Mixed Finite Element Methods for Acoustic Waves

We are interested in studying the use of both mixed finite element and discontinuous Galerkin methods for modeling both acoustic and elastic waves. There is significant interest in simulating the effects of wave propagation in heterogenous media to aid in the interpretation of field data and to predict the damage patterns due to earthquakes.

Simulated waveform data (seismograms) computed for an assumed earth model are compared against the recorded data. If the match is unacceptable, the model is perturbed, the simulation is redone and compared again. This procedure is implemented formally by global optimization techniques resulting in a description of an earth model (with its associated uncertainties) that explains the observations. Thus there is a need for a fast and accurate simulation technique that can be used for real time analysis of seismograms.

Associated References:

• Lawrence C. Cowsar and Todd F. Dupont and Mary F. Wheeler, A Priori Estimates for Mixed Finite Element Methods for the Wave Equation, Computer Methods in Applied Mechanics and Engineering, pp. 205-222, 1990(82).

• Garth A. Baker, Error Estimates for Finite Element Methods for Second Order Hyperbolic Equations, SIAM Journal on Numerical Analysis, pp. 564-576, 1976(13), no. 4.

• Beatrice M. Riviere, Discontinuous Galerkin Finite Element Methods for Solving the Miscible Displacement Problem in Porous Media, Ph.D. Thesis, University of Texas at Austin, May 2000.

• E. W. Jenkins, B. Riviere, and M. F. Wheeler, A priori error estimates for mixed finite element formulations of the acoustic wave equation , Tech. Rep. TI01-09, Texas Institute for Computational and Applied Mathematics, University of Texas at Austin, March, 2001.