Archive : Numerical Analysis and PDE

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Dr. Hong Zhang, Argonne National Lab152Dr. Hong Zhang, Argonne National Lab4/22/2022 3:00:00 PM4/22/2022 4:00:00 PM
Professor Peter Monk, University of Delaware148Professor Peter Monk, University of Delaware<p>‚Äč<span class="wrap-text"><strong>Title:</strong> Development of a coupled Trefftz and finite element method for approximating Maxwell's equations<br><strong>Abstract:</strong> The talk focuses on a discontinuous Galerkin method for the time harmonic Maxwell system. This method is based on the use of a finite element grid, but uses plane wave solutions of Maxwell's equations on each element to approximate the global field. Because each basis function satisfies Maxwell's equations, the problem can be reduced to a coupled linear system on the faces of the grid. Arbitrarily high order of convergence can be achieved by taking more planes waves in suitable directions element by element, although ill-conditioning must be carefully controlled. Unfortunately this method has severe deficiencies when applied to some resonant problems. To remedy this, we have tried to couple polynomial finite element methods with the plane wave scheme. I shall report on the current state of this effort.</span><br></p>2/18/2022 4:00:00 PM2/18/2022 5:00:00 PM
Dr. William Taitano, Air Force Research Lab151Dr. William Taitano, Air Force Research Lab2/11/2022 4:00:00 PM2/11/2022 5:00:00 PM
Dr. Jingmei Qiu, University of Delaware150Dr. Jingmei Qiu, University of Delaware<p>‚ÄčTitle: A Conservative Low Rank Tensor Representation of Nonlinear Vlasov-Poisson and Vlasov-Maxwell Solutions</p><p><br>Abstract: We propose a low-rank tensor approach to approximate nonlinear Vlasov solutions and their associated flow maps. The approach takes advantage of the fact that the differential operators in the Vlasov equation is tensor friendly, based on which we propose to dynamically and adaptively build up low-rank solution basis by adding new basis functions from discretization of the PDE, and removing basis from an SVD-type truncation procedure. For the discretization, we adopt a high order finite difference spatial discretization and a second order strong stability preserving multi-step time discretization. We apply the same procedure to evolve the dynamics of the flow map in a low-rank fashion, which proves to be advantageous when the flow map enjoys the low rank structure, while the solution suffers from high rank or displays filamentation structures. Local conservation properties are built in by a set of special projections. Hierarchical Tucker decomposition is adopted for high dimensional problems. An extensive set of linear and nonlinear Vlasov test examples are performed to show the high order spatial and temporal convergence of the algorithm with mesh refinement up to SVD-type truncation, the significant computational savings of the proposed low-rank approach especially for high dimensional problems, the improved performance of the flow map approach for solutions with filamentations.<br></p>12/3/2021 4:00:00 PM12/3/2021 5:00:00 PM

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