A New Approach for Error Reduction in the Volume Penalization Method

A new approach for reducing error of the volume penalization method is proposed.The mask function is modified by shifting the interface between solid and fluid by√νηtoward the fluid region,whereνandηare the viscosity and the permeability,respectively.The shift length√νηis derived from the analytical solution of the one-dimensional diffusion equation with a penalization term.The effect of the error reduction is verified numerically for the one-dimensional diffusion equation,Burgers’equation,and the two-dimensional Navier-Stokes equations.The results show that the numerical error is reduced except in the vicinity of the interface showing overall second-order accuracy,while it converges to a non-zero constant value as the number of grid points increases for the original mask function.However,the new approach is effective when the grid resolution is sufficiently high so that the boundary layer,whose width is proportional to√νη,is resolved.Hence,the approach should be used when an appropriate combination ofνandηis chosen with a given numerical grid.

MULTIPLICITY AND CONCENTRATION BEHAVIOUR OF POSITIVE SOLUTIONS FOR SCHRDINGER-KIRCHHOFF TYPE EQUATIONS INVOLVING THE p-LAPLACIAN IN R^N

In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of SchrSdinger-Kirchhoff type -εpM（εp-N∫RN｜△u｜p）△pu＋v（x｜u｜p-2u=f（u）in RN, where △p is the p-Laplacian operator, 1 〈 p 〈 N, M ： R＋ → R＋ and V ： RN →R＋ are continuous functions, ε is a positive parameter, and f is a continuous function with subcritical growth. We assume that V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and Lyusternik- Schnirelmann theory, we prove the existence, multiplicity, and concentration of solutions for the above equation.

Entropy Unilateral Solution for Some Noncoercive Nonlinear Parabolic Problems Via a Sequence of Penalized Equations

We give an existence result of the obstacle parabolic equations3b（x,u） div（a（x,t,u, Vu））＋div（（x,t,u））=f in QT, 3twhere b（x,u） is bounded function ot u, the term atva,x,r,u, v u）） is a Letay type operator and the function is a nonlinear lower order and satisfy only the growth condition. The second term belongs to L1 （QT）. The proof of an existence solution is based on the penalization methods.

Influence of wing flexibility on the aerodynamic performance of a tethered flapping bumblebee

The sophisticated structures of flapping insect wings make it challenging to study the role of wing flexibility in insect flight.In this study,a mass-spring system is used to model wing structural dynamics as a thin,flexible membrane supported by a network of veins.The vein mechanical properties can be estimated based on their diameters and the Young's modulus of cuticle.In order to analyze the effect of wing flexibility,the Young's modulus is varied to make a comparison between two different wing models that we refer to as flexible and highly flexible.The wing models are coupled with a pseudo-spectral code solving the incompressible Navier–Stokes equations,allowing us to investigate the influence of wing deformation on the aerodynamic efficiency of a tethered flapping bumblebee.Compared to the bumblebee model with rigid wings,the one with flexible wings flies more efficiently,characterized by a larger lift-to-power ratio.

THE MULTIPLICITY AND CONCENTRATION OF POSITIVE SOLUTIONS FOR THE KIRCHHOFF-CHOQUARD EQUATION WITH MAGNETIC FIELDS

In this paper,we study the multiplicity and concentration of positive solutions for the following fractional Kirchhoff-Choquard equation with magnetic fields:(aε^(2s)+bε^(4 s-3)[u]_(ε)^(2),A/ε)(-Δ)_(A/ε)^(s)u+V(x)u=ε^(-α)(Iα*F(|u|^(2)))f(|u|^(2))u in R^(3).Hereε>0 is a small parameter,a,b>0 are constants,s E(0,1),(-Δ)As is the fractional magnetic Laplacian,A:R^(3)→R^(3) is a smooth magnetic potential,Iα=Γ(3-α/2)/2απ3/2Γ(α/2)·1/|x|^(α) is the Riesz potential,the potential V is a positive continuous function having a local minimum,and f:R→R is a C^(1) subcritical nonlinearity.Under some proper assumptions regarding V and f,we show the multiplicity and concentration of positive solutions with the topology of the set M:={x∈R^(3):V(x)=inf V}by applying the penalization method and LjusternikSchnirelmann theory for the above equation.

Optimal obstacle control problem

In the paper we discuss some properties of the state operators of the optimal obstacle control problem for elliptic variational inequality.Existence,uniqueness and regularity of the optimal control,problem are established.In addition,the approximation of the optimal obstacle problem is also studied.

HERMITE SCATTERED DATA FITTING BY THE PENALIZED LEAST SQUARES METHOD

Given a set of scattered data with derivative values. If the data is noisy or there is an extremely large number of data, we use an extension of the penalized least squares method of von Golitschek and Schumaker [Serdica, 18 （2002）, pp.1001-1020] to fit the data. We show that the extension of the penalized least squares method produces a unique spline to fit the data. Also we give the error bound for the extension method. Some numerical examples are presented to demonstrate the effectiveness of the proposed method.