This work is concerned with the construction and the hp non-conforming a priori error analysis of a Discontinuous Galerkin DG numerical scheme applied to the hypersingular integral equation related to the Helmholtz problem in 3D. The main results of this article are an error bound in a norm suited to the problem and in the $L^2$-norm. Those bounds are quasi-optimal for the $h$-convergence and the $p$-convergence. Various formulation choices and penalty functions are theoretically discussed. In particular we show that a penalty function of the shape $\frac{h^2}{p}$ leads to a quasi-optimal convergence of the scheme. Some numerical experiments confirm the expected rates of convergence and the effect of the penalty function.

这项工作涉及不连续的Galerkin DG数值方案的构造和hp不合格先验误差分析，该数值方案应用于与3D中的亥姆霍兹问题有关的超奇异积分方程。本文的主要结果是在适用于该问题的规范中以及在${\mathrm{大号}}^2$-规范。这些边界对于$H$-收敛和 $p$-收敛。理论上讨论了各种公式选择和惩罚函数。特别地，我们表明形状的惩罚函数$\frac{H^2}{p}$导致该方案的准最优收敛。一些数值实验证实了预期的收敛速度和惩罚函数的效果。