This paper proposes an $O(N)$ fast direct solver for two-dimensional elastic wave scattering problems. The proxy surface method is extended to elastodynamics to obtain shared coefficients for low-rank approximations from discretized integral operators. The proposed method is a variant of the Martinsson-Rokhlin-type fast direct solver. Our variant avoids the explicit computation of the inverse of the coefficient matrix, thereby reducing the required number of matrix-matrix multiplications. Numerical experiments demonstrate that the proposed solver has a complexity of $O(N)$ in the low-frequency range and has a highly parallel computation efficiency with a strong scaling efficiency of 70\%. Furthermore, multiple right-hand sides can be solved efficiently; specifically, when solving problems with 180 right-hand side vectors, the processing time per vector from the second vector onward was approximately 28,900 times faster than that for the first vector. This is a key advantage of fast direct methods.

翻译：本文提出了一种针对二维弹性波散射问题的$O(N)$快速直接求解器。通过将代理表面法扩展至弹性动力学领域，可从离散化积分算子中获得用于低秩近似的共享系数。所提出的方法是Martinsson-Rokhlin型快速直接求解器的变体。我们的变体避免显式计算系数矩阵的逆矩阵，从而减少了所需的矩阵-矩阵乘法次数。数值实验表明，该求解器在低频范围内具有$O(N)$的计算复杂度，并具备高度并行计算效率，其强可扩展效率达到70%。此外，该方法能高效求解多右端项问题：具体而言，在求解包含180个右端项向量的问题时，从第二个向量开始每个向量的处理时间比第一个向量快约28,900倍。这是快速直接求解方法的关键优势。