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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716304514&_mathId=si11.gif&_user=111111111&_pii=S0377042716304514&_rdoc=1&_issn=03770427&md5=bc684eaa5ad1482806810b70e53100ae" title="Click to view the MathML source">b=Zaclass="mathContainer hidden">class="mathCode">$b=Za$ with a map class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716304514&_mathId=si12.gif&_user=111111111&_pii=S0377042716304514&_rdoc=1&_issn=03770427&md5=651bc797f43f9c1a7841ecb18c57e7ea">class="imgLazyJSB inlineImage" height="14" width="91" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0377042716304514-si12.gif">class="mathContainer hidden">class="mathCode">$Z:{\ell }_2\to {\ell }_2$ 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 class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716304514&_mathId=si13.gif&_user=111111111&_pii=S0377042716304514&_rdoc=1&_issn=03770427&md5=9a7ccb4556d5c9b5a401bd74ec445912">class="imgLazyJSB inlineImage" height="16" width="49" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0377042716304514-si13.gif">class="mathContainer hidden">class="mathCode">$\tilde{b}=\tilde{Z}\tilde{a}$ is uniquely solvable and is unconditionally stable, i.e. the stability can be made to depend on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716304514&_mathId=si14.gif&_user=111111111&_pii=S0377042716304514&_rdoc=1&_issn=03770427&md5=21936eeffaece1ac2a5bb705e6b217f6" title="Click to view the MathML source">Zclass="mathContainer hidden">class="mathCode">$Z$ only, not on the discretization. Convergence rates of finite approximations class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716304514&_mathId=si15.gif&_user=111111111&_pii=S0377042716304514&_rdoc=1&_issn=03770427&md5=ac08c201e07e8c29a06737830ec0710a">class="imgLazyJSB inlineImage" height="16" width="10" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0377042716304514-si15.gif">class="mathContainer hidden">class="mathCode">$\tilde{b}$ of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716304514&_mathId=si16.gif&_user=111111111&_pii=S0377042716304514&_rdoc=1&_issn=03770427&md5=55d8df201bddfc8a8378e1a65e9aedbe" title="Click to view the MathML source">bclass="mathContainer hidden">class="mathCode">$b$ then carry over to convergence rates of finite approximations class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716304514&_mathId=si17.gif&_user=111111111&_pii=S0377042716304514&_rdoc=1&_issn=03770427&md5=b3843f1600602410b4c63f162d2b628f">class="imgLazyJSB inlineImage" height="13" width="10" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0377042716304514-si17.gif">class="mathContainer hidden">class="mathCode">$\tilde{a}$ of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716304514&_mathId=si18.gif&_user=111111111&_pii=S0377042716304514&_rdoc=1&_issn=03770427&md5=00cd533cf1475e22030da78bf99be3b4" title="Click to view the MathML source">aclass="mathContainer hidden">class="mathCode">$a$. Spectral convergence is a special case. Some examples are added for illustration.