Jamie Bramwell, University of Texas at Austin

KAUST-IAMCS Workshop on Multiscale Modeling, Advanced Discretization Techniques, and Simulation of Wave Propagation

May 7-8, 2011

King Abdullah University of Science and Technology (KAUST)

Thuwal, Kingdom of Saudi Arabia

A Discontinuous Petrov-Galerkin Method for Elastic Wave Propagation

Authors: Jamie Bramwell, Leszek Demkowicz, Jay Gopalakrishnan, and Weifeng Qiu

In previous research, a Discontinuous Petrov-Galerkin (DPG) finite element method with optimal test functions for static linear elasticity was introduced. A key result was the proof of quasi-optimality of the method without the use of the exact sequence. This implies that both h and p convergence can occur without the use of element spaces which conform to an exact sequence. From this work, I will present an outline of the quasi-optimality proof as well as make comparisons between our methods and the mixed method of Arnold, Falk, and Winther.

Additionally, the two methods can be extended to time-harmonic elastic wave propagation problems. A key feature of the DPG method is the reduction of pollution error, and can therefore be used to solve problems with a large number of wavelengths. Due to this fact, I will present numerical results for elastic wave scattering problems with high wave numbers. Also, since the DPG method is equipped with an a priori error estimator, I will present results from various 'greedy' adaptive schemes.

The principal contribution of this research is a practical adaptive 2D time-harmonic elasticity finite element code with a priori error estimation that can be used for high wave number problems. A particular application where this method could be applied is large-scale seismic wave propagation. In this poster, I will present an overview of the theoretical DPG framework, the specic formulation for both static and time-harmonic elasticity, and the numerical results for both cases.

References:

1. J. Bramwell, L. Demkowicz, and W. Qiu. Solution of Dual-Mixed Elasticity Equations using Arnold-Falk-Winther Element and Discontinuous Petrov-Galerkin Method, a Comparison. Technical Report 10-23, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, June 2010.
2. J. Bramwell, L. Demkowicz, J. Gopalakrishnan, and W. Qiu. An hp DPG Method for Linear Elasticity with Symmetric Stresses, in preparation.