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Learn about matrix-valued finite elements in this seminar presentation from the FEM@LLNL series. Explore how matrix-valued finite elements naturally arise in continuum models, with stress elements in structural mechanics serving as typical examples. Discover a recent element for viscous stresses in fluid mechanics, motivated by H(div)-conforming approximations of fluid velocity where the divergence-free constraint of viscous incompressible flows can be enforced exactly. Examine the development of simple elements with nt-continuity (normal and tangential components on element interfaces), corresponding to shear continuity of traceless viscous stresses, and understand the associated Sobolev space H(curl div) of matrix-valued functions. Investigate the Mass-Conserving Stress-yielding (MCS) method, which is exactly mass conserving, pressure robust, yields optimal-order approximations for velocity, pressure, stress, and vorticity, and relies only on facet-based coupling for hybridization compatibility. Gain insight into the broader context of other matrix-valued finite elements, including nn-continuous Hellan–Herrmann–Johnson elements and tt-continuous Regge elements. Understand recent developments connecting nn-, nt-, and tt-continuous matrix-valued finite elements, and explore challenges in seeking unifying principles for tensor-valued discretizations beyond the de Rham complex. This presentation by Jay Gopalakrishnan from Portland State University is part of the MFEM project's seminar series focusing on finite element research and applications.
