In this paper, we present a pseudospectral scheme for solving 2D elastic wave equations. We start by analyzing boundary operators leading to the well-posedness of the problem. In addition, equivalent characteristic boundary conditions of common physical boundary conditions are discussed. These theoretical results are further employed to construct a Legendre pseudospectral penalty scheme based on a tensor product formulation for approximating waves on a general curvilinear quadrilateral domain. A stability analysis of the scheme is conducted for the case where a straight-sided quadrilateral element is used. The analysis shows that, by properly setting the penalty parameters, the scheme is stable at the semi-discrete level. Numerical experiments for testing the performance of the scheme are conducted, and the expected p- and h- convergence patterns are observed. Moreover, the numerical evidences also show that the scheme is time stable, which makes the scheme suitable for long time simulations.

[1] Carcion, J. M. (1994). The wave equation in generalized coordinates. Geo-physics. 59, 1911-1919.
[2] Carcion, J. M. (1996). A 2D Chebyshev di®erential operator for the elastic wave equaton. Comput. Methods Applied. Mech. Engrg., 130, 33-45.
[3] Carpenter, M. H., and Kennedy, C. A. (1994). Fourth order 2N-storage Runge-Kutta scheme. NASA-TM-109112 (NASA Langley Research Center, VA).
[4] Carpenter, M. H., and Gottlieb, D. (1996). Spectral methods on arbitary grids. J. Comput. Phys., 129, 74-86.
[5] Don, W. S., and Gottlieb, D. (1994). The Chebyshev-Legendre method: Implementing Legendre methods on Chebyshev points. SIAM J. Numer. Anal., 31, 1519-1534.
[6] Dumbser, M., and KÄaser M. (2006). An arbitray high-order discontinuous Galerkin method for elatic waves on unstructured meshes II. The three-dimension isotropic case. Geophys. J. Int. 167, 319-336.
[7] Funaro, D., and Gottlieb, D. (1988). A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations. Math. Comp. 51, 599-613.
[8] Funaro, D., and Gottlieb, D. (1991). Convergence results for pseudospectral approximations of hyperbolic systems by a penalty-type boundary treatment. Math. Comp. 57, 585-596.
[9] Gordon, W. J., and Hall, C. A. (1973). Trans¯nite element methods: Blending-function interpolation over arbitrary curved element domains. Numer. Math., 21, 109-129.
[10] Gordon, W. J., and Hall, C. A. (1973). Construction of curvilinear coordinate systems and applications to mesh generations. Int. J. Numer. Meth. Eng., 7, 461-477.
[11] Gottlieb, D., and Hesthaven, J. S. (2001). Spectral methods for hyperbolic problems. J. Comput. Appl. Math., 128, 83-131.
[12] Hesthaven, J. S. (1997). A stable penalty method for the compressible Navier-Stokes equations. II. One-dimensional domain decomposition schemes. SIAM J. Sci. Comp., 18, 658-685.
[13] Hesthaven, J. S. (1999). A stable penalty method for the compressible Navier-Stokes equations. III. Multi dimensional domain decomposition schemes. SIAM J. Sci. Comp., 20, 63-92.
[14] Hesthaven, J. S. (2000). Spectral penalty methods. Appl. Numer. Math., 33, 23-41.
[15] Hesthaven, J. S., and Gottlieb, D. (1996) A stable penalty method for the compressible Navier-Stokes Equations. I. Open boundary conditions. SIAM J. Sci. Comp., 17, 579-612.
[16] Hesthaven, J. S., and D. Gottlieb, D. (1999) Stable spectral methods for
conservation laws on triangles with unstructured grids. Comput. Methods Appl. Mech. Engrg., 175, 361-381.
[17] Hesthaven, J. S., Rasumssen, J. Juul, Bergμe, L. and Wyller J. (1997) Numerical studies of localized wave ¯elds governed by the Raman-extented derivative nonlinear SchrÄodinger equation. J. Phys. A: Math. Gen., 30, 8207-8224.
[18] Hesthaven, J. S., and Teng, C. H. (2000). Stable spectral methods on
tetrahedral elements. SIAM J. Sci. Comput., 21, 2353-2380.
[19] Hesthaven, J. S., and Warburton, T. (2002). Nodal high-order methods on unstructured grids. I. Time-domain solution of Maxwell's equations. J. Comput. Phys., 181, 186-221.
[20] Igel, H. (1999).Wave propagation in three-dimensional spherical sections by the Chebyshev spectral method. J. Comput. Phys., 57, 585-596.
[21] KÄaser, M., and Dumbser, M. (2006). An arbitray high-order discontinuous Galerkin method for elatic waves on unstructured meshes-I. the two-dimensional isotropic case with external source terms. Geophys. J. Int., 166, 855-877.
[22] KÄaser, M., Dumbser, M., Puente, J., and Igel, H. (2007). An arbitray high-order discontinuous Galerkin method for elatic waves on unstructured meshes-III. Viscoelastic attenuation. Geophys. J. Int., 168, 224-242.
[23] Komatitsch, D., and Tromp, J. (1999). Introduction to the spectral element method for 3-D seismic wave propagation. Geophysics. J. Int.,
139, 806-822.
[24] Komatitsch, D., and Tromp, J. (2002). Spectral-element simulations of global siesmic wave propagation{I. Validation. Geophysics. J. Int., 149, 390-412.
[25] Komatitsch, D., and Vilotte, J. P. (1998). The spectral-element method: an e±cients tool to simulate the seismic response of 2D and 3D geological structures. Bull. seism. Soc. Am., 88, 368-392.
[26] Nielsen, S. A., and Hesthaven, J. S. (2002). A multi-domain Chebyshev collocation method for predicting ultrasonic ¯eld parameters in complex material geometries. Ultrasonics, 40, 177-180.
[27] Priolo, E., Carcione, J. M., and Seriani G. (1994) Numerical simulation of interface waves by high-order spectral modeling techniques. J. Acoust. Soc. Am., 95, 681-693.
[28] Zeng, Y. Q., and Liu, Q. H. (2004). A multidomain PSTD method for 3D elastic wave equations. Bull. seism. Soc. Am., 94, 1002-1015.
[29] Achenbach, J. D. (1973). Wave Propagation in Elastic Solids , Amsterdam: North-Holland Pub. Co.; New York: American Elsevier Pub. Co.
[30] Brekhovskikh, L. M., and Goncharov, V. (1994). Mechanics of Continua and Wave Dynamics, 2nd ed., (Springer series on wave phenoma; 1), Springer-Verlag, New York.
[31] Canuto, C., Hussaini M. Y., Quarteroni A., and Zang T. A. (1986). Spectral Methods in Fluid Dynamis, Springer Seies in Computational Physics, Springer-Verlag, New York.
[32] Funaro, D. (1992). Polynomial Approximation of Di®erential Equations, Springer-Verlag, New York.
[33] Funaro, D. (1993). FORTRAN Routines for Spectral Methods, Report no. 891, I.A.N.-C.N.R., Pavia. Available with software on anoymous ftp server: ian.pv.cnr.it, in the directory /pub/splib.
[34] Funaro, D. (1997). Spectral Elements for Transport-Dominated Equations, Springer-Verlag, New York.