异构高频亥姆霍兹问题的高效 $C^1$ Bernstein 拟-Trefftz 离散化
Efficient $C^1$ Bernstein Quasi-Trefftz Discretization for Heterogeneous High-Frequency Helmholtz Problems
- Texas A&M University（德克萨斯农工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对异构高频亥姆霍兹问题，提出一种基于 Bernstein 拟-Trefftz 的 $C^1$ 离散化方法，通过局部残差约束压缩空间，实现高阶精度与计算成本降低。
AI中文摘要：
我们针对非结构化三角网格上的异构高频亥姆霍兹问题，开发了一种高效的 $C^1$ Bernstein 拟-Trefftz 离散化方法。该方法首先通过 Bernstein--Bézier 光滑性关系在局部宏块上施加强 $C^1$ 连续性，然后通过强制执行投影亥姆霍兹残差约束来缩减所得的一致性多项式空间。在两三角形块上，在自然秩条件下，这将局部维度从多项式次数的二次增长压缩为大小为 $2p+1$ 的迹规模空间。变系数矩阵值系数通过局部多项式投影处理，使得系数逼近效应与离散误差分离。图残差公式耦合缩减的块空间，并在压缩坐标中产生稀疏全局系统。实现结合了显式 $C^1$ 延拓、批量系数投影、批量残差构建以及通过 QR 分解的稳定局部核提取。在异构非结构化网格上的数值实验证明了高阶精度、舍入级 $C^1$ 一致性、鲁棒的高频分辨率以及局部和全局计算成本的大幅降低。转折点和可穿透散射示例进一步说明了该方法的灵活性。
英文摘要：
We develop an efficient $C^1$ Bernstein quasi-Trefftz discretization for heterogeneous high-frequency Helmholtz problems on unstructured triangular meshes. The method first imposes strong $C^1$ continuity through Bernstein--Bézier smoothness relations on local macro-patches and then reduces the resulting conforming polynomial space by enforcing projected Helmholtz residual constraints. On two-triangle patches, this compresses the local dimension from quadratic growth in the polynomial degree to a trace-sized space of dimension $2p+1$ under the natural rank condition. Variable matrix-valued coefficients are handled through local polynomial projection, allowing coefficient-approximation effects to be separated from discretization error. A graph-residual formulation couples the reduced patch spaces and produces a sparse global system in compressed coordinates. The implementation combines explicit $C^1$ continuation, batched coefficient projection, batched residual construction, and stable local kernel extraction by QR factorization. Numerical experiments on heterogeneous unstructured meshes demonstrate high-order accuracy, roundoff-level $C^1$ conformity, robust high-frequency resolution, and substantial reductions in local and global computational cost. Turning-point and penetrable-scattering examples further illustrate the flexibility of the approach.
