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Archive : Numerical Analysis and PDE

 

 

Sara Pollock, Wright State UniversitySara Pollock, Wright State UniversityEWG 336Title. Adaptive regularization for nonlinear diffusion Abstract. Nonlinear elliptic problems featuring solution-dependent diffusion appear in many applications in physical modeling. The numerical approximation of solutions to these problems, however, presents unique challenges not encountered in linear problems, or in the related problems of nonlinear monotone type. The first main challenge is the instability of the iterative solution process. Ill-conditioned and potentially indefinite Jacobians may cause spurious modes to grow from one solution iterate to the next, leading to the divergence of direct linearization methods. The second main challenge is assuring the consistency of the discrete problem with respect to the continuous model. In particular, highly oscillatory layers in the diffusion coefficient may need to be uncovered during the solution process. While finite element solutions are known in theory to have good approximation properties assuming a globally fine discretization, the computability and in particular the efficient computability of the numerical solutions has yet to be established. We will discuss adaptive strategies based on incomplete solves of regularized problems, to solve the discrete nonlinear equations induced by finite element discretizations of these nonmonotone quasilinear PDE. Strategies for regularization parameter selection and exit criteria for the nonlinear iterations will be discussed in the context of running efficient adaptive simulations starting from a coarse mesh. Numerical examples will illustrate the ideas and demonstrate the presented adaptive algorithm. 3/23/2017 3:00:00 PM3/23/2017 4:00:00 PMFalse
Tom H Anderson, University of DelawareTom H Anderson, University of DelawareEWG336Title: Optoelectronic Simulations of Nonhomogeneous Thin Film Solar Cells Abstract: In order to reduce dependence on fossil fuel based power generation, development of highly efficient solar cells is essential. However, for large scale terrestrial power generation the installed cost per watt is usually a superior figure of merit. To this end, thin film techbologies may provide a way to mass produce low efficiency, yet ultra cheap solar cells. The 'thin' in 'thin film' refers to the optical depth of the material: the solar cell is only a few multiples of a wavelength of light thick. On this scale, photonic structure may be included to couple incident light into guided wave modes and surface plasmon-polariton waves. Therefore, in order to predict the behaviour of these thin film solar cells, it is necessary to solve Maxwells equations to predict where the light will be absorbed. Once the generation profile for the whole solar spectrum has been calculated, drift/diffusion equations must be solved in order to predict the resulting solar power conversion efficiency. 2/23/2017 4:00:00 PM2/23/2017 5:00:00 PMFalse
Jingmei Qiu, University of HoustonJingmei Qiu, University of HoustonEWG 336<b>High order Semi-Lagrangian Methods for Transport Problems with Applications to Vlasov Simulations and Global Transport</b> <br><br> The semi-Lagrangian (SL) scheme for transport problems gains more and more popularity in the computational science community due to its attractive properties. For example, the SL scheme, compared with the Eulerian approach, allows extra large time step evolution by incorporating characteristics tracing mechanism, hence achieving great computational efficiency. In this talk, we introduce a family of high order SL methods coupled with the finite element discontinuous Galerkin (DG) method. The proposed SL schemes are applied to transport problems, the Vlasov model arising from the plasma physics and the global transport problems based on the cubed-sphere geometry from the operational climate model. The methods have been extensively tested and benchmarked with classical test problems in the literature. 1/6/2017 4:00:00 PM1/6/2017 5:00:00 PMFalse
Amanda Diegel, Louisiana State University Amanda Diegel, Louisiana State University Ewg 336<div class="WordSection1"><div><p class="MsoNormal" style="">Title: Phase Field Models and the Analysis of Their Numerical Schemes <br><br>Abstract: We investigate phase field modeling and some numerical schemes designed to solve the resulting systems of equations. Due to the vastness in applicability, phase field models, also know as diffuse interface models, have drawn the attention of a number of researchers in recent years. The upshot to phase field modeling is that the modeler does not need to explicitly track the motion of interface between the two phases in the system as the motion is captured by the an indicator function. Generally, this is done by coupling the popular Cahn-Hilliard equation to other multi-phase or multi-physics models. One of the most well known examples of these pairings is the coupling of the Cahn-Hilliard equation with fluid flow models to describe two-phase flows: for example, the Cahn-Hilliard-Navier-Stokes equations and the Cahn-Hilliard-Darcy-Stokes equations. However, the Cahn-Hilliard equation has also been used in pairings to study vesicle membranes, viod-electro migration, and most recently, liquid crystals. <br><br>Phase field modeling allows us to overcome the hardships found while attempting to derive stable and convergent computational schemes for the traditional sharp interface models. The numerical schemes we will discuss in this talk mimic the energy dissipation laws which are inherent to models using a phase field approach. Creating numerical schemes in this way makes it possible to rigorously prove three key properties: unconditional stability, unconditional unique solvability, and optimal convergence. Convergence results provide valuable feedback concerning the approximation properties of a numerical scheme and unconditional stability leads to enhanced convergence estimations which leads to high confidence that the numerical schemes accurately estimate solutions to the equations upon which they are designed. </p></div></div>12/8/2016 4:00:00 PM12/8/2016 5:00:00 PMFalse

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