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An unconditionally stable discontinuous Galerkin method for the elastic Helmholtz equations with large frequency

In this paper we propose and analyze an interior penalty discontinuous Galerkin (IP-DG) method using piecewise linear polynomials for the elastic Helmholtz equations with the first order absorbing boundary condition. It is proved that the sesquilinear form for the problem satisfies a generalized weak coercivity property, which immediately infers a stability estimate for the solution of the differential problem in all frequency regimes. It is also proved that the proposed IP-DG method is unconditionally stable with respect to both frequency $ω$ and mesh size $h$. Sub-optimal order (with respect to $h$) error estimates in the broken $H^1$-norm and in the $L^2$-norm are obtained in all mesh regimes. These estimate improve to optimal order when the mesh size $h$ is restricted to the pre-asymptotic regime (i.e., $ω^βh =O(1)$ for some $1\leq β<2$). The novelties of the proposed IP-DG method include: first, the method penalizes not only the jumps of the function values across the element edges but also the jumps of the normal derivatives; second, the penalty parameters are taken as complex numbers with positive imaginary parts. In order to establish the desired unconditional stability estimate for the numerical solution, the main idea is to exploit a (simple) property of linear functions to overcome the main difficulty caused by non-Hermitian nature and strong indefiniteness of the Helmholtz-type problem. The error estimate is then derived using a nonstandard technique adapted from \cite{Feng_Wu09}. Numerical experiments are also presented to validate the theoretical results and to numerically examine the pollution effect (with respect to $ω$) in the error bounds.