A first order system least squares method for the Helmholtz equation

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• 作者：Huangxin Chen^a ; ^{chx@xmu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Weifeng Qiu^b ; ^{weifeqiu@cityu.edu.hk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：65N30 ; 65L12
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：309
• 期：Complete
• 页码：145-162
• 全文大小：1733 K

文摘

We present a first order system least squares (FOSLS) method for the Helmholtz equation at high wave number k$k$, which always leads to a Hermitian positive definite algebraic system. By utilizing a non-trivial solution decomposition to the dual FOSLS problem which is quite different from that of the standard finite element methods, we give an error analysis to the hp$hp$-version of the FOSLS method where the dependence on the mesh size h$h$, the approximation order p$p$, and the wave number k$k$ is given explicitly. In particular, under some assumption of the boundary of the domain, the L²$L^2$ norm error estimate of the scalar solution from the FOSLS method is shown to be quasi optimal under the condition that kh/p$kh/p$ is sufficiently small and the polynomial degree p$p$ is at least O(logk)$O(logk)$. Numerical experiments are given to verify the theoretical results.