625 research outputs found
Adaptive Finite Element Approximations for Kohn-Sham Models
The Kohn-Sham equation is a powerful, widely used approach for computation of
ground state electronic energies and densities in chemistry, materials science,
biology, and nanosciences. In this paper, we study the adaptive finite element
approximations for the Kohn-Sham model. Based on the residual type a posteriori
error estimators proposed in this paper, we introduce an adaptive finite
element algorithm with a quite general marking strategy and prove the
convergence of the adaptive finite element approximations. Using D{\" o}rfler's
marking strategy, we then get the convergence rate and quasi-optimal
complexity. We also carry out several typical numerical experiments that not
only support our theory,but also show the robustness and efficiency of the
adaptive finite element computations in electronic structure calculations.Comment: 38pages, 7figure
The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations
The cutoff method, which cuts off the values of a function less than a given
number, is studied for the numerical computation of nonnegative solutions of
parabolic partial differential equations. A convergence analysis is given for a
broad class of finite difference methods combined with cutoff for linear
parabolic equations. Two applications are investigated, linear anisotropic
diffusion problems satisfying the setting of the convergence analysis and
nonlinear lubrication-type equations for which it is unclear if the convergence
analysis applies. The numerical results are shown to be consistent with the
theory and in good agreement with existing results in the literature. The
convergence analysis and applications demonstrate that the cutoff method is an
effective tool for use in the computation of nonnegative solutions. Cutoff can
also be used with other discretization methods such as collocation, finite
volume, finite element, and spectral methods and for the computation of
positive solutions.Comment: 19 pages, 41 figure
The Mathematical Analysis of a Micro Scale Model for Lithium-Ion Batteries
In this dissertation we consider a quasi-linear elliptic-parabolic system of pdes arising in the modeling of Li-ion batteries. We establish local existence of weak solutions by exploiting the properties of the elliptic subproblem. In three or less spatial dimensions we obtain uniqueness. As auxillary results we discuss two finite element discretizations of the elliptic subproblem and present some numerical results
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