This paper detailedly discusses the locally one-dimensional numerical methods for efficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional diffusion equation. The second order finite difference scheme is used to discretize the space fractional derivative and the Crank-Nicolson procedure to the time derivative. We theoretically prove and numerically verify that the presented numerical methods are unconditionally stable and second order convergent in both space and time directions. In particular, for the Riesz fractional diffusion equation, the idea of reducing the splitting error is used to further improve the algorithm, and the unconditional stability and convergency are also strictly proved and numerically verified for the improved scheme.

In this paper, we investigate numerical methods for high order differential equations. We propose new spectral and spectral element methods for high order problems with mixed inhomogeneous boundary conditions, and prove their spectral accuracy by using the recent results on the Jacobi quasi-orthogonal approximation. Numerical results demonstrate the high accuracy of suggested algorithm, which also works well even for oscillating solutions.

We are concerned, in a static regime, with an imaging approach of the locations in a three-dimensional bounded domain of certain electromagnetic imperfections. This approach is related to Electrical Impedance Tomography and makes use of a new perturbation formula in the electric fields. We present two localization procedures, from a Current Projection method that deals with the single imperfection context and an inverse Fourier process that is devoted to multiple imperfections configurations. These procedures extend those that were described in our previous work, since operating for a broader class of settings. Namely, the localization is additionally performed for certain purely electric imperfections, as established from numerical simulations.

A new HSS-like iterative method is first proposed based on HSS-like splitting of non-Hermitian (1,1) block for solving saddle point problems. The convergence analysis for the new method is given. Meanwhile, we consider the solution of saddle point systems by preconditioned Krylov subspace method and discuss some spectral properties of the preconditioned saddle point matrices. Numerical experiments are given to validate the performances of the preconditioners.  

In this work, we solve a long-standing open problem: Is it true that the convergence rate of the Lions' Robin-Robin nonoverlapping domain decomposition (DD) method can be constant, independent of the mesh size $h?$ We closed this old problem with a positive answer. Our theory is also verified by numerical tests.

Based on the nonmonotone line search technique proposed by Gu and Mo (Appl. Math. Comput. 55, (2008) pp. 2158-2172), a new nonmonotone trust region algorithm is proposed for solving unconstrained optimization problems in this paper. The new algorithm is developed by resetting the ratio $ρ_k$ for evaluating the trial step $d_k$ whenever acceptable. The global and superlinear convergence of the algorithm are proved under suitable conditions. Numerical results show that the new algorithm is effective for solving unconstrained optimization problems.