SUMMARY We present a new approach to the solution of the Poisson equation present in the coupled gravito-elastic equations of motion for global seismic wave propagation in time domain aiming at the inclusion of the full gravitational response into spectral element solvers. We leverage the Salvus meshing software to include the external domain using adaptive mesh refinement and high order shape mapping. Together with Neumann boundary conditions based on a multipole expansion of the right-hand side this minimizes the number of additional elements needed. Initial conditions for the iterative solution of the Poisson equation based on temporal extrapolation from previous time steps together with a polynomial multigrid method reduce the number of iterations needed for convergence. In summary, this approach reduces the extra cost for simulating full gravity to a similar order as the elastic forces. We demonstrate the efficacy of the proposed method using the displacement from an elastic global wave propagation simulation (decoupled from the Poisson equation) at $200\, \mbox{s}$ dominant period to compute a realistic right-hand side for the Poisson equation.

中文翻译：

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全三维全球地震波传播的自引力模型

总结我们提出了一种求解泊松方程的新方法，该方程存在于时域中全球地震波传播的耦合重力-弹性运动方程中，旨在将全重力响应包含在谱元求解器中。我们利用 Salvus 网格划分软件使用自适应网格细化和高阶形状映射来包含外部域。再加上基于右手边多极展开的 Neumann 边界条件，可以最大限度地减少所需的附加元素数量。Poisson 方程的迭代解的初始条件基于从先前时间步长的时间外推以及多项式多重网格方法减少了收敛所需的迭代次数。总之，这种方法将模拟完全重力的额外成本降低到与弹性力相似的顺序。我们使用来自弹性全局波传播模拟（与泊松方程解耦）在 $200\,\mbox{s}$ 主导周期的位移来计算泊松方程的现实右侧，证明了所提出方法的有效性。