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Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation. (English) Zbl 1102.76032

Summary: Discontinuous Galerkin finite element methods (DGFEM) offer certain advantages over standard continuous finite element methods when applied to the spatial discretisation of the acoustic wave equation. For instance, the mass matrix has a block diagonal structure which, used in conjunction with an explicit time stepping scheme, gives an extremely economical scheme for time domain simulation. This feature is ubiquitous and extends to other time-dependent wave problems such as Maxwell’s equations.
An important consideration in computational wave propagation is the dispersive and dissipative properties of the discretisation scheme in comparison with those of the original system. We investigate these properties for two popular DGFEM schemes: the interior penalty discontinuous Galerkin finite element method applied to the second-order wave equation, and a more general family of schemes applied to the corresponding first-order system. We show how the analysis of the multi-dimensional case may be reduced to consideration of one-dimensional problems. We derive the dispersion error for various schemes and conjecture on the generalisation to higher-order approximation in space

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76Q05 Hydro- and aero-acoustics
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[1] Ainsworth, M., Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J, Numer. Anal., 42, 2, 553-575 (2004) · Zbl 1074.65112
[2] Ainsworth, M., Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comp, Phys., 198, 106-130 (2004) · Zbl 1058.65103
[3] Arnold, D. N., An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19, 742-760 (1982) · Zbl 0482.65060
[4] Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 1749-1779 (2002) · Zbl 1008.65080
[5] Baker, G. A., Finite element methods for elliptic equations using non-conforming elements, Math. Comput., 31, 45-59 (1977) · Zbl 0364.65085
[6] Cockburn, B., Discontinuous Galerkin methods, ZAMM Z. Angew. Math. Mech., 83, 731-754 (2003) · Zbl 1036.65079
[7] Cockburn, B., Karniadakis, G., and Shu, C. W. (eds.), (2000). First International Symposium on Discontinuous Galerkin methods, volume 11 of Lecture notes in Computational Science and Engineering. Springer.
[8] Cockburn, B.; Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35, 2440-2463 (1998) · Zbl 0927.65118
[9] Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. SIAM J, Sci. Comput., 16, 193-261 (2001) · Zbl 1065.76135
[10] Cohen, G. C., Higher-order Numerical Methods for Transient Wave Equations (2002), Berlin: Springer, Berlin · Zbl 0985.65096
[11] Douglas J., and Dupont T. (1976). Interior Penalty Procedures for Elliptic and Hyperbolic Discontinuous Galerkin Procedures, volume 58 of Lecture notes in physics. Springer-Verlag.
[12] Hesthaven, J. S.; Warburton, T., Nodal high-order methods on unstructured grids - I. Time-domain solution of Maxwell’s equations, J. Comput. Phys., 181, 186-221 (2002) · Zbl 1014.78016
[13] Hu, F. Q.; Atkins, H. L., Eigensolution analysis of the discontinuous Galerkin method with non-uniform grids, Part, 1, One space dimension-J Comput Phys 1822516545 (2002)
[14] Hu, F. Q., and Atkins, H. L. (2002). Two-dimensional wave analysis of the discontinuous Galerkin method with non-uniform grids and boundary conditions. In Proc. 8th AIAA/CEAS Aeronautics Conference, Breckenridge, Colorado, June 2002. AIAA paper no. 2002-2514
[15] Odeh, F.; Keller, J. B., Partial differential equations with periodic coefficients and Bloch waves in crystals. J. Math, Phys., 5, 1499-504 (1964) · Zbl 0129.46004
[16] Reed, W. H.; Hill, T. R., Triangular mesh methods for the neutron transport equation (1973), Los Alamos, New Mexico, USA: Los Alamos National Laboratory, Los Alamos, New Mexico, USA
[17] Riviere, B., and Wheeler, M. (2003). Discontinuous finite element methods for acoustic and elastic wave problem. In ICM2002-Beijing Satellite Conference on Scientific Computing, volume 329 of Contemporary Mathematics, AMS, pp. 271-282 · Zbl 1080.76039
[18] Vichnevetsky, R., and Bowles, J. B. (1982). Fourier Analysis of Numerical Approximations of Hyperbolic Equations, volume 5 of CMBS Regional Conference Series in Applied Mathematics. SIAM, Philadelphia. · Zbl 0495.65041
[19] Vila, J.-P.; Villedieu, P., Convergence of an explicit finite volume scheme for first order symmetric systems. Numer, Math., 94, 573-602 (2003) · Zbl 1030.65110
[20] Wheeler, M. F., An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal., 15, 152-161 (1978) · Zbl 0384.65058
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