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Based on the Boolean sum technique, we introduce and analyze in this paper a class of multi-level iterative corrections for finite dimensional approximations. This type of multi-level corrections is adaptive and can produce highly accurate approximations. For illustration, we present some old and new finite element correction schemes for an elliptic boundary value problem.

A two-grid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations. With this new scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid, and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.

This paper studies a simple postprocessing of two-scale finite element discretizations of elliptic partial differential operators, where boundary value problems and eigenvalue problems on tensor-product domains Ω ⊂ ℝ d are both examined and conforming piecewise d-linear elements are used. For d ≥ 2, suppose that the two-scale method is applied using a coarse mesh and some univariate fine meshes satisfying H = O(h²/³), where h and H are the fine and coarse mesh widths. It is shown for both classes of problem that the postprocessed two-scale finite element solution achieves the same order of accuracy in the H¹(Ω)norm as the postprocessed standard finite element solution, but the former solution uses only O(h− (2d+1)/3 ) degrees of freedom compared with the O(h −d ) degrees of freedom required by the latter.

To produce highly accurate approximations for self-adjoint elliptic eigenvalues, we propose a finite element defect correction scheme. Its efficiency is proven by theoretical and numerical evidence. Very satisfying applications in computational quantum chemistry are also presented.

In this paper, some two-scale Boolean Galerkin discretizations are proposed and analyzed for a class of Fredholm integral equations of the second kind in multidimensions. It is shown by both theory and numerics that this type of multiscale discretization algorithm not only significantly reduces the number of degrees of freedom but also produces very accurate approximations.

Based on globally and locally coupled discretizations, some three-scale finite element schemes are proposed in this paper for a class of quantum eigenvalue problems. It is shown that the solution of a quantum eigenvalue problem on a fine grid may be reduced to the solution of an eigenvalue problem on a relatively coarse grid, and the solutions of linear algebraic systems on a globally mesoscopic grid and the locally fine grid, and the resulting solution is still very satisfactory.

The density functional theory (DFT) in electronic structure calculations can be formulated as either a nonlinear eigenvalue or a direct minimization problem. The most widely used approach for solving the former is the so-called self-consistent field (SCF) iteration. A common observation is that the convergence of SCF is not clear theoretically, while approaches with convergence guarantees for solving the latter are often not competitive with SCF numerically. In this paper, we study gradient type methods for solving the direct minimization problem by constructing new iterations along the gradient on the Stiefel manifold. Global convergence rate (i.e., convergence to a stationary point from any initial solution) as well as local convergence rate can be obtained from the standard theory for gradient methods on manifold directly. A major computational advantage is that the computation of linear eigenvalue problems is no longer needed. The main costs of our approaches arise from the assembling of the total energy functional and its gradient on the manifold and the projection onto the manifold. These tasks are cheaper than eigenvalue computation and are often more suitable for parallelization as long as the evaluation of the total energy functional and its partial derivative is efficient. Numerical results show that they can outperform SCF consistently on many practical large systems. [PUBLICATION ABSTRACT]

The Kohn--Sham model is a powerful, widely used approach for computation of ground state electronic energies and densities in chemistry, materials science, biology, and nanoscience. In this paper, we study 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 a Dorfler marking strategy, we then get the convergence rate and quasi-optimal complexity. Moreover, we demonstrate 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.

In this article, we propose a two-grid based adaptive proper orthogonal decomposition (POD) method to solve the time dependent partial differential equations. Based on the error obtained in the coarse grid, we propose an error indicator for the numerical solution obtained in the fine grid. Our new method is cheap and easy to be implement. We apply our new method to the solution of time-dependent advection–diffusion equations with the Kolmogorov flow and the ABC flow. The numerical results show that our method is more efficient than the existing POD methods.

Some high accuracy finite element methods for hyperbolic problems are studied in this paper. It is proven that over a finite element mesh, the convergence order of linear finite element solutions for both linear and nonlinear equations can be higher than one even if the exact solutions are discontinuous. The theoretical tools for the convergence analysis are some superclose error estimates that are also developed in this paper for nonsmooth solutions.