In this article the authors present a new solver for the Poisson problem on spherical shells. The method is derived by application of the Fast Fourier Transform (FFT) in all three spherical variables. Through a particular change of variables valid on spherical shells, this new approach avoids a variable coefficient on the radial differential operator, resulting in a constant-coefficient formulation that can be fast-diagonalized via FFT. The authors describe the Fourier expansion in the angular variables and the fast diagonalization of the radial derivative using the FFT, including second-, fourth- and sixth-order finite difference approximations. An algorithmic description of the sixth-order solver is described, and numerical tests that demonstrate the theoretical rate of convergence are presented.

We propose a simple and efficient finite-difference method for the incompressible MHD equation. The numerical method combines the advantage of the MAC scheme for the Navier–Stokes equation and Yee's scheme for the Maxwell equation. In particular, the semi-discrete version of our scheme introduces no numerical dissipation and preserves the energy identity exactly.

In this paper, we propose a time-frequency analysis method to obtain instantaneous frequencies and the corresponding decomposition by solving an optimization problem. In this optimization problem, the basis that is used to decompose the signal is not known a priori. Instead, it is adapted to the signal and is determined as part of the optimization problem. In this sense, this optimization problem can be seen as a dictionary adaptation problem, in which the dictionary is adaptive to one signal rather than a training set in dictionary learning. This dictionary adaptation problem is solved by using the augmented Lagrangian multiplier (ALM) method iteratively. We further accelerate the ALM method in each iteration by using the fast wavelet transform. We apply our method to decompose several signals, including signals with poor scale separation, signals with outliers and polluted by noise and a real signal. The results show that this method can give accurate recovery of both the instantaneous frequencies and the intrinsic mode functions.

In this paper, we consider the time dependent Maxwell’s equations in dispersive media on a bounded three-dimensional domain. Global superconvergence is obtained for semi-discrete mixed finite element methods for three most popular dispersive media models: the isotropic cold plasma, the one-pole Debye medium, and the two-pole Lorentz medium. Global superconvergence for a standard finite element method is also presented. To our best knowledge, this is the first superconvergence analysis obtained for Maxwell’s equations when dispersive media are involved.

In this paper, we develop a mass conservative multiscale method for coupled flow and transport in heterogeneous porous media. We consider a coupled system consisting of a convection-dominated transport equation and a flow equation. We construct a coarse grid solver based on the Generalized Multiscale Finite Element Method (GMsFEM) for a coupled system. In particular, multiscale basis functions are constructed based on some snapshot spaces for the pressure and the concentration equations and some local spectral decompositions in the snapshot spaces. The resulting approach uses a few multiscale basis functions in each coarse block (for both the pressure and the concentration) to solve the coupled system. We use the mixed framework, which allows mass conservation. Our main contributions are:(1) the development of a mass conservative GMsFEM for the coupled flow and transport;(2) the development of a robust multiscale method for convection-dominated transport problems by choosing appropriate test and trial spaces within Petrov-Galerkin mixed formulation. We present numerical results and consider several heterogeneous permeability fields. Our numerical results show that with only a few basis functions per coarse block, we can achieve a good approximation. View Full-Text