Document Zbl 1372.65276

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Numer. Math.</a> <a href="/?q=in%3A369605" title="Articles in this Issue">121, 82-95 (2017)</a>. </div> <div class="abstract">Summary: In this paper, a new Crank-Nicolson finite element method for the time-fractional subdiffusion equation is developed, in which a novel time discretization called the modified $\mathrm{L}1$ method is used to discretize the Riemann-Liouville fractional derivative. The present method is unconditionally stable and convergent of order $O(\tau^{1 + \beta} + h^{r + 1})$, where $\beta \in(0, 1)$, $\tau$ and $h$ are the step sizes in time and space, respectively, and $r$ is the degree of the piecewise polynomial space. The derived method is reduced to the classical Crank-Nicolson method when $\beta \rightarrow 1$. The new time discretization is also used to solve the fractional cable equation. And the unconditional stability and convergence are given. Numerical examples are provided which support the theoretical analysis. The comparison with the existing methods are also given, which shows good performances of the present methods.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6770902">Cited in <strong>30</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M60" title="MSC2020">65M60</a> </td> <td class="space"> Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M06" title="MSC2020">65M06</a> </td> <td class="space"> Finite difference methods for initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M12" title="MSC2020">65M12</a> </td> <td class="space"> Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35K05" title="MSC2020">35K05</a> </td> <td class="space"> Heat equation </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35R11" title="MSC2020">35R11</a> </td> <td class="space"> Fractional partial differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afinite+element+method">finite element method</a>; <a href="/?q=ut%3Asubdiffusion">subdiffusion</a>; <a href="/?q=ut%3Athe+time-fractional+subdiffusion+equation">the time-fractional subdiffusion equation</a>; <a href="/?q=ut%3Aunconditional+stability">unconditional stability</a>; <a href="/?q=ut%3Aconvergence">convergence</a>; <a href="/?q=ut%3ACrank-Nicolson+method">Crank-Nicolson method</a>; <a href="/?q=ut%3Anumerical+example">numerical example</a></div> <div class="software"> <h3>Software:</h3><a href="/software/20918">ma2dfc</a>; <a href="/software/8377">FODE</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1372.65276" data-ciurl="/ci/06770902" data-biburl="/bibtex/06770902.bib" data-amsurl="/amsrefs/06770902.bib" data-xmlurl="/xml/06770902.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06770902.pdf" title="Zbl 1372.65276 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.apnum.2017.06.011" aria-label="DOI for “A new Crank-Nicolson finite element method for the time-fractional subdiffusion equation”" title="10.1016/j.apnum.2017.06.011">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Agrawal, O. 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