We propose a novel a posteriori error estimator for conforming finite element discretizations of two- and three-dimensional Helmholtz problems. The estimator is based on an equilibrated flux that is computed by solving patchwise mixed finite element problems. We show that the estimator is reliable up to a prefactor that tends to one with mesh refinement or with polynomial degree increase. We also derive a fully computable upper bound on the prefactor for several common settings of domains and boundary conditions. This leads to a guaranteed estimate without any assumption on the mesh size or the polynomial degree, though the obtained guaranteed bound may lead to large error overestimation. We next demonstrate that the estimator is locally efficient, robust in all regimes with respect to the polynomial degree, and asymptotically robust with respect to the wavenumber. Finally we present numerical experiments that illustrate our analysis and indicate that our theoretical results are sharp.
翻译:我们为符合两维和三维Helmholtz问题的有限元素分解提出了一个新颖的事后误差估计器。 估计器基于一种均衡的通量, 该通量的计算方法是解决不完全的混合的有限元素问题。 我们显示, 估计器可靠到预设物, 其预设物往往带有网状精细或多元度的提高。 我们还在预设器上得出一个完全可比较的数个域和边界条件共同设置的上限。 这导致在没有网目大小或多维度任何假设的情况下进行有保证的估计, 尽管获得的保证约束可能会导致大误测。 我们接下来要证明, 估计器在当地效率高, 在所有机制中, 与多元度有关, 且与波数有关, 且不具有任意性强力。 我们最后提出一个数字实验, 以说明我们的分析, 并表明我们的理论结果是尖锐的。