We give here an overview of the orbital-free density functional
theory that is used for modeling atoms and molecules. We review
typical approximations to the kinetic energy, exchange-correlation
corrections to the kinetic and Hartree energies, and constructions
of the pseudopotentials. We discuss numerical discretizations for
the orbital-free methods and include several numerical results for
illustrations.
We prove convergence for a meshfree first-order system least squares (FOSLS)
partition of unity finite element method (PUFEM). Essentially, by virtue of the partition
of unity, local approximation gives rise to global approximation in $\mathrm{H}(div)\cap\mathrm{H}(curl)$.
The FOSLS formulation yields local a posteriori error estimates to guide the judicious
allotment of new degrees of freedom to enrich the initial point set in a meshfree discretization. Preliminary numerical results are provided and remaining challenges are
discussed.
We discuss estimates for the rate of convergence of the method of successive
subspace corrections in terms of condition number estimate for the method of parallel
subspace corrections. We provide upper bounds and in a special case, a lower bound
for preconditioners defined via the method of successive subspace corrections.
In this paper, we are interested in HSS preconditioners for saddle
point linear systems with a nonzero $(2,2)$-th block. We study an
approximation of the spectra of HSS preconditioned matrices and
use these results to illustrate and explain the spectra obtained
from numerical examples, where the previous spectral analysis of
HSS preconditioned matrices does not cover.
In this paper we investigate several solution algorithms for the convex feasibility problem (CFP) and the best approximation problem (BAP) respectively. The
algorithms analyzed are already known before, but by adequately reformulating the
CFP or the BAP we naturally deduce the general projection method for the CFP from
well-known steepest decent method for unconstrained optimization and we also give a
natural strategy of updating weight parameters. In the linear case we show the connection of the two projection algorithms for the CFP and the BAP respectively. In addition,
we establish the convergence of a method for the BAP under milder assumptions in the
linear case. We also show by examples a Bauschke's conjecture is only partially correct.
We propose an efficient and robust algorithm to solve the steady
Euler equations on unstructured grids. The new algorithm is a
Newton-iteration method in which each iteration step is a linear
multigrid method using block lower-upper symmetric Gauss-Seidel
(LU-SGS) iteration as its smoother. To regularize the Jacobian
matrix of Newton-iteration, we adopted a local residual dependent
regularization as the replacement of the standard time-stepping
relaxation technique based on the local CFL number. The proposed
method can be extended to high order approximations and three
spatial dimensions in a nature way. The solver was tested on a
sequence of benchmark problems on both quasi-uniform and local
adaptive meshes. The numerical results illustrated the efficiency
and robustness of our algorithm.
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