SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3313-A3339, January 2020.
We study for the first time Schwarz domain decomposition methods for the solution of the Navier equations modeling the propagation of elastic waves. These equations in the time-harmonic regime are difficult to solve by iterative methods, even more so than the Helmholtz equation. We first prove that the classical Schwarz method is not convergent when applied to the Navier equations and can thus not be used as an iterative solver, only as a preconditioner for a Krylov method. We then introduce more natural transmission conditions between the subdomains and show that if the overlap is not too small, this new Schwarz method is convergent. We illustrate our results with numerical experiments, both for situations covered by our technical two subdomain analysis and situations that go far beyond, including many subdomains, cross points, heterogeneous materials in a transmission problem, and Krylov acceleration. Our numerical results show that the Schwarz method with adapted transmission conditions leads systematically to a better solver for the Navier equations than the classical Schwarz method.

中文翻译：

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时域弹性波解的自然域分解算法

SIAM科学计算杂志，第42卷，第5期，第A3313-A3339页，2020年1月。
我们首次研究了Schwarz域分解方法，以求解模拟弹性波传播的Navier方程。在时谐状态下，这些方程很难用迭代方法求解，甚至比亥姆霍兹方程还难。我们首先证明经典的Schwarz方法在应用于Navier方程时不会收敛，因此不能用作迭代求解器，而只能用作Krylov方法的前提。然后，我们在子域之间引入更自然的传输条件，并表明如果重叠不是太小，则此新的Schwarz方法是收敛的。我们通过数值实验说明了我们的结果，无论是针对技术性两个子域分析所涵盖的情况，还是远远超出了其他情况（包括许多子域，交叉点，传输问题中的非均质材料和Krylov加速度。我们的数值结果表明，与经典Schwarz方法相比，具有适应传输条件的Schwarz方法系统地导致了Navier方程的更好求解器。