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Archive : Numerical Analysis and PDE

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Michael Neilan, University of PittsburghMichael Neilan, University of PittsburghEWG 336Title: Inf-sup stable Stokes pairs on barycentric refinements producing divergence-free approximations <br><br> Abstract: We construct several stable finite element pairs for the Stokes problem on barycentric refinements in arbitrary dimensions and for any polynomial degree. A key feature of the spaces is that the divergence maps the discrete velocity space onto the discrete pressure space; thus, when applied to models of incompressible flows, the pairs yield divergence-free velocity approximations. The key ingredients to prove these results are local inf-sup stability estimates and a modification of Bernardi-Raugel bubble functions. This is joint work with Johnny Guzman. 5/17/2018 3:00:00 PM5/17/2018 4:00:00 PMFalse
DelMar Numerics Day 2018DelMar Numerics Day 2018TBAKeynote speaker: Mark Ainsworth, Brown University5/5/2018 12:00:00 AM5/5/2018 11:59:00 PMTrue
Ana Maria Soane, U.S. Naval AcademyAna Maria Soane, U.S. Naval AcademyEWG 336Title. Computational analysis of problems on domains with small holes <br></br> Abstract: The modeling challenges arising when the problem domain has small holes in it are considered through a representative membrane problem. We propose a numerical method which combines analytic knowledge of the solution singularities with finite element approximation of its smooth components. This allows us to perform the finite element modeling on a "filled in" domain without discs, thus obviating the need to use curvilinear elements or implement any special meshing. Theoretical and numerical results are provided to establish the efficacy and robustness of the method, both in the the energy norm and in extracting a representative quantity of interest. The method converges both with respect to the size of the holes and the mesh discretization parameter, and provides a more accurate alternative to using the asymptotic limit. This is joint work with Ivo Babuska (UT Austin) and Manil Suri (UMBC).4/19/2018 3:00:00 PM4/19/2018 4:00:00 PMFalse
Yanlai Chen, University of Massachusetts at DartmouthYanlai Chen, University of Massachusetts at DartmouthEwing 336Title: <br> Reduced Basis Method: Applications, recent improvements on efficiency and robustness, and a new iterative scheme Abstract:<br> Models of reduced computational complexity is indispensable in scenarios where a large number of numerical solutions to a parametrized problem are desired in a fast/real-time fashion. Thanks to an offline-online procedure and the recognition that the parameter-induced solution manifolds can be well approximated by finite-dimensional spaces, reduced basis method (RBM) and reduced collocation method (RCM) can improve efficiency by several orders of magnitudes. The accuracy of the RBM/RCM solutions is maintained through a rigorous a posteriori error estimator whose efficient and accurate evaluation is critical. After a brief introduction of the RBM, this talk will show our recent work on novel approaches for speeding up the offline portion of the approaches by around 6-fold, and new residual-based and residual-free strategies for circumventing error stagnation that is traditional of the classical RBM. If time permits, we will introduce a new iterative scheme inspired by RBM that outperforms conjugate gradient method with multigrid as a preconditioner for linear systems discretizing Poisson's equation. 4/12/2018 3:00:00 PM4/12/2018 4:00:00 PMFalse

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