Using HDG+ to Compute Solutions of the 3D Linear Elastic and Poroelastic Wave Equations
Date
2019
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Publisher
University of Delaware
Abstract
We are interested in the numerical simulation of elastic and poroelastic waves in three dimensions on polyhedral domains. First we tackle the frequency-domain case for elasticity, proving that our HDG+ method's solution converges at O(h^{k+2}) to the exact displacement solution and O(h^{k+1}) to the exact stress solution, where k is the polynomial degree used in the approximation and h is the maximum length of an edge of our tetrahedra. Next we show numerical experiments to verify these results. We then extend our results to the time-domain, proving that the system is conservative and showing numerical results that match our predictions. Then we introduce an extended method by adding a third variable corresponding to the strain, and show numerical results that match our predictions. We next go on to explore HDG+ for Biot's poroelastic system in 3D, proving dissipativity of our method and showing numerical results of the same convergence rates as well as O(h^{k+2}) for pressure and O(h^{k+1}) for pressure flux in both the frequency domain and the time-domain.