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首页> 外文会议>International Conference on Domain Decomposition Methods in Science and Engineering >A Discrete Domain Decomposition Method for Acoustics with Uniform Exponential Rate of Convergence Using Non-local Impedance Operators
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A Discrete Domain Decomposition Method for Acoustics with Uniform Exponential Rate of Convergence Using Non-local Impedance Operators

机译：使用非局部阻抗运算符统一呈指数率的离散域分解方法

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摘要

We consider the Helmholtz equation in harmonic regime in a domain Ω(∩)R~ d, d = 2 or 3, and a first order absorbing condition on its boundary F with unit outward normal vector n. Let k∈R be a constant wave number and f∈L~2(Ω), we seek u∈H~1(Ω) such that{-Δu-k~2u=f,inΩ,(∂_n+ik)u=0,onΓ.(1) In previous works [2, 3,5], a domain decomposition method (DDM) using non-local transmission operator with suitable properties was described. The relaxed Jacobi algorithm written at the continuous level was proven to converge exponentially. However, it was only a conjecture, hinted at by numerical experiments in [5, Section 8], that the discretized algorithm using finite elements has a rate of convergence uniformly bounded with respect to the discretization parameter, hence does not deteriorate when the mesh is refined. In this work we prove this conjecture for the case of Lagrange finite elements. Numerical experiments in [5, Section 8.3] highlighted that this important property is not shared by DDM based on local operators [4] or rational fractions of local operators [1].
机译：我们考虑在域ω（∩）R〜D，D = 2或3中的谐波状态下的Helmholtz方程，以及具有单元向外普通向量n的边界F上的第一订单吸收条件。让k∈r是恒定波数和f∈l〜2（ω），我们寻求U∈h〜1（ω），使得{-Δu-k〜2u = f，inω，（∂_n+ ik）u = 0，ONγ。（1）在先前的作品中，描述了使用具有合适性质的非本地传输操作员的域分解方法（DDM）。被证明在连续水平上写入的轻松Jacobi算法以指数增长。然而，只有一个猜想，在[5，第8节]中，通过数值实验暗示，使用有限元的离散算法具有相对于离散化参数均匀界定的会聚速率，因此当网格是时不会恶化精制。在这项工作中，我们证明了这种猜想为拉格朗日有限元的情况。 [5，第8.3节]中的数值实验强调，基于本地运营商[4]或本地运营商的合理分数，DDM不共享这一重要性[1]。

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