An adaptive finite element DtN method for the elastic wave scattering problem
An adaptive finite element DtN method for the elastic wave scattering problem
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DOI：
10.1007/s00211-022-01273-4
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发表时间：
2022-03
影响因子：
2.1
通讯作者：
Peijun Li;Xiaokai Yuan
中科院分区：
文献类型：
--
作者：
Peijun Li;Xiaokai Yuan
考虑二维均匀各向同性弹性介质中刚性障碍物对入射波的散射。基于Dirichlet-to-Neumann（DtN）算子，引入一个精确的透明边界条件，将散射问题转化为有界区域上弹性波方程的边值问题.通过发展一种新的对偶方法，利用截断DtN算子的有限元方法，得到了离散问题的后验误差估计。后验误差估计由有限元逼近误差和DtN算子的截断误差组成，后者相对于截断参数呈指数衰减。提出了一种求解弹性障碍物散射问题的自适应有限元算法，其中截断参数由截断误差确定，局部加密网格单元由有限元离散误差选择。数值实验证明了该方法的有效性。
Consider the scattering of an incident wave by a rigid obstacle, which is immersed in a homogeneous and isotropic elastic medium in two dimensions. Based on a Dirichlet-to-Neumann (DtN) operator, an exact transparent boundary condition is introduced and the scattering problem is formulated as a boundary value problem of the elastic wave equation in a bounded domain. By developing a new duality argument, an a posteriori error estimate is derived for the discrete problem by using 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 decays exponentially with respect to the truncation parameter. An adaptive finite element algorithm is proposed to solve the elastic obstacle scattering problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are chosen through the finite element discretization error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.