The multigrid method solves the finite element equations in optimal order, i.e., solving a linear system of O(N) equations in O(N) arithmetic operations. Based on low level solutions, we can use finite element extrapolation to obtain the high-level finite element solution on some coarse-level element boundary, at an higher accuracy O(h^4_i). Thus, we can solve higher level (h_j, j\lesssim2i) finite element problems locally on each such coarse-level element. That is, we can skip the finite element problem on middle levels, h_{i+1}, h{_i+2},..., h_{j-1}. Loosely speaking, this jumping multigrid method solves a linear system of O(N) equations by a memory of O(p\sqrt{N}), and by a parallel computation of O(p\sqrt{N}).

Based on the structure-preserving characteristics of the symplectic algorithm, the wave propagation problem of auxetic cellular structures is analyzed and compared with conventional cellular materials. The dispersion relations along the first Brillouin zone boundary and the contour plots of phase constant surfaces are obtained using the finite element method. Numerical results reveal the superiority of auxetic re-entrant honeycombs in sound reduction applications compared with conventional hexagon lattices. Band structures of the chiral honeycomb are developed to illustrate the unique feature of the symplectic algorithm at higher frequencies calculation. The highly-directional wave propagation properties of auxetic cellular structures are also analyzed, which will provide invaluable guidelines for the future application of auxetic cellular structures in sound insulation.

IMplicit Pressure Explicit Saturation (IMPES) scheme with treating buoyancy and capillary forces is used to solve the two-phase water-CO_2 flow problem. In most of the previous studies of the two-phase flow, the buoyancy force term was ignored; however, in the case of liquidgas systems such as water-CO_2, the gravity term is very important to express the buoyancy effect. In this paper, we present three numerical examples to study the CO_2 plume in homogeneous, layered, and fractured porous media. In each numerical example, we tested four different models by ignoring both gravity and capillary pressure, considering only gravity, considering only capillary pressure, and considering both gravity and capillary pressure. The cell-centered finite difference (CCFD) method is used to discretize the problem under consideration. Furthermore, we also present the stability analysis of the IMPES scheme. The numerical results demonstrate the effects of the gravity and the capillary pressure on the flow for the four different cases.

We derive localized pointwise error estimates for Nitsche's method applied to an elliptic second order problem in R^n (n=2,3). Using these results, we also prove quasi-optimal global L^p error estimates as well. Numerical experiments are provided which back up the theoretical findings.

We study a finite element method for the 3D Navier-Stokes equations in velocity-vorticity-helicity formulation, which solves directly for velocity, vorticity, Bernoulli pressure and helical density. Moreover, the algorithm strongly enforces solenoidal constraints on both the velocity (to enforce the physical law for conservation of mass) and vorticity (to enforce the mathematical law that div(curl)= 0). We prove unconditional stability of the velocity, and with the use of a (consistent) penalty term on the difference between the computed vorticity and curl of the computed velocity, we are also able to prove unconditional stability of the vorticity in a weaker norm. Numerical experiments are given that confirm expected convergence rates, and test the method on a benchmark problem.

Youth gang activity is on the rise again since declining from its peak in the early 2000s. We focus on contagion of youth gang membership and delinquency among adolescents who are at risk to peer pressure by creating an epidemiological model with differential equations. The model seeks to examine dynamics of the system through stability analysis. A tipping point that spreads youth gang activity is identified, and sensitivity analysis on the threshold condition is performed to discuss the prevention strategies and the effectiveness of juvenile arrest and sanction. All parameters are approximated and results are also exploited by simulations. Analysis indicates that the system is most sensitive to prevention, early intervention and an effective juvenile system with treatment and rehabilitation.

This paper is concerned with a class of nonlocal Fisher-KPP type reaction-diffusion equations in n-dimensional space with time-delay. It is proved that, all noncritical planar wavefronts are exponentially stable in the form of t^{-\frac{n}{2}}e^{-ν_τt} for some constant ν_τ=ν(τ)> 0, where τ≥ 0 is the time-delay, while the critical planar wavefronts are algebraically stable in the form of t^{-\frac{n}{2}}. These convergent rates are optimal in the sense with L^1-initial perturbation. The adopted approach is the weighted energy method combining Fourier transform. It is also realized that, the effect of time-delay essentially causes the decay rate of the solution slowly down. These results significantly generalize and develop the existing study [37] for 1-D time-delayed Fisher-KPP type reaction-diffusion equations. When the time-delay τ vanishes, we automatically obtain the exponential stability for the noncritical planar traveling waves and the algebraic stability for the critical planar traveling waves to the regular Fisher-KPP equations.

This paper deals with monotone relaxation iterates for solving nonlinear monotone difference schemes of elliptic type. The monotone ω-Jacobi and SUR (Successive Under-Relaxation) methods are constructed. The monotone methods solve only linear discrete systems at each iterative step and converge monotonically to the exact solution of the nonlinear monotone difference schemes. Convergent rates of the monotone methods are estimated. The proposed methods are applied to solving semilinear singularly perturbed reaction-diffusion problems. Uniform convergence of the monotone methods is proved. Numerical experiments complement the theoretical results.

We present the dynamic instability of smooth compactly supported stationary solutions to the nonlinear Vlasov equations with self-consistent attractive forces. For this, we explicitly construct a one-parameter family of perturbed solutions via the method of the Galilean boost. Initially, these perturbations can be close to the given stationary solution as much as possible in any L^p-norm, p∈[1, ∞], and have the same local mass density profile as a stationary solution, but a different bulk velocity profile. At the macroscopic level, these perturbations correspond to the traveling waves with compact supports. However in finite-time, the phase-space supports of these perturbations will be disjoint from the support of the given stationary solution. This establishes the dynamic instability of stationary solutions in any L^p-norm.