We introduce an arbitrary order, stabilized finite element method for solving a unique continuation problem subject to the time-harmonic elastic wave equation with variable coefficients. Based on conditional stability estimates we prove convergence rates for the proposed method which take into account the noise level and the polynomial degree. A series of numerical experiments corroborates our theoretical results and explores additional aspects, e.g. how the quality of the reconstruction depends on the geometry of the involved domains. We find that certain convexity properties are crucial to obtain a good recovery of the wave displacement outside the data domain and that higher polynomial orders can be more efficient but also more sensitive to the ill-conditioned nature of the problem.

翻译：本文针对变系数时谐弹性波方程的唯一延拓问题，提出了一种任意阶的稳定化有限元方法。基于条件稳定性估计，我们证明了所提方法的收敛速率，该速率同时考虑了噪声水平与多项式阶次。系列数值实验验证了理论结果，并探讨了重建质量与区域几何特征的关联性等附加问题。研究发现：特定凸性性质对波位移在数据域外的重建效果具有关键作用，高阶多项式虽能提升计算效率，但会加剧对问题病态特性的敏感度。