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An adaptive finite element DtN method for the three-dimensional acoustic scattering problem
Discrete and Continuous Dynamical Systems-Series B ( IF 1.2 ) Pub Date : 2020-11-18 , DOI: 10.3934/dcdsb.2020351
Gang Bao , , Mingming Zhang , Bin Hu , Peijun Li ,

This paper is concerned with a numerical solution of the acoustic scattering by a bounded impenetrable obstacle in three dimensions. The obstacle scattering problem is formulated as a boundary value problem in a bounded domain by using a Dirichlet-to-Neumann (DtN) operator. An a posteriori error estimate is derived for the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator, where the latter is shown to decay exponentially with respect to the truncation parameter. Based on the a posteriori error estimate, an adaptive finite element method is developed for the obstacle scattering problem. The truncation parameter is determined by the truncation error of the DtN operator and the mesh elements for local refinement are marked through the finite element approximation error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

中文翻译:

三维声散射问题的自适应有限元DtN方法

本文涉及有界不可穿透障碍物在三个维度上的声散射数值解。通过使用Dirichlet-to-Neumann(DtN)运算符将障碍物散射问题表述为有界域中的边值问题。使用截断后的DtN运算符可得出有限元方法的后验误差估计。后验误差估计包括有限元逼近误差和DtN算子的截断误差,其中DtN算子显示为相对于截断参数呈指数衰减。基于后验误差估计,针对障碍物散射问题,提出了一种自适应有限元方法。截断参数由DtN运算符的截断误差确定,并且用于局部细化的网格元素通过有限元逼近误差进行标记。数值实验表明了该方法的有效性。
更新日期:2020-12-15
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