We present a class of hybridizable discontinuous Galerkin (HDG) methods for the numerical simulation of wave phenomena in acoustics and elastodynamics. The methods are fully implicit and high-order accurate in both space and time, yet computationally attractive owing to their following distinctive features. First, they reduce the globally coupled unknowns to the approximate trace of the velocity, which is defined on the element faces and single-valued, thereby leading to a significant saving in the computational cost. In addition, all the approximate variables (including the approximate velocity and gradient) converge with the optimal order of k+1 in the L2-norm, when polynomials of degree k≥0 are used to represent the numerical solution and when the time-stepping method is accurate with order k+1. When the time-stepping method is of order k+2, superconvergence properties allows us, by means of local postprocessing, to obtain better, yet inexpensive approximations of the displacement and velocity at any time levels for which an enhanced accuracy is required. In particular, the new approximations converge with order k+2 in the L2-norm when k≥1 for both acoustics and elastodynamics. Extensive numerical results are provided to illustrate these distinctive features.
Bibliographical noteFunding Information:
J. Peraire and N.C. Nguyen would like to acknowledge the Singapore–MIT Alliance and the Air Force Office of Scientific Research under the MURI project on Biologically Inspired Flight and the AFOSR Grant No. FA9550-08-1-0350 for partially supporting this work. B. Cockburn would like to acknowledge the National Science Foundation for partially supporting this work through Grant No. DMS-0712955 .
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- Discontinuous Galerkin methods
- Finite element method
- Hybrid/mixed methods