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The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. <a href="https://mathscinet.ams.org/mathscinet/search/journal/profile?groupId=3285" target="new">Subscribers to MathSciNet</a> may click through for more detailed information. </div> </p> </section> </div> </header> <main> <section class="row"> <nav class="col-md-3">  <nav class="journals-list-group"> <script type="text/javascript" src="/journals/javascript/mcom_nav.js"></script> <a href="/journals/journals"><i class="fa-sharp fa-solid fa-house"></i> Journals Home</a> <a href="/epubsearch">eContent Search</a> <a href="/aboutmcom" class="">About Mathematics of Computation</a> <a href="/mcomedit" class="">Editorial Board</a> <a href="/mcom/mcomsubmit.html" class="">Author and Submission Information</a> <a href="/policies/policies" class="" target="_blank">Journal Policies</a> <a href="/subscriberinfo" class="" target="_blank">Subscription Information</a> <a href="/jemail" class="" target="_blank"> Journal Email Notifications</a> <hr> <a href="/pairing/pair_my_device.html"><i class="fa fa-globe" aria-hidden="true"></i> Remote Access</a> </nav> </nav> <article class="col-md-9"> <section>   <div class="row">  <ul class="nav nav-tabs"> <li role="presentation" class=""> <script type="text/javascript" src="/journals/javascript/mcom_nav.js"></script> <a href="javascript: goto_most_recent_mcom()" aria-controls="currissue" role="tab"> Current volume </a> </li> <li role="presentation" class="">  <a href="/cgi-bin/mstrack/accepted_papers/mcom?active=press" aria-controls="press" role="tab">Articles in press </a> </li> <li role="presentation" class=""> <a href="/journals/mcom/0000-000-00?active=recent"> Recently published</a> </li> <li role="presentation" class=""> <a href="/journals/mcom/all_issues.html?active=allissues"> All issues : 1943 – Present </a> </li> </ul> <script src="/js/abstracts/ajaxcall.js"></script> </div> <p> </p> <style> #navCell { display: none;} </style> <script src="/journals/javascript/references.js" type="text/javascript"></script> <script async src="/js/mj.js"></script>  <script src="https://platform-api.sharethis.com/js/sharethis.js#property=598c93d1e2587a001253a011&product=unknown" async="async"></script>  <div class="sharethis-inline-share-buttons"></div>   <article class="col-md-12"> <header> <h2>Finite element/holomorphic operator function method for the transmission eigenvalue problem<br> <small>HTML articles powered by AMS MathViewer <a href="/AMSMathViewer"><i class="fa fa-info-circle" aria-label="about MathViewer"></i></a></small></h2> <dl> <dd>by <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Gong%2C%20Bo">Bo Gong</a>, <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?mrauthid=738684">Jiguang Sun</a>, <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?mrauthid=995099">Tiara Turner</a> and <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Zheng%2C%20Chunxiong">Chunxiong Zheng</a>;  </dd> <dd>Math. Comp. <strong>91</strong> (2022), 2517-2537 </dd> <dd> DOI: https://doi.org/10.1090/mcom/3767 </dd> <dd> Published electronically: August 11, 2022 </dd> <dd class="mathviewerPDFLink"> <a href = "viewer"><strong>HTML</strong></a> | <a href="S0025-5718-2022-03767-4.pdf"><small>PDF</small></a> | <a href="https://www.copyright.com/openurl.do?issn=1088-6842&WT.mc.it=American%20Mathematical%20Society" target="_blank">Request permission</a> </dd> </dl> </header> <section id="Abstract"> <h2>Abstract:</h2> The transmission eigenvalue problem arises from the inverse scattering theory for inhomogeneous media. It plays a key role in the unique determination of inhomogeneous media. Furthermore, transmission eigenvalues can be reconstructed from the scattering data and used to estimate the material properties of the unknown object. The problem is posted as a system of two second order partial differential equations and is nonlinear and non-selfadjoint. It is challenging to develop effective numerical methods. In this paper, we formulate the transmission eigenvalue problem as the eigenvalue problem of a holomorphic operator function. The Lagrange finite elements are used for the discretization and the convergence is proved using the abstract approximation theory for holomorphic Fredholm operator functions. The spectral indicator method is employed to compute the eigenvalues. Numerical examples are presented to validate the proposed method. </section> <section id="references"> <details> <summary>References</summary> <ul><li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Ammari%2C%20Kais">Kaïs Ammari</a> and <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Amrouche%2C%20Cherif">Chérif Amrouche</a>, <em>Resolvent estimates for wave operators in Lipschitz domains</em>, Calc. Var. 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China Math. <strong>59</strong> (2016), no. 8, 1613–1622. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3528506">MR <strong>3528506</strong></a>, DOI <a href="https://doi.org/10.1007/s11425-016-0289-8">10.1007/s11425-016-0289-8</a></li> </ul> </details> </section> <section id="similar"> <details> <summary>Similar Articles</summary> <ul> <li> Retrieve articles in <em>Mathematics of Computation</em> with MSC (2020): <a href="/joursearch/servlet/DoSearch?f1=msc&v1=35P30&jrnl=one&onejrnl=mcom">35P30</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=65N30&jrnl=one&onejrnl=mcom">65N30</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=78A46&jrnl=one&onejrnl=mcom">78A46</a> </li> <li> Retrieve articles in all journals with MSC (2020): <a href="/joursearch/servlet/DoSearch?f1=msc&v1=35P30">35P30</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=65N30">65N30</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=78A46">78A46</a> </ul> </details> </section> <section id="additionalinformation"> <details open> <summary>Bibliographic Information</summary> <ul> <li><strong>Bo Gong</strong></li><li>Affiliation: Beijing University of Technology, Beijing 100124, People’s Republic of China</li> <li>Email: gongbo@bjut.edu.cn</li> <li><strong>Jiguang Sun</strong></li><li>Affiliation: Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931</li> <li>MR Author ID: <a href='https://mathscinet.ams.org/mathscinet/search/authors.html?mrauthid=738684'>738684</a></li><li>ORCID: <a href='https://orcid.org/0000-0003-4105-8463'>0000-0003-4105-8463</a></li><li>Email: jiguangs@mtu.edu</li> <li><strong>Tiara Turner</strong></li><li>Affiliation: Department of Mathematics and Computer Science, University of Maryland Eastern Shore, Princess Anne, Maryland 21853</li> <li>MR Author ID: <a href='https://mathscinet.ams.org/mathscinet/search/authors.html?mrauthid=995099'>995099</a></li><li>Email: tdturner@umes.edu</li> <li><strong>Chunxiong Zheng</strong></li><li>Affiliation: College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, People’s Republic of China; and Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China</li> <li>Email: czheng@tsinghua.edu.cn</li> <li>Received by editor(s): June 2, 2021</li> <li>Received by editor(s) in revised form: March 28, 2022</li> <li>Published electronically: August 11, 2022</li> <li>Additional Notes: The first author’s work was partially supported by NSFC 11871092 and NSAF U1930402. The second author’s work was partially supported by Simons Foundation 711922. The fourth author’s work was supported by Natural Science Foundation of Xinjiang Autonomous Region under No. 50500-600112096, and National Natural Science Foundation under No. 11771248.</li> <li>© Copyright 2022 American Mathematical Society</li> <li>Journal: Math. Comp. <strong>91</strong> (2022), 2517-2537 </li> <li>MSC (2020): Primary 35P30, 65N30, 78A46 <li>DOI: https://doi.org/10.1090/mcom/3767</li> <li>MathSciNet review: <a href = "https://mathscinet.ams.org/mathscinet-getitem?mr=4473095">4473095</a></li> </ul> </details> </section> </article> </section> </article> </section> </main> </div>   <hr class="mx-auto"> <footer id="pagefooter" class="text-center"> <div class="container text-center"> <div class="row">  <div class="col-12 mb-1"> <p> American Mathematical Society · <a href="https://goo.gl/maps/ZvJdkQEC79p" title="Google map 201 Charles Street Providence, Rhode Island" aria-label="Google map to AMS headquarters"> 201 Charles Street Providence, Rhode Island 02904-2213 </a> · <a href="/reach" aria-label="Contact Us">Contact Us</a> </p> </div>  <script src="https://kit.fontawesome.com/2ad6d7fbd6.js" crossorigin="anonymous" defer></script>  <nav class="col-12 mb-1" aria-label="Social media links"> <ul class="list-inline"> <li class="list-inline-item"> <a href="https://www.facebook.com/amermathsoc" class="link-dark" title="Follow us on Facebook" aria-label="View the AMS on Facebook"> <i class="fa-brands fa-facebook-f" aria-hidden="true"></i></a></li> <li class="list-inline-item"> <a href="https://bsky.app/profile/amermathsoc.bsky.social" class="link-dark" title="Follow us on Bluesky" aria-label="View the AMS on Bluesky"> <i class="fa-brands fa-bluesky"></i></a></li> <li class="list-inline-item"> <a href="https://twitter.com/amermathsoc" class="link-dark" title="Follow us on X" aria-label="View the AMS on X"> <i class="fa-brands fa-x-twitter" aria-hidden="true"></i></a></li> <li class="list-inline-item"> <a href="https://www.linkedin.com/company/51001" class="link-dark" title="Follow us on LinkedIn" aria-label="View the AMS on LinkedIn"> <i class="fa-brands fa-linkedin-in" aria-hidden="true"></i></a></li> <li class="list-inline-item"> <a href="https://www.instagram.com/amermathsoc" class="link-dark" title="Follow us on Instagram" aria-label="View the AMS on Instagram"> <i class="fa-brands fa-instagram" aria-hidden="true"></i></a></li> <li class="list-inline-item"> <a href="https://www.youtube.com/amermathsoc" class="link-dark" title="Follow us on YouTube" aria-label="View the AMS on YouTube"> <i class="fa-brands fa-youtube" aria-hidden="true"></i></a></li> </ul> </nav>  <div class="col-12"> <p><small> © <a href="/copyright" aria-label="Copyright policy">Copyright</a> <span id="copyYear">2024</span> American Mathematical Society · <a href="/privacy" aria-label="Privacy statement">View Our Privacy Statement</a> · <a href="/tou" aria-label="Terms of use">Terms of Use</a> · <a href="/accessibility" aria-label="Accessibility policy">Accessibility and AMS Online Content</a> </small></p> <p><small> AMS, American Mathematical Society, the tri-colored AMS logo, and Advancing research, Creating connections, are trademarks and service marks of the American Mathematical Society and registered in the U.S. Patent and Trademark Office. </small></p> </div> </div> </div> <script> // Dynamically update the copyright year document.getElementById('copyYear').textContent = new Date().getFullYear(); </script> </footer>