稳态问题混合边界积分方程的高精度求积法与分裂外推

提出了求积法解稳态问题的混合边界积分方程,它拥有高精度,低复杂度.通过并行地解粗网格上的离散方程,根据误差的多参数渐近展开,应用分裂外推算法得到高精度的近似解,同时获得后验误差估计.     

第二类弱奇异积分方程的高精度Nystrom方法与外推

借助奇异函数的新型求积公式，建立了解第二类弱奇异积分方程的高精度Ｎｙｓｔｒｏｍ算法及渐近展开式。数值试验表明本文的算法较常用的投影法计算量大大减少，崦精度却很高，其外推法打丰了Ｆ．Ｃｈａｔｅｌｉｎ认为非光滑核积分方程的 解外推缺少理论依据的断定。     

定常Stokes问题的介积分方程的高精度求积方法与外推

借助Ｓｉｄｅ＝Ｉｓｒａｅｌｉ求积公式,给出解Ｓｔｏｋｅｓ问题的边界积分方程的机械求积方法。此方法精度较高、计算量少,近似解的误差有奇数幂的渐近展开,这表明使用Ｒｉｃｈａｒｄｏｎ外推能够改善近似解的精度阶。     

第二类边界积分方程Nystrom解的高精度组合方法

第二类边界积分方程常用配置法或Ｇａｌｅｒｋｉｎ法计算，主要困难有：计算积分耗去大量机时；离散方程是满阵且不对称，计算量随剖分精细而急剧增加。本文提出Ｎｙｓｔｒｏｍ近似解的高精度组合法能有效克服上述困难。组合方法是并行地解ｍ个具有ｎ个不同结点的方程组，对得到的ｍ个内点值取算术平均就得到了组合近似，本文证明组合近似精度几乎与解ｍｎ个结点近似方程达到精度同阶，数值结果表明本文方法简单、有效、并且算法高度     

弹性力学的重构核粒子边界无单元法

将重构核粒子法(RKPM)和边界积分方程方法结合，提出了一种新的边界积分方程无网格方法——重构核粒子边界无单元法(RKP-BEFM).对弹性力学问题，推导了其重构核粒子边界无单元法的公式，研究其数值积分方案，建立了重构核粒子边界无单元法离散化边界积分方程，并推导了重构核粒子边界无单元法的内点位移和应力积分公式.重构核粒子法形成的形函数具有重构核函数的光滑性，且能再现多项式在插值点的精确值，所以本方法具有更高的精度.最后给出了数值算例，验证了本方法的有效性和正确性. 关键词： 重构核粒子法 弹性力学 边界无单元法     

基于积分方程法和奇异值分解的磁性目标磁场延拓技术研究

对磁性目标磁场延拓技术进行了研究,提出了一种基于积分方程法和奇异值分解的新方法.应用该方法只需要采用积分方程法对磁性目标的结构进行较为粗略的单元划分,利用目标下方大平面上的磁场测量值,得到相应的线性方程组.采用基于奇异值分解的截断奇异值方法和修正奇异值方法对该线性方程组进行正则化求解,可实现磁性目标磁场的三维磁场重建、向上或向下延拓.该方法较以前的方法,提高了磁性目标磁场延拓的精度和可靠性,并且解决了磁性目标磁场在一定范围内向上延拓的技术难题. 关键词： 磁性目标磁场 延拓 积分方程法 奇异值分解     

The Collocation Method and the Splitting Extrapolation for the First Kind of Boundary Integral Equations on Polygonal Regions

In this paper, the collocation methods are used to solve the boundary integral equations of the first kind on the polygon. By means of Sidi's periodic transformation and domain decomposition, the errors are proved to possess the multi-parameter asymptotic expansion at the interior point with the powers $h_{i}^{3}(i=1,...,d)$, which means that the approximations of higher accuracy and a posteriori estimation of the errors can be obtained by splitting extrapolations. Numerical experiments are carried out to show that the methods are very efficient.     

Navier-Stokes方程的线性化微分求积法

提出一种新的线性化微分求积法(LDQM),将这种目的应用到流函数和涡量形式的Navier-Stokes方程.通过LDQM,非线性方程很容易被解出来,并且容易处理压力的边界条件.为检验本目的,计算了两个数值算例.     

移动边界问题的时空域正则区域迭代配点法

考虑热传导方程的移动边界问题,其定解区域随着时间而变化.构造一种时空域上的高精度数值算法求解1+1维移动边界问题.在时空域上假设一个初始移动边界位置,构成移动边界问题的不规则计算区域,选择一个适当的正则区域(矩形区域)完全覆盖所计算的不规则区域,在正则区域上利用移动边界约束条件和固定边界条件,采用时空域重心插值配点法求...     

三维位势问题边界元法中几乎奇异积分的正则化

采用一种半解析正则化算法,计算了三维位势问题边界元法中近边界点的几乎强奇异和几乎超奇异面积分.该算法适用于三角形线性等参元.对高次单元将其细分为几个三节点三角形单元即可应用该算法.由于几乎奇异性,与内点邻近的单元上的积分,采用半解析正则化积分算法计算;而远处单元的积分仍保持常规高斯积分.对三维热传导算例,计算了近边界点的温度和热流.数值结果证明了该算法的有效性和精确性.     

Equivalent a Posteriori Error Estimator of Spectral Approximation for Control Problems with Integral Control-State Constraints in One Dimension

In this paper, we investigate the Galerkin spectral approximation for elliptic control problems with integral control and state constraints. Firstly, an a posteriori error estimator is established, which can be acted as the equivalent indicator with explicit expression. Secondly, appropriate base functions of the discrete spaces make it probable to solve the discrete system. Numerical test indicates the reliability and efficiency of the estimator, and shows the proposed method is competitive for this class of control problems. These discussions can certainly be extended to two- and three-dimensional cases.     

A Set of Exact Solutions for Singular Integral Equation and a Divergence Pcreoblem of 1D Hydrogen Atom in Moment Represent

The divergence problem in momentum representation of some singular potentials in Schrodinger equation is discussed. A set of exact solutions for a singular integral equation and the criteria determining the solutions in momentum space are obtained. The problems on Loudon's symmetric solutions and counter example of nondegeneracy theorem are solved.     

A Note on the GMRES Method for Linear Discrete Ill-Posed Problems

In this paper, we are presenting a proposal for new modified algorithms for RRGMRES and AGMRES. It is known that RRGMRES and AGMRES are viable methods for solving linear discrete ill-posed problems. In this paper we have focused on the residual norm and have come up with two improvements where successive updates and the stabilization of decreases for the residual norm improve performance respectively. Our numerical experiments confirm that our improved algorithms are effective for linear discrete ill-posed problems.     

二维半线性扩散反应方程的高精度全隐格式及其多重网格方法

针对二维非定常半线性扩散反应方程，空间导数项采用四阶紧致差分公式离散，时间导数项采用四阶向后Euler公式进行离散，提出一种无条件稳定的高精度五层全隐格式.格式截断误差为O（τ⁴+τ²h²+h⁴），即时间和空间均具有四阶精度.对于第一、二、三时间层采用Crank-Nicolson方法进行离散，并采用Richardson外推公式将启动层时间精度外推到四阶.建立适用于该格式的多重网格方法，加快在每个时间层上迭代求解代数方程组的收敛速度，提高计算效率.最后通过数值实验验证格式的精确性和稳定性以及多重网格方法的高效性.     

A Posteriori Error Estimates of a Combined Mixed Finite Element and Discontinuous Galerkin Method for a Kind of Compressible Miscible Displacement Problems

A kind of compressible miscible displacement problems which include molecular diffusion and dispersion in porous media are investigated. The mixed finite element method is applied to the flow equation, and the transport one is solved by the symmetric interior penalty discontinuous Galerkin method. Based on a duality argument, employing projection estimates and approximation properties, a posteriori residual-type $hp$ error estimates for the coupled system are presented, which is often used for guiding adaptivity. Comparing with the error analysis carried out by Yang (Int. J. Numer. Meth. Fluids, 65(7) (2011), pp. 781-797), the current work is more complicated and challenging.