Convergence analysis of general spectral methods

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• 作者：M. Mohammadi^a ; ^{m.mohammadi@khu.ac.ir" class="auth_mail" title="E-mail the corresponding author} ; R. Schaback^b ; ^{schaback@math.uni-goettingen.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：65M12 ; 65M70 ; 65N12 ; 65N35 ; 65M15 ; 65M22 ; 65J10 ; 65J20 ; 35D30 ; 35D35 ; 35B65 ; 41A25 ; 41A63
• 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：284-293
• 全文大小：402 K

文摘

If a spectral numerical method for solving ordinary or partial differential equations is written as a biinfinite linear system e53100ae" title="Click to view the MathML source">b=Za$b=Za$ with a map e7ea">$Z:{\ell }_2\to {\ell }_2$ that has a continuous inverse, this paper shows that one can discretize the biinfinite system in such a way that the resulting finite linear system 9a7ccb4556d5c9b5a401bd74ec445912">$\tilde{b}=\tilde{Z}\tilde{a}$ is uniquely solvable and is unconditionally stable, i.e. the stability can be made to depend on aece1ac2a5bb705e6b217f6" title="Click to view the MathML source">Z$Z$ only, not on the discretization. Convergence rates of finite approximations 9a06737830ec0710a">$\tilde{b}$ of 8df201bddfc8a8378e1a65e9aedbe" title="Click to view the MathML source">b$b$ then carry over to convergence rates of finite approximations $\tilde{a}$ of a$a$. Spectral convergence is a special case. Some examples are added for illustration.