A fast trigonometric collocation method for some elliptic pseudodifferential equations

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• 作者：Guangqing Long ; ^{longgq@amss.ac.cn" class="auth_mail" title="E-mail the corresponding author} ; Pailing Li ^{lipailing@126.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Elliptic pseudodifferential equations ; Trigonometric collocation method ; Fast methods
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：313
• 期：Complete
• 页码：294-306
• 全文大小：1078 K

文摘

In this paper, we propose and analyze a fast trigonometric collocation method for a class of periodic elliptic pseudodifferential equations, whose pseudodifferential operators can always be represented as the sum of a principal part and a smoothing operator. We show that the whole matrix representation for the principal part in our discrete linear system can be generated by only computing md5=5e8b0c4eb2233df9a08b9eddd6dcf2d8" title="Click to view the MathML source">O(n)$\mathcal{O}\left(n\right)$ entries rather than computing all entries of the matrix, where md5=99d8340416768920013ee91071806bd7" title="Click to view the MathML source">2n$2n$ or md5=c767ef08ec07a7cd9be5666578a462d0" title="Click to view the MathML source">2n+1$2n+1$ is the size of the matrix. The dense matrix for the smoothing operator can be compressed into a sparse matrix with only md5=469cff8db36043657c20bd3f088544bf" title="Click to view the MathML source">O(nlogn)$\mathcal{O}(nlogn)$ nonzero entries. We also prove that our proposed method preserves the optimal convergent order the same as without compression. Some numerical experiments for solving three cases of boundary integral equations are presented to demonstrate its approximate accuracy and computational efficiency, verifying the theoretical estimates.