Recently, the use of special local test functions other than polynomials in Discontinuous Galerkin (DG) approaches has attracted a lot of attention and became known as DG-Trefftz methods. In particular, for the 2D Helmholtz equation plane waves have been used in [10] to derive an Interior Penalty (IP) type Plane Wave DG (PWDG) method and to provide an a priori error analysis of its p-version with respect to equidistributed plane wave directions. However, the dependence on the distribution of the plane wave directions has not been studied. In this contribution, we study this dependence by formulating the choice of the directions as an optimal control problem with a tracking type objective functional and the variational formulation of the PWDG method as a constraint. The necessary optimality conditions are derived and numerically solved by a projected gradient method. Numerical results are given which illustrate the benefits of the approach.

中文翻译：

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亥姆霍兹方程平面波不连续伽辽金方法中平面波方向的优化

最近，在不连续伽辽金 (DG) 方法中使用多项式以外的特殊局部测试函数引起了很多关注，并被称为 DG-Trefftz 方法。特别是，对于 2D Helmholtz 方程，平面波已在 [10] 中用于推导内部惩罚 (IP) 类型的平面波 DG (PWDG) 方法，并提供其 p 版本相对于等分布的先验误差分析平面波方向。然而，尚未研究对平面波方向分布的依赖性。在这个贡献中，我们通过将方向的选择表述为具有跟踪类型目标函数的最优控制问题和 PWDG 方法的变分公式作为约束来研究这种依赖性。必要的优化条件是通过投影梯度法推导出和数值求解的。给出的数值结果说明了该方法的好处。