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.