机械求积法及其外推算法求解Helmholtz非线性边界积分方程

当Helmholtz微分方程转化为非线性边界积分方程后,可以利用机械求积法求得近似解,此方法具有较高的收敛精度阶O(h3)和较低的计算复杂度.构造机械求积法时,一个非线性方程系统通过离散非线性积分方程得到.此外,每个矩阵元素的值都不需要计算任何奇异积分.根据渐近紧理论和Stepleman定理,整个系统的稳定性和收敛性得到了证明.利用h3-Richardson外推算法,收敛精度阶可以提高到O(h5).为了求解非线性方程组,利用Ostrowski不动点定理研究了Newton的解的收敛性.几个算例从数值上说明了本算法的有效性.     

高精度特征解及其外推求解位势方程

根据位势理论,基本边界特征值问题可转化为具有对数奇性的边界积分方程.利用机械求积方法求解特征值和特征向量,以及利用这些特征解求解Laplace方程.特征解和Laplace方程的解具有高精度和低的计算复杂度.利用Anselone聚紧和渐近紧理论,证明了方法的收敛性和稳定性.此外,还给出了误差的奇数阶渐近展开.利用h3-Richardson外推,不仅误差近似的精度阶大为提高,而且,得到的后验误差估计可以构造自适应算法.具体的数值例子说明了算法的有效性.     

解第一类边界积分方程的高精度机械求积法与外推

０．引言使用单层位势理论把Ｄｉｒｉｃｈｌｅｔ问题：转化为具有对数核的边界积分方程：这里Г假设为简单光滑闭曲线．熟知,若Г的容度Ｃｒ≠１,（０．２）有唯一解存在［１］．借助参数变换这里的数值解法有Ｇａｌｅｒｋｉｎ法［２］,配置法［３］,和谱方法~［４］,这些方法有一个共同缺点就是矩阵元素的生成要计算反常积分,由于离散方程的系数矩阵是满阵,使矩阵生成的工作量很庞大,甚至超过了解方程组的工作量．显然,如能找到适当求积公式离散（０．２）,则可节省大量计算．使用求积公式法解（０．２）的文献不多,［５］中提…     

POISSON方程新的边界积分方程

POISSON方程边界值问题边界元法所应用的边界积分方程,其类型,关于未知位势导数是第一类积分方程,关于未知位势是第二类积分方程。本本文从格林公式出发,通过建立位势的单、双场守恒积分公式,推导出POISSON方程新的边界积分方程,其类型与经典方程相反,关于未知位势是第一类积分方程,关于未知位势导数是第二类积分方程。     

三维重调和方程的双方程边界积分方程法

以守恒积分为工具,推导了三维重调和方程的新的边界积分方程,所得出的新方程与传统的边界积分方程相比较,降低了奇异性,避免了传统边界元方法中的强奇异积分的计算.对不同边界都采用第二类积分方程,得到了三维重调和方程的双方程方法.     

弹性力学问题解唯一的边界积分方程

从积分方程式出发,应用基本解的特性分析,说明在力边值问题中,位移边界积分方程和面力边界积分方程的位移解不唯一.提出了位移解唯一的条件,建立了唯一解的位移边界积分方程和面力边界积分方程.实例计算结果表明唯一解的边界积分方程是有效的.     

位势平面问题的新的规则化边界积分方程

广泛实践集中在直接变量边界积分方程的规则化研究,其本质是利用简单解消除边界积分的奇异性．然而,至今关于平面位势问题的第一类边界积分方程的规则化研究尚未涉足．致力于间接变量边界积分方程的规则化方法研究,基于一种新的思想和观点,确立平面位势问题的间接变量规则边界积分方程,它不包含CPV强奇异积分和HFP超奇异积分．数值算例表明现在的方法可取得很好的精度和效率,特别是边界量的计算．     

含曲线裂纹圆柱扭转问题的新边界元法

研究含曲线裂纹圆柱的Saint-Venant扭转,将问题化归为裂纹上边界积分方程的求解．利用裂纹尖端的奇异元和线性元插值模型,给出了扭转刚度和应力强度因子的边界元计算公式．对圆弧裂纹、曲折裂纹以及直线裂纹的典型问题进行了数值计算,并与用Gauss-Chebyshev求积法计算的直裂纹情形结果进行了比较,证明了方法的有效性和正确性．     

平面定常Stokes问题的无奇异第一类边界积分方程

对无奇异边界积分方程归化法的研究,已有的结果都是针对直接变量的,其核心思想是利用刚体位移(包括刚体的转动和平移)或均匀场．然而,对第一类边界积分方程的无奇异边界归化法的研究,至今还未涉足．本文提交一种新方法,归化出平面定常Stokes问题的第一类无奇异边界积分方程,并建立完整的数值求解体系．一个简单的算例表明本文方法可获得理想的数值结果,特别是边界量的数值结果。     

解带有Hilbert核的奇异积分方程的高精度组合算法

该文给出了用求积法解带Hilber核的奇异积分方程的高精度组合算法.把网格点分成互不相交的子集合, 在子集合上并行求解离散方程组, 再利用组合算法求得全局网格点的逼近.如果积分方程的系数属于B_δ, 则求积法的精度可达O(e^-nδ). 此外, 使用组合算法不仅能得到更高的精度阶, 而且能够得到后验误差估计. 数值算例的结果表明组合算法是极其有效的.     

A fast interface solver for the biharmonic Dirichlet problem on polygonal domains

Summary. In this paper we propose and analyze an efficient discretization scheme for the boundary reduction of the biharmonic Dirichlet problem on convex polygonal domains. We show that the biharmonic Dirichlet problem can be reduced to the solution of a harmonic Dirichlet problem and of an equation with a Poincaré-Steklov operator acting between subspaces of the trace spaces. We then propose a mixed FE discretization (by linear elements) of this equation which admits efficient preconditioning and matrix compression resulting in the complexity . Here is the number of degrees of freedom on the underlying boundary, is an error reduction factor, or for rectangular or polygonal boundaries, respectively. As a consequence an asymptotically optimal iterative interface solver for boundary reductions of the biharmonic Dirichlet problem on convex polygonal domains is derived. A numerical example confirms the theory. Received September 1, 1995 / Revised version received February 12, 1996     

Numerical solution of a second biharmonic boundary value problem

The second boundary value problem for the biharmonic equation is equivalent to the Dirichlet problems for two Poisson equations. Several finite difference approximations are defined to solve these Dirichlet problems and discretization error estimates are obtained. It is shown that the splitting of the biharmonic equation produces a numerically efficient procedure.     

Biharmonic equation in a highly oscillating domain and homogenization of an associated control problem

We consider a Dirichlet boundary control problem posed in an oscillating boundary domain governed by a biharmonic equation. Homogenization of a PDE with a non-homogeneous Dirichlet boundary condition on the oscillating boundary is one of the hardest problems. Here, we study the homogenization of the problem by converting it into an equivalent interior control problem. The convergence of the optimal solution is studied using periodic unfolding operator.     

On a nonlocal generalization of the biharmonic Dirichlet problem

We consider a mixed problem with the Dirichlet boundary conditions and integral conditions for the biharmonic equation. We prove the existence and uniqueness of a generalized solution in the weighted Sobolev space W ₂². We show that the problem can be viewed as a generalization of the Dirichlet problem.     

On solvability of a boundary value problem for a nonhomogeneous biharmonic equation with a boundary operator of a fractional order

This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.     

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR A NONHOMOGENEOUS BIHARMONIC EQUATION WITH A BOUNDARY OPERATOR OF A FRACTIONAL ORDER

This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.     

An optimization problem for the Biharmonic equation with Sobolev conditions

We state and solve an optimization problem about distribution of several supporting points under a Kirchhoff plate clamped along the boundary: the biharmonic equation is supplied with the Dirichlet boundary conditions and point Sobolev conditions. Some open questions are formulated. Bibliography: 23 titles.     

Polynomial solutions of the Dirichlet problem for the biharmonic equation in the ball

We find a polynomial solution of the Dirichlet problem for the inhomogeneous biharmonic equation with polynomial right-hand side and polynomial boundary data in the unit ball. To this end, we use the closed-form representation of harmonic functions in the Almansi formula.     

Smoothness of solutions of the Dirichlet problem for the biharmonic equation in nonsmooth 2D domains

We study the smoothness of a generalized solution of the Dirichlet problem for the biharmonic equation in a two-dimensional domain. We introduce a weighted test function and derive an estimate for the absolute value of the solution in a neighborhood of an irregular boundary point.     

High Accuracy Solution of Three-Dimensional Biharmonic Equations

In this paper, we consider several finite-difference approximations for the three-dimensional biharmonic equation. A symbolic algebra package is utilized to derive a family of finite-difference approximations for the biharmonic equation on a 27 point compact stencil. The unknown solution and its first derivatives are carried as unknowns at selected grid points. This formulation allows us to incorporate the Dirichlet boundary conditions automatically and there is no need to define special formulas near the boundaries, as is the case with the standard discretizations of biharmonic equations. We exhibit the standard second-order, finite-difference approximation that requires 25 grid points. We also exhibit two compact formulations of the 3D biharmonic equations; these compact formulas are defined on a 27 point cubic grid. The fourth-order approximations are used to solve a set of test problems and produce high accuracy numerical solutions. The system of linear equations is solved using a variety of iterative methods. We employ multigrid and preconditioned Krylov iterative methods to solve the system of equations. Test results from two test problems are reported. In these experiments, the multigrid method gives excellent results. The multigrid preconditioning also gives good results using Krylov methods.