We consider interior penalty discontinuous Galerkin discretizations of time-harmonic wave propagation problems modeled by the Helmholtz equation, and derive novel a priori and a posteriori estimates. Our analysis classically relies on duality arguments of Aubin-Nitsche type, and its originality is that it applies under minimal regularity assumptions. The estimates we obtain directly generalize known results for conforming discretizations, namely that the discrete solution is optimal in a suitable energy norm and that the error can be explicitly controlled by a posteriori estimators, provided the mesh is sufficiently fine.

翻译：本文考虑由亥姆霍兹方程建模的时谐波传播问题的内罚间断伽辽金离散方法，并推导了新型的先验和后验估计。我们的分析经典地依赖于奥宾-尼切型对偶论证，其创新之处在于该方法适用于极小正则性假设。所得估计直接推广了已知的协调离散结果，即在网格足够精细的条件下，离散解在适当能量范数下达到最优，且误差可通过后验估计器显式控制。