This work presents a high-order Bernstein-Bézier finite element (FE) discretisation to accurately solve time harmonic elastic wave problems on unstructured triangular mesh grids. Although high-order FEs possess many advantages over standard FEs, the computational cost of matrix assembly is a major issue in high-order computations. A key ingredient to address this drawback is to resort to low complexity procedures in building the local high order FE matrices. This is achieved in this work by exploiting the tensorial property of Bernstein polynomials and applying the sum factorisation method for curved elements. An efficient implementation of the analytical rules for affine elements is also proposed. Furthermore, element-level static condensation of the interior degrees of freedom is performed to reduce the memory requirements. Additionally, the applicability of the method with a variable polynomial order, based on a simple a priori indicator, is investigated.

The computational complexities of sum factorisation, analytical rules and standard quadrature are first evaluated, in terms of the CPU time against the polynomial order. The analysis shows that the achieved numerical complexities compare favourably to those expected theoretically. A significant runtime saving is also obtained by using analytical rules and sum factorisation. The performance of the Bernstein-Bézier FEs is then assessed on various benchmark tests, over a wide range of frequencies. Results from the elastic wave scattering problem demonstrate the effectiveness of this method in coping with the pollution error, and its accuracy in resolving high order evanescent wave modes. Additionally, a wave transmission problem with high wave speeds contrast and a curved interior interface is considered, where a simple a priori indicator is proposed to assign the variable polynomial order. The study provides evidence of the great benefit of a non uniform $p$-refinement in reducing the computational cost and enhancing accuracy.

这项工作提出了一种高阶 Bernstein-Bézier 有限元 (FE) 离散化，以准确解决非结构三角形网格上的时间谐波弹性波问题。尽管高阶 FE 比标准 FE 具有许多优势，但矩阵组装的计算成本是高阶计算中的一个主要问题。解决这个缺点的一个关键因素是在构建局部高阶有限元矩阵时采用低复杂度的程序。这是在这项工作中通过利用伯恩斯坦多项式的张量性质并应用曲线元素的总和分解方法来实现的。还提出了仿射元素分析规则的有效实现。此外，执行内部自由度的元素级静态压缩以减少内存需求。此外，

根据多项式阶次的 CPU 时间，首先评估求和因式分解、分析规则和标准求积的计算复杂性。分析表明，所实现的数值复杂性与理论上预期的相比是有利的。通过使用分析规则和求和因式分解，还可以显着节省运行时间。然后在各种基准测试中评估 Bernstein-Bézier FE 的性能，在很宽的频率范围内。弹性波散射问题的结果证明了该方法在处理污染误差方面的有效性及其在解析高阶倏逝波模式方面的准确性。此外，还考虑了具有高波速对比和弯曲内部界面的波传输问题，其中一个简单的建议先验指标来分配可变多项式阶数。该研究提供了非制服的巨大好处的证据$磷$- 在降低计算成本和提高准确性方面的改进。