoo Sheen

Journal:
Math. Comp. **61** (1993), 595-606

MSC:
Primary 65M60

DOI:
https://doi.org/10.1090/S0025-5718-1993-1192975-5

MathSciNet review:
1192975

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study finite element methods for the wave equation in a rectangular domain with a second-order absorbing boundary condition imposed on the boundary. For this problem there seems to be no known finite element method, although many finite difference methods have been proposed. A third-order energy, however, will be introduced which will be utilized to reduce our original second-order problem to a first-order symmetric dissipative hyperbolic system. Then, for this first-order system a weak formulation will be given and continuous-time and discrete-time Galerkin procedures will be investigated. Error estimates will also be given.

**[1]**A. Bamberger, P. Joly, and J. E. Roberts,*Second-order absorbing boundary conditions for the wave equation*:*A solution for the corner problem*, SIAM J. Numer. Anal.**27**(1990), 323-352. MR**1043609 (91b:35066)****[2]**J. F. Claerbout,*Fundamentals of geophysical data processing*, McGraw-Hill, New York, 1976.**[3]**R. Clayton and B. Engquist,*Absorbing boundary conditions for acoustic and elastic wave equations*, Bull. Seismol. Soc. Amer.**67**(1977), 1529-1540.**[4]**S. H. Emerman and R. A. Stephen,*Comment on "Absorbing boundary conditions for acoustic and elastic wave equations" by R. Clayton and B. Engquist*, Bull. Seismol. Soc. Amer.**73**(1983), 661-665.**[5]**B. Engquist and A. Majda,*Absorbing boundary conditions for the numerical simulation of waves*, Math. Comp.**31**(1977), 629-651. MR**0436612 (55:9555)****[6]**-,*Radiation boundary conditions for acoustic and elastic wave calculations*, Comm. Pure Appl. Math.**32**(1979), 313-357. MR**517938 (80e:76041)****[7]**T. HaDuong and P. Joly,*On the stability analysis of boundary conditions for the wave equation by energy methods, Part*I:*The homogeneous case*, Report No. 224, Ecole Polytechnique, Centre de Mathématiques Appliquées, 1990.**[8]**R. L. Higdon, Absorbing boundary conditions for difference approximations to the multidimensional wave equation, Math. Comp.**47**(1986), 437-459. MR**856696 (87m:65131)****[9]**-,*Numerical absorbing boundary conditions for the wave equation*, Math. Comp.**49**(1987), 65-90. MR**890254 (88f:65168)****[10]**R. G. Keys,*Absorbing boundary conditions for acoustic media*, Geophys.**50**(1985), 892-902.**[11]**W. J. Layton,*Stable Galerkin methods for hyperbolic systems*, SIAM J. Numer. Anal.**20**(1983), 221-223. MR**694515 (85c:65120)****[12]**A. Majda and S. Osher,*Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary*, Comm. Pure Appl. Math.**28**(1975), 607-675. MR**0410107 (53:13857)****[13]**J. Rauch,*Symmetric positive systems with boundary characteristic of constant multiplicity*, Trans. Amer. Math. Soc.**291**(1985), 167-187. MR**797053 (87a:35122)****[14]**D. Sheen,*On the wave equation in a half plane with second order absorbing boundary conditions*(to appear).**[15]**B. Swartz and B. Wendroff,*Generalized finite-difference schemes*, Math. Comp.**23**(1969), 37-49. MR**0239768 (39:1125)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65M60

Retrieve articles in all journals with MSC: 65M60

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1993-1192975-5

Keywords:
Finite element method,
absorbing boundary conditions,
higher-order energy

Article copyright:
© Copyright 1993
American Mathematical Society