A two-grid algorithm is presented and discussed for a finite volume element method to a nonlinear parabolic equation in a convex polygonal domain. The two-grid algorithm consists of solving a small nonlinear system on a coarse-grid space with grid size $H$ and then solving a resulting linear system on a fine-grid space with grid size $h$. Error estimates are derived with the $H^1$-norm $O(h+H^2)$ which shows that the two-grid algorithm achieves asymptotically optimal approximation as long as the mesh sizes satisfy $h=O(H^2)$. Numerical examples are presented to validate the usefulness and efficiency of the method.

An efficient time second-order characteristic finite element method for solving the nonlinear multi-component aerosol dynamic equations is developed. While a highly accurate characteristic method is used to treat the advection multi-component condensation/evaporation process, a time high-order extrapolation along the characteristics is applied to approximate the nonlinear multi-component coagulation terms. The scheme is of second order accuracy in time for the multi-component problems. We study the theoretical analysis and obtain the time second-order error estimate of the scheme. Numerical experiments are further given to confirm the theoretical results. The dynamic behaviours of multi-component aerosol distributions are also simulated for the multi-component aerosol problems of aerosol water, black carbon and sulfate components with different tri-modal log-normal initial distributions.

In this paper, we propose and analyze a linear time stepping finite element method for abstract second order linear evolution problems. For such methods, we derive optimal order a posteriori error estimates and sharp a posteriori nodal error estimates using the energy approach and the duality argument. Based on these estimates, we further design an adaptive time stepping strategy for the previous discretization in time. Several numerical experiments are provided to show the reliability and efficiency of the a-posteriori error estimates and to assess the effectiveness of the proposed adaptive time stepping method.

Dispersion properties of Rayleigh-type surface waves are widely used in environmental and engineering geophysics to image and characterize a shallow subsurface. In this paper, we numerically study the Rayleigh-type surface waves and their properties in 2D viscoelastic media. A finite difference method in a time-space domain is proposed, with an unsplit convolutional perfectly matched layer (C-PML) absorbing boundary condition. For two models that have analytical expressions of wave fields/dispersion curves, we calculate their wave fields and compare the analytical and numerical solutions to demonstrate the validity of this method. For the case where a medium has a high Poisson's ratio, say 0.49, traditional finite difference methods with a PML boundary condition are not stable when modeling Rayleigh waves but the proposed method is stable. For a laterally heterogeneous viscoelastic media model (Model 1) and a two-layer viscoelastic media model (Model 2) with a cavity, we use this method to obtain their corresponding Rayleigh waves. For several quality factors, the dispersion properties of these Rayleigh waves are analyzed. The results of Model 1 show that in a shallow subsurface, the phase velocity of a fundamental mode of the Rayleigh waves increases considerably with a quality factor $Q$ decreasing; the phase velocity increases with Poisson's ratio increasing. The results of Model 2 indicate that the energy of higher modes of the Rayleigh waves become strong when $Q$ decreases.

We numerically investigate the dynamics of drop formation when a Newtonian fluid is injected through a tube into another immiscible, co-flowing Newtonian fluid with different density and viscosity using the phase field method. The two phase system is modeled by a coupled three dimensional Cahn-Hilliard and Navier-Stokes equation in cylindrical coordinates. And the contribution from the chemical potential has been taken into account in the classical Navier-Stokes equation. The numerical method involves a convex splitting scheme for the Cahn-Hilliard equation and a projection type scheme for the momentum equation. Our study of the dynamics of the drop formation is motivated by the experimental work by Utada et al [Phys. Rev. Lett. 99(2007), 094502] on dripping and jetting transition. The simulation results demonstrate that the process of drop formation can be reasonably predicated by the phase field model we used. Our simulations also identify two classes of dripping to jetting transition, one controlled by the Capillary number of the outer fluid and another one controlled by the Weber number of the inner fluid. The results match well with the experimental results in Utada et al [A. S. Utada, A. Fernandez-Nieves, H. A. Stone, and D. A. Weitz, Phys. Rev. Lett. 99(2007), 094502] and Zhang [Chem. Eng. Sci. 54(1999), 1759-1774]. We also study how the dynamics of the drop formation depends on the various physical parameters of the system. Similar behaviors with existing results are obtained for most parameters, yet different behavior is observed for density ratio $\lambda_{\rho}$ and viscosity ratio $\lambda_{\eta}$.

The size-modified Poisson-Boltzmann equation (SMPBE) has been developed to consider ionic size effects in the calculation of electrostatic potential energy, but its numerical solution for a biomolecule remains a challenging research issue. To address this challenge, in this paper, we propose a solution decomposition formula and then develop an effective minimization protocol for solving the nonlinear SMPBE model by using finite element approximation techniques. As an application, a particular SMPBE numerical algorithm is constructed and programmed as a finite element program package for a biomolecule (e.g., protein and DNA) in a symmetric 1:1 ionic solvent. We also construct a nonlinear SMPBE ball model with an analytical solution and use it for validation of our new SMPBE numerical algorithm and program package. Furthermore, numerical experiments are made on a central charged ball model to show some physical features captured by the SMPBE model. Finally, they are made for six biomolecules with different net charges to demonstrate the computer performance of our SMPBE finite element program package. Numerical results show that the SMPBE model can capture some physical properties of an ionic solvent more reasonably, and can be solved more efficiently than the classic PBE model. As an application of the SMPBE model, free solvation energies were calculated and compared to the case of the PBE model.

This paper presents a hybridized formulation for the weak Galerkin finite element method for the biharmonic equation based on the discrete weak Hessian recently proposed by the authors. The hybridized weak Galerkin scheme is based on the use of a Lagrange multiplier defined on the element interfaces. The Lagrange multiplier is verified to provide a numerical approximation for certain derivatives of the exact solution. An error estimate of optimal order is established for the numerical approximations arising from the hybridized weak Galerkin finite element method. The paper also derives a computational algorithm (Schur complement) by eliminating all the unknowns associated with the interior variables on each element, yielding a significantly reduced system of linear equations for unknowns on the element interfaces.

In this paper, some inequality relations between the Laplacian/signless Laplacian H-eigenvalues and the clique/coclique numbers of uniform hypergraphs are presented. For a connected uniform hypergraph, some tight lower bounds on the largest Laplacian $H^+$-eigenvalue and signless Laplacian H-eigenvalue related to the clique/coclique numbers are given. And some upper and lower bounds on the clique/coclique numbers related to the largest Laplacian/signless Laplacian H-eigenvalues are obtained. Also some bounds on the sum of the largest/smallest adjacency/Laplacian/signless Laplacian H-eigenvalues of a hypergraph and its complement hypergraph are showed. All these bounds are consistent with what we have known when $k$ is equal to 2.

In this paper we consider the discrete constrained least squares problem coming from numerical approximation by hybrid scheme on the sphere, which applies both radial basis functions and spherical polynomials. We propose a novel $l_2-l_1$ regularized least square model for this problem and show that it is a generalized model of the classical "saddle point" model. We apply the alternating direction algorithm to solve the $l_2-l_1$ model and propose a convenient stopping criterion for the algorithm. Numerical results show that our model is more efficient and accurate than other models.

We consider time domain formulations of Maxwell's equations for the Lorentz model for metamaterials. The field equations are considered in two different forms which have either six or four unknown vector fields. In each case we use arguments tuned to the physical laws to derive data-stability estimates which do not require Gronwall's inequality. The resulting estimates are, in this sense, sharp. We also give fully discrete formulations for each case and extend the sharp data-stability to these. Since the physical problem is linear it follows (and we show this with examples) that this stability property is also reflected in the constants appearing in the a priori error bounds. By removing the exponential growth in time from these estimates we conclude that these schemes can be used with confidence for the long-time numerical simulation of Lorentz metamaterials.

We develop in this paper an efficient and robust spectral-Galerkin method for solving the three-dimensional time-harmonic Maxwell equations in exterior domains. We first reduce the problem to a bounded domain by using the capacity operator which characterizes the transparent boundary condition (TBC). Then, we adopt the transformed field expansion (TFE) approach to reduce the problem to a sequence of Maxwell equations in a spherical shell. Finally, we develop an efficient spectral algorithm by using Legendre approximation in the radial direction and vector spherical harmonic expansion in the tangential directions.

In this paper, we first consider the numerical method that Lin and Xu proposed and analyzed in [Finite difference/spectral approximations for the time-fractional diffusion equation, JCP 2007] for the time-fractional diffusion equation. It is a method based on the combination of a finite different scheme in time and spectral method in space. The numerical analysis carried out in that paper showed that the scheme is of $(2-\alpha)$-order convergence in time and spectral accuracy in space for smooth solutions, where $\alpha$ is the time-fractional derivative order. The main purpose of this paper consists in refining the analysis and providing a sharper estimate for both time and space errors. More precisely, we improve the error estimates by giving a more accurate coefficient in the time error term and removing the factor in the space error term, which grows with decreasing time step. Then the theoretical results are validated by a number of numerical tests.