Numerical analysis of a wave equation for lossy media obeying a frequency power law
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We study a wave equation with a nonlocal time fractional damping term that models the effects of acoustic attenuation characterized by a frequency dependence power law. First we prove existence of unique solutions to this equation with particular attention paid to the handling of the fractional derivative. Then we derive an explicit time stepping scheme based on the finite element method in space and a combination of convolution quadrature and second order central differences in time. We conduct a full error analysis of the mixed time discretization and in turn the fully space time discretized scheme. A number of numerical results are presented to support the error analysis for both smooth and nonsmooth solutions.
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