Mixed finite element method

<h2 id="variations">Variations</h2>
<h3>Constraints</h3>
<p>In constrained mixed methods, the Lagrange multiplier fields inside the elements, usually enforcing the applicable partial differential equations.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> This results in a saddle point system having negative pivots and eigenvalues, rendering the system matrix to be non-definite which results in complications in solving for it. In <a href="/facts/Sparse_matrix/FFtN0XjR">sparse</a> direct solvers, pivoting may be needed, where ultimately the resulting matrix has 2x2 blocks on the diagonal,<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> rather than a  working towards a completely pure LLH <a href="/facts/Cholesky_decomposition/Q3IjkuTs">Cholesky</a> decomposition for positive definite symmetric or Hermitian systems. Pivoting may result in unpredictable memory usage increases.<a href="https://www.eng-tips.com/viewthread.cfm?qid=446658">[1]</a> For iterative solvers, only <a href="/facts/Gmres/Ak7gbYzI">GMRES</a> based solvers work, rather than slightly "cheaper" <a href="/facts/MINRES/rtIaK9Am">MINRES</a> based solvers. 
</p>
<h3>Hybrid Methods</h3>
<p>In hybrid methods, the Lagrange fields are for jumps of fields between elements, living on the boundary of the elements, weakly enforcing continuity; continuity from fields in the elements does not need to be enforced through shared degrees of freedom between elements anymore. Both mixing and hybridization can be applied simultaneously.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> These enforcements are "<a href="/facts/Weak_formulation/hXd2jSoW">weak</a>", i.e. occur upon having the solutions and possibly only at some points or e.g. matching moment integral conditions, rather than "strong" in which case the conditions are fulfilled directly in the type of solutions sought. Apart from the harmonics (usually semi-trivial local solution to the homogeneous equations at zero loads), hybridization allows for static Guyan condensation of the discontinuous fields internal to the elements, reducing the number of degrees of freedom, and moreover reducing or eliminating the number of negative eigenvalues and pivots resulting from application of the mixed method. 
</p>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Olek C Zienkiewicz, Robert L Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Elsevier. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Arnold, Douglas. "Douglas Arnold 2016 Woudschoten Conference" (PDF). <a href="https://wsc.project.cwi.nl/woudschoten-conferences/2016-woudschoten-conference/DAtalk1.pdf" target="_blank">https://wsc.project.cwi.nl/woudschoten-conferences/2016-woudschoten-conference/DAtalk1.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Lungten, S.; Schilders, W.H.A.; Maubach, J.M.L. (2016-08-01). "Sparse block factorization of saddle point matrices". Linear Algebra and Its Applications. 502: 214–242. doi:10.1016/j.laa.2015.07.042. ISSN 0024-3795. <a href="https://doi.org/10.1016%2Fj.laa.2015.07.042" target="_blank">https://doi.org/10.1016%2Fj.laa.2015.07.042</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>"2019 Feb 1, Bernardo Cockburn, University of Minnesota, Variational principles for hybridizable discontinuous Galerkin methods: A short story". PSU Media Space. Retrieved 2021-05-02. <a href="https://media.pdx.edu/media/t/0_znpv6bco" target="_blank">https://media.pdx.edu/media/t/0_znpv6bco</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Mixed finite element method open-in-new

Mixed finite element method