Stabilized mixed finite element methods for the Navier‐Stokes equations with damping

In this paper, the stabilized mixed finite element methods are presented for the Navier‐Stokes equations with damping. The existence and uniqueness of the weak solutions are proven by use of the Brouwer fixed‐point theorem. Then, optimal error estimates for the H¹‐norm and L²‐norm of the velocity and the L²‐norm of the pressure are derived. Moreover, on the basis of the optimal L²‐norm error estimate of the velocity, a stabilized two‐step method is proposed, which is more efficient than the usual stabilized methods. Finally, two numerical examples are implemented to confirm the theoretical analysis.     

On the finite element approximation of elliptic QVIs with noncoercive operators

In this paper, we extend the approach developed by the author for the standard finite element method in the L^∞‐norm of the noncoercive variational inequalities (VI) (Numer Funct Anal Optim.2015;36:1107‐1121.) to impulse control quasi‐variational inequality (QVI). We derive the optimal error estimate, combining the so‐called Bensoussan‐Lions Algorithm and the concept of subsolutions for VIs.     

Multiscale finite element discretizations based on local defect correction for the biharmonic eigenvalue problem of plate buckling

In this paper, we study the multiscale finite element discretizations about the biharmonic eigenvalue problem of plate buckling. On the basis of the work of Dai and Zhou (SIAM J. Numer. Anal. 46[1] [2008] 295‐324), we establish a three‐scale scheme, a multiscale discretization scheme, and the associated parallel version based on local defect correction. We first prove a local priori error estimate of finite element approximations, then give the error estimates of multiscale discretization schemes. Theoretical analysis and numerical experiments indicate that our schemes are suitable and efficient for eigenfunctions with local low smoothness.     

Singular solutions of the Poisson equation at edges of three‐dimensional domains and their treatment with a predictor–corrector finite element method

Solutions of boundary value problems in three‐dimensional domains with edges may exhibit singularities which are known to influence both the accuracy of the finite element solutions and the rate of convergence in the error estimates. This paper considers boundary value problems for the Poisson equation on typical domains Ω ? ?³ with edge singularities and presents, on the one hand, explicit computational formulas for the flux intensity functions. On the other hand, it proposes and analyzes a nonconforming finite element method on regular meshes for the efficient treatment of the singularities. The novelty of the present method is the use of the explicit formulas for the flux intensity functions in defining a postprocessing procedure in the finite element approximation of the solution. A priori error estimates in H¹(Ω) show that the present algorithm exhibits the same rate of convergence as it is known for problems with regular solutions.     

An efficient spectral method and rigorous error analysis based on dimension reduction scheme for fourth order problems

In this paper, we propose an effective spectral method based on dimension reduction scheme for fourth order problems in polar geometric domains. First, the original problem is decomposed into a series of one‐dimensional fourth order problems by polar coordinate transformation and the orthogonal properties of Fourier basis function. Then the weak form and the corresponding discrete scheme of each one‐dimensional fourth order problem are derived by introducing polar conditions and appropriate weighted Sobolev spaces. In addition, we define the projection operators in the weighted Sobolev space and give its approximation properties, and further prove the error estimation of each one‐dimensional fourth order problem. Finally, we provide some numerical examples, and the numerical results show the effectiveness of our algorithm and the correctness of the theoretical results.     

Discontinuous Galerkin method for the coupled Stokes‐Biot model

In this paper, a semi‐discrete scheme and a fully discrete scheme of the Stokes‐Biot model are proposed, and we analyze the semi‐discrete scheme in detail. First of all, we prove the existence and uniqueness of the semi‐discrete scheme, and a‐priori error estimates are derived. Then, we present the same conclusions for the fully discrete scheme. Finally, under both matching and non‐matching meshes some numerical tests are given to validate the analysis of convergence, which well support the theoretical results.     

The Virtual Element Method for an Elliptic Hemivariational Inequality with Convex Constraint

An abstract framework of numerical method is devised for solving an elliptic hemivariational inequality with convex constraint. Convergence of the method is explored under the minimal solution regularity available from the well-posedness of the hemivariational inequality. A Céa-type inequality is derived for error estimation. As a typical example, a virtual element method is proposed to solve a frictionless unilateral contact problem and its optimal error estimates are obtained as well. Numerical results are reported to show the performance of the proposed method.     

向后迭代区间选择算法及其在近红外光谱模型转移中的应用

近红外光谱（NIR）具有快速、无损、操作方便的特点,故广泛用于食品分析。作为一种间接的分析技术,NIR需要建立光谱与待测浓度之间的统计模型来实现检测。故模型的维护有助于保证NIR的预测准确性。在外界条件发生变化的情况下,诸如样品性状的改变、仪器对理化指标函数关系的变化、湿度和温度等环境因素的改变,会导致相同样品的光谱信号发生偏移,进而使得原有模型的预测精度下降。此时,如果重新建模,虽然可以解决光谱偏移对建模的影响,但是重新建模将耗费大量的人力物力。对此,模型转移可以在避免重新建模的情况下,校正光谱的偏移,进而提高模型预测精度。通常模型转移算法多用全光谱进行模型转移,这种方法计算量较大,且不能找到合适的有化学意义的波段。故提出一种基于模型转移中的变量选择方法：向后迭代区间选择法（IIBS）,通过计算主光谱（用于建模的那组光谱）和从光谱（发生偏移,需要通过模型转移算法将其校正的光谱）中,变量区间的重要性信息（回归系数（β）、残差向量（Res）以及变量重要性投影（VIP））。进而通过计算该区间变量重要性信息的几何平均数,并以此作为该区间的区间重要性指标。接着根据区间的重要性,删除重要性信息较小的变量区间。然后对主光谱和从光谱重复迭代上述过程：计算变量的重要性信息,计算区间的重要性信息,删除重要性信息较小的区间。最后,比较不同的主光谱和从光谱区间组合的验证均方根误差（RMSEV）,选择RMSEV最小的主光谱和从光谱区间作为最优区间。玉米、小麦两套NIR数据测试了该算法。结果显示,与全波段相比,β,Res以及VIP均可以从主光谱和从光谱中选择较少的,有化学意义的区间,提高模型转移的精度。在比较不同变量重要性向量方面,基于β的变量选择算法,模型转移的计算误差较小。     

北京低边界层雨滴谱的垂直分布特征

基于安装在气象铁塔不同高度的3台Parsivel激光雨滴谱仪, 研究2020年6—8月北京地区31次降雨过程低边界层的雨滴谱垂直分布特征及其对雷达测量降雨的影响。结果表明, 在雨滴从280 m下落到140 m的过程中, 雨滴破碎占据主导地位, 大尺度雨滴减少, 小尺度雨滴大量增加; 从140 m下落到地面的过程中, 雨滴合并占据主导地位, 大尺度和小尺度雨滴均减少, 中等尺度雨滴增加; 对流型降雨中, 随着高度降低, 大尺度雨滴迅速减少, 雨滴破碎对雨滴谱分布的影响较大。北京地区对流型降雨的雨滴谱特征介于海洋型与大陆型之间。由于雨滴谱的分布随高度变化, 不同降雨特征量也随高度发生非线性变化, 因此使用地面的降雨测量结果对气象雷达进行校准以及联合反演降雨会产生Z-R错配误差, 这种误差在稳定型降雨和弱降雨中更显著。     

信道状态信息存在误差情况下不可信中继系统的功率分配

研究在目的节点进行协作干扰的全双工不可信中继传输系统中,在信道状态信息(channel state information, CSI)存在随机误差的情况下,以最大化系统可达保密速率为目标的功率分配问题。为了研究提高全双工不可信中继传输系统的保密性能的功率分配方案,先推导系统可达保密速率的上界,针对随机信道状态信息的误差再求该保密速率上界的期望,即平均保密速率(average secrecy rate, ASR)。在此基础上,考虑源节点发送信息信号和目的节点发送人工噪声总功率的约束,以最大化ASR为优化目标,通过黄金分割法优化源节点和目的节点的功率分配。仿真结果表明,相比于固定功率分配因子和不考虑随机信道系数误差影响的功率分配方案,考虑信道的随机误差优化功率分配方案能有效提高系统的可达保密速率。