Motivated by the above results, the third part of this paper considers the equivalence problems that every stationary point or kuhn-tucker point is an efficient solution . we define i-quasi-invex vector function ., i-strictly quasi-invex vector function and kt-i-strictly quasi invex vector function, and derive the above equivalent condition for unconstrained or constrained multiobjective programming 于是,在本文的第三部分,我們定義了類不變擬凸、類嚴格不變擬凸、kt-類嚴格不變擬凸的向量值函數(shù),并且在無約束或約束多目標規(guī)劃中,獲得了每個駐點(或k-t點)是有效解的等價條件。
Motivated by the above results, the third part of this paper considers the equivalence problems that every stationary point or kuhn-tucker point is an efficient solution . we define i-quasi-invex vector function ., i-strictly quasi-invex vector function and kt-i-strictly quasi invex vector function, and derive the above equivalent condition for unconstrained or constrained multiobjective programming 于是,在本文的第三部分,我們定義了類不變擬凸、類嚴格不變擬凸、kt-類嚴格不變擬凸的向量值函數(shù),并且在無約束或約束多目標規(guī)劃中,獲得了每個駐點(或k-t點)是有效解的等價條件。
The characteristic approximation is used to handle the convection part along the direc-tion of fluid namely characteristic direction to ensure the high stability of the method in approximating the sharp fronts and reduce the numerical diffusion; the mixed finite element spatial approximation is employed to deal with diffusion part and approximate the scalar unknown and the adjoint vector function optimally and simultaneously; in order to preserve the integral conservation of the method, we introduce the modified characteristic method 該方法對方程的對流部分沿流體流動的方向即特征方向離散以保證格式在流動的鋒線前沿逼近的高穩(wěn)定性,消除數(shù)值彌散現(xiàn)象;對方程的擴散部分采用最低次混合有限元方法離散、同時以高精度逼近未知函數(shù)及未知函數(shù)的梯度;為保證方法的整體守恒性,在格式中引入修正項
Using the regularized greens functions and a duality argument, it is proved that the mixed finite element method proposed in this paper possesses the superconvergence by almost one order maximum norm estimates for the l2 projection of the function and quasi-optimal maximum norm estimates for the associated vector function for a strongly nonlinear second order elliptic problem 本文利用正規(guī)格林函數(shù)及對偶論證技術(shù)證明了一類強非線性二階橢圓問題混合元方法對函數(shù)的l2投影具有幾乎超收斂一階的最大模誤差估計,對伴隨向量函數(shù)具有擬最優(yōu)最大模誤差估計
The new method is a combination of characteristic approximation to handle the convection part, to ensure the high stability of the method in approximating the sharp fronts and reduce the numerical diffusion, a smaller time truncation is gained at the same time, and a mixed finite element spatial approximation to deal with the diffusion part, the sealer unknown and the adjoint vector function are approximated optimally and simultaneously 此方法即為對方程的對流項沿流體流動的方向即特征方向進行離散,從而保證格式在流動鋒線前沿逼近的高穩(wěn)定性,消除了數(shù)值彌散現(xiàn)象,并得到了較小的時間截斷誤差;另一方面,對方程的擴散項采用混合元離散,可同時高精度逼近未知函數(shù)及其伴隨向量函數(shù),理論分析表明,此方法是穩(wěn)定的,具有最優(yōu)的l~2逼近精度。