Abstract
In this work we apply the discontinuous Galekin (dG) spectral element method on meshes made of simplicial elements for the approximation of the elastodynamics equation. Our approach combines the high accuracy of spectral methods, the geometrical flexibility of simplicial elements and the computational efficiency of dG methods. We analyze the dissipation, dispersion and stability properties of the resulting scheme, with a focus on the choice of different sets of basis functions. Finally, we apply the method on benchmark as well as realistic test cases.
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Adams, R.A., Fournier, J.J.F.: Sobolev spaces, volume 140 of pure and applied mathematics, 2nd edn. Elsevier/Academic Press, Amsterdam (2003)
Antonietti, P.F., Ayuso de Dios, B., Mazzieri, I., Quarteroni, A.: Stability analysis for discontinuous Galerkin approximations of the elastodynamics problem. MOX-Report, 56/2013 (2013)
Antonietti, P.F., Houston, P.: A class of domain decomposition preconditioners for hp-discontinuous Galerkin finite element methods. J. Sci. Comput. 46(1), 124–149 (2011)
Antonietti, P.F., Mazzieri, I., Quarteroni, A., Rapetti, F.: Non-conforming high order approximations of the elastodynamics equation. Comput. Methods Appl. Mech. Engrg. 209(/212), 212–238 (2012)
Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)
Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2001/02)
Berg, P., If, F., Nielsen, P., Skovgaard, O.: Analytical reference solutions. Modeling the earth for oil exploration, 421–427 (1994)
Bos, L., Taylor, M.A., Wingate, B.A.: Tensor product Gauss-Lobatto points are Fekete points for the cube. Math. Comp. 70(236), 1543–1547 (2001)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods. Fundamentals in single domains. Scientific Computation. Springer, Berlin (2006)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods. Evolution to complex geometries and applications to fluid dynamics. Scientific Computation. Springer, Berlin (2007)
Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann. 100(1), 32–74 (1928)
De Basabe, J.D., Sen, M.K.: Stability of the high-order finite elements for acoustic or elastic wave propagation with high-order time stepping. Geophys. J. Int. 181(1), 577–590 (2010)
Douglas Jr, J., Dupont, T.: Interior penalty procedures for elliptic and parabolic Galerkin methods. In: Computing methods in applied sciences (Second Internat. Sympos., Versailles, 1975), pp. 207–216. Lecture Notes in Phys., Vol. 58. Springer, Berlin (1976)
Dubiner, M.: Spectral methods on triangles and other domains. J. Sci. Comput. 6(4), 345–390 (1991)
Dunavant, D.A.: High degree efficient symmetrical Gaussian quadrature rules for the triangle. Internat. J. Numer. Methods Engrg. 21(6), 1129–1148 (1985)
Faccioli, E., Maggio, F., Paolucci, R., Quarteroni, A.: 2d and 3d elastic wave propagation by a pseudo-spectral domain decomposition method. J. Seismol. 1(3), 237–251 (1997)
Gassner, G.J., Lörcher, F., Munz, C.-D., Hesthaven, J.S.: Polymorphic nodal elements and their application in discontinuous Galerkin methods. J. Comput. Phys. 228(5), 1573–1590 (2009)
Georgoulis, E.H., Hall, E., Houston, P.: Discontinuous Galerkin methods on hp-anisotropic meshes. I, A priori error analysis. Int. J. Comput. Sci. Math. 1(2–4), 221–244 (2007)
Georgoulis, E.H., Hall, E., Houston, P.: Discontinuous Galerkin methods for advection-diffusion-reaction problems on anisotropically refined meshes. SIAM J. Sci. Comput. 30(1), 246–271 (2007/08)
Harriman, K., Houston, P., Senior, B., Süli, E.: hp-version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form. In: Recent advances in scientific computing and partial differential equations (Hong Kong, 2002), volume 330 of Contemp. Math. 89–119. Amer. Math. Soc., Providence, RI (2003)
Hesthaven, J.S.: From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J. Numer. Anal. 35(2), 655–676 (1998)
Hesthaven, J.S., Teng, C.-H.: Stable spectral methods on tetrahedral elements. SIAM J. Sci. Comput. 21(6), 2352–2380 (2000)
Hesthaven, J.S., Warburton, T.: Nodal discontinuous Galerkin methods, volume 54 of Texts in Applied Mathematics. Springer, New York (2008)
Karniadakis, G.E., Sherwin, S.J.: Spectral/hp element methods for computational fluid dynamics. Numerical Mathematics and Scientific Computation, 2nd edn. Oxford University Press, New York (2005)
Klöckner, A., Warburton, T., Bridge, J., Hesthaven, J.S.: Nodal discontinuous Galerkin methods on graphics processors. J. Comput. Phys. 228(21), 7863–7882 (2009)
Komatitsch, D., Tromp, J.: Spectral-element simulations of global seismic wave propagation-i. validation. Geophys. J. Int. 149(2), 390–412 (2002)
Koornwinder, T.: Two-variable analogues of the classical orthogonal polynomials. In: Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), 435–495. Math. Res. Center, Univ. Wisconsin, Publ. No. 35. Academic Press, New York (1975)
Lamb, H.: On the propagation of tremors over the surface of an elastic solid. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character 203, 1–42 (1904)
Lyness, J.N., Jespersen, D.: Moderate degree symmetric quadrature rules for the triangle. J. Inst. Math. Appl. 15, 19–32 (1975)
Mazzieri, I.: Non-conforming high order methods for the elastodynamics equation. PhD thesis, Politecnico di Milano (2012)
Mazzieri, I., Rapetti, F.: Dispersion analysis of triangle-based spectral element methods for elastic wave propagation. Numer. Algorithms 60(4), 631–650 (2012)
Mercerat, E.D., Glinsky, N.: A nodal high-order discontinuous Galerkin method for elastic wave propagation in arbitrary heterogeneous media. Geophys. J. Int. 201 (2), 1101–1118 (2015)
Mercerat, E.D., Vilotte, J.-P., Sánchez-Sesma, F.J.: Triangular spectral element simulation of two-dimensional elastic wave propagation using unstructured triangular grids. Geophys. J. Int. 166(2), 679–698 (2006)
Pasquetti, R., Rapetti, F.: Spectral element methods on unstructured meshes: which interpolation points? Numer. Algorithms 55(2–3), 349–366 (2010)
Patera, A.T.: A spectral element method for fluid dynamics: Laminar flow in a channel expansion. J. Comput. Phys. 54(3), 468–488 (1984)
Pelties, C., de la Puente, J., Ampuero, J.-P., Brietzke, G.B., Käser, M.: Three-dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes. J. Geophys. Res. Solid Earth 117(B2) (2012)
Peter, D., Komatitsch, D., Luo, Y., Martin, R., Le Goff, N., Casarotti, E., Le Loher, P., Magnoni, F., Liu, Q., Blitz, C., Nissen-Meyer, T., Basini, P., Tromp, J.: Forward and adjoint simulations of seismic wave propagation on fully unstructured hexahedral meshes. Geophys. J. Int. 186(2), 721–739 (2011)
Quarteroni, A., Sacco, R., Saleri, F.: Numerical mathematics, volume 37 of Texts in Applied Mathematics. Springer, Berlin, second edition (2007)
Rapetti, F., Sommariva, A., Vianello, M.: On the generation of symmetric Lebesgue-like points in the triangle. J. Comput. Appl. Math. 236(18), 4925–4932 (2012)
Raviart, P.-A., Thomas, J.-M.: Introduction à l’analyse numérique des équations aux dérivées partielles. Collection Mathématiques Appliquées pour la Maî trise. Masson, Paris (1983)
Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory (1973)
Rivière, B., Wheeler, M.F.: Discontinuous finite element methods for acoustic and elastic wave problems. In Current trends in scientific computing (Xi’an, 2002), volume 329 of Contemp. Math., 271–282. Amer. Math. Soc., Providence, RI, 2003
Rivière, B., Wheeler, M.F., Girault, V.: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. I. Comput. Geosci. 3(3–4), 1999 (2000)
Ryland, B.N., Munthe-Kaas, H.Z.: On multivariate Chebyshev polynomials and spectral approximations on triangles. In: Spectral and High Order Methods for Partial Differential Equations, pp. 19–41. Springer (2011)
Seriani, G., Priolo, E., Pregarz, A.: Modelling waves in anisotropic media by a spectral element method. Proceedings of the third international conference on mathematical and numerical aspects of wave propagation, pp. 289–298 (1995)
Sherwin, S.J., Karniadakis, G.E.: A new triangular and tetrahedral basis for high-order (h p) finite element methods. Internat. J. Numer. Methods Engrg. 38(22), 3775–3802 (1995)
Stupazzini, M., Paolucci, R., Igel, H.: Near-fault earthquake ground-motion simulation in the Grenoble valley by a high-performance spectral element code. Bull. Seismol. Soc. Am. 99(1), 286–301 (2009)
Szego, G.: Orthogonal polynomials. American Mathematical Society, Providence, R.I., fourth edition, 1975. American Mathematical Society Colloquium Publications, Vol. XXIII
Taylor, M.A., Wingate, B.A., Bos, L.P.: A cardinal function algorithm for computing multivariate quadrature points. SIAM J. Numer. Anal. 45(1), 193–205 (2007). electronic
Wandzura, S., Xiao, H.: Symmetric quadrature rules on a triangle. Comput. Math. Appl. 45(12), 1829–1840 (2003)
Warburton, T.: An explicit construction of interpolation nodes on the simplex. J. Engrg. Math. 56(3), 247–262 (2006)
Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15(1), 152–161 (1978)
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Antonietti, P.F., Marcati, C., Mazzieri, I. et al. High order discontinuous Galerkin methods on simplicial elements for the elastodynamics equation. Numer Algor 71, 181–206 (2016). https://doi.org/10.1007/s11075-015-0021-7
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DOI: https://doi.org/10.1007/s11075-015-0021-7