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

## Jamie Bramwell, University of Texas at Austin A Discontinuous Petrov-Galerkin Method for Elastic Wave Propagation

### Authors

• Jamie Bramwell
• Leszek Demkowicz
• Jay Gopalakrishnan
• Weifeng Qiu

### Abstract

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 specific 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, .
2. J. Bramwell, L. Demkowicz, J. Gopalakrishnan, and W. Qiu, An hp DPG Method for Linear Elasticity with Symmetric Stresses, in Preparation.