用于亥姆霍兹方程的方向自适应平面波间断伽略金方法
Direction-Adaptive Plane-Wave Discontinuous Galerkin Methods for the Helmholtz Equation
中文总结 AI 辅助
该研究提出方向自适应平面波间断伽略金方法，通过最小化残差选择传播方向，经数值实验验证其可准确恢复亥姆霍兹方程解的主导相位方向，对低方向复杂度解效果最优。
中文摘要 AI 辅助
我们考虑带有自适应局部传播方向的亥姆霍兹方程的平面波间断伽略金（PWDG）近似，这些方向通过最小化网格骨架上的加权残差来选择。我们研究两种格式：在A部分中，针对固定方向求解PWDG系统的系数；而在B部分中，通过变量投影消去系数后，联合最小化残差关于系数和方向的取值。复角将传播型和倏逝型Trefftz波统一起来。离散系统在Trefftz-DG范数下归一化，且使用局部柯西迹格拉姆矩阵去除数值相关的方向。在直边上，残差积分是精确的。对于全狄利克雷问题，残差等于DG误差的平方，并提供局部自适应指标。我们证明了降阶方向泛函在固定秩的可识别零残差解附近呈局部二次增长。数值实验中，精确的圆形DtN测试表明，即使少量平面波扇区会产生更大的场误差，仍可准确恢复汉克尔波的主导相位方向。我们还分离了基函数、迹截断和迹谱算术对高p阶DtN基底的影响。最后，对于平面波的有限和，基于残差的ENRICH--MOVE延拓可在M=1到19时将所有方向恢复至舍入精度；当M=20时，自动生成步骤会陷入虚假盆地，而附近的生成步骤可再次达到舍入精度。结果表明，方向自适应对低方向复杂度的解最为有效。
英文摘要
We consider plane-wave discontinuous Galerkin (PWDG) approximations of the Helmholtz equation with adaptive local propagation directions. The directions are chosen by minimizing a weighted residual on the mesh skeleton. We study two formulations: in Part A the PWDG system is solved for the coefficients at fixed directions, while in Part B the same residual is minimized jointly over coefficients and directions, with the coefficients eliminated by variable projection. Complex angles unify propagating and evanescent Trefftz waves. The discrete system is normalized in the Trefftz-DG norm, and a local Cauchy-trace Gramian is used to remove numerically dependent directions. On straight edges the residual integrals are exact. For all-Dirichlet problems, the residual equals the squared DG error and provides local adaptive indicators. We prove local quadratic growth of the reduced direction functional near an identifiable zero-residual solution of fixed rank. Numerically, an exact circular DtN test shows that the dominant phase direction of a Hankel wave can be recovered accurately even when a small plane-wave fan gives a larger field error. We also separate the effects of the coefficient basis, trace cutoff, and trace-spectrum arithmetic on the high-$p$ DtN floor. Finally, for finite sums of plane waves, residual-based ENRICH--MOVE continuation recovers all directions to roundoff for $M=1,\ldots,19$. At $M=20$ the automatic birth step enters a false basin, whereas a nearby birth again reaches roundoff. The results indicate that direction adaptation is most effective for solutions of low directional complexity.
