椭圆偏微分方程边值与逆边值问题的数值方法及稳定性分析

发布时间：2018-04-11 17:26

  本文选题：椭圆方程边值问题 + 光滑点插值法　； 参考：《太原理工大学》2017年硕士论文

【摘要】：椭圆方程边值问题描述了工程应用中大量的定常态问题,例如弹性力学中平衡问题,导体中的电子密度等。由于问题域及边值条件的复杂性,精确解的求解非常困难,因此对椭圆方程的精确解进行数值近似并且对数值近似的方法进行收敛性分析具有实际意义。而椭圆方程的逆边值问题是在声波散射,层析成像及无损检测等领域出现的一类不适定问题,即测量数据的微小误差会引起解的巨大震荡。因此建立稳定的数值算法并对其进行收敛性分析对实际问题具有指导意义。本论文的工作集中于将基于节点的光滑点插值法和超定Kansa方法应用于求解椭圆方程边值和逆边值问题并且研究所提出数值算法的收敛性。基于节点的径向基函数光滑点插值法被用来求解椭圆方程边值问题。节点形函数通过径向基函数的点插值法构造。基于三角形和四边形背景网格,两种基于节点的光滑域被构造。光滑伽辽金弱形式用来构造离散的系统方程。数值结果显示,和有限元相比,在网格变形严重时,该方法可以得到更高精度,更高收敛率的解。对于能量范数,基于节点的光滑点插值法和有限元分别得到了精确解的上界解和下界解。这说明当精确能量范数未知时,我们可以结合这两种方法对其进行估计。对于椭圆方程逆边值问题,我们基于超定的Kansa方法提出了两种数值算法并证明了算法的收敛性。通过施加等式约束条件来控制柯西边界计算误差,基于三种带权重最小二乘公式的自适应重构算法被提出,这种算法最多只需要三步。自适应算法的收敛性定理在对称正定径向基函数的本核空间内离散点近似理论下得到了证明。为了保证数值稳定性的Tikhnov正则化项在算法建立过程中自然出现,且在收敛性分析过程中可以得到其取值公式。通过在柯西边界上施加二次型约束条件来控制计算误差,椭圆逆边值问题的优化重构方法被建立。逆边值问题的半离散解首先被定义为含有二次型约束的优化问题。半离散解的收敛性定理基于径向基函数的重构希尔伯特空间理论和柯西问题的条件稳定性得到证明。通过在问题域和可测边界上配置点处对半离散解进行离散,我们定义了柯西问题离散的数值解。离散的解定义为有二次型约束的最小二乘优化问题(LSQI问题)。离散解的收敛性定理通过分数阶的抽样不等式得到证明。二维和三维数值结果表明,两种数值算法均可以在不同噪声情况下重构出来稳定的、高精度的数值解。
[Abstract]:The elliptic equation boundary value problem describes a large number of stationary normal problems in engineering applications, such as equilibrium problems in elastic mechanics, electron density in conductors, and so on.Due to the complexity of the problem domain and the boundary conditions, it is very difficult to solve the exact solution. Therefore, it is of practical significance to analyze the convergence of the exact solution of the elliptic equation and the method of numerical approximation.The inverse boundary value problem of elliptic equation is a kind of ill-posed problem in the fields of acoustic wave scattering, tomography and nondestructive testing.Therefore, the establishment of a stable numerical algorithm and its convergence analysis have a guiding significance for practical problems.This paper focuses on the node-based smooth point interpolation method and the overdetermined Kansa method for solving the boundary value and inverse boundary value problems of elliptic equations and studies the convergence of the numerical algorithm.The nodal radial basis function smoothing point interpolation method is used to solve the boundary value problem of elliptic equations.The nodal function is constructed by the point interpolation method of the radial basis function.Based on triangular and quadrilateral background meshes, two node-based smooth domains are constructed.The smooth Galerkin weak form is used to construct discrete system equations.The numerical results show that compared with the finite element method, the solution with higher accuracy and higher convergence rate can be obtained when the mesh deformation is serious.For the energy norm, the upper bound solution and the lower bound solution of the exact solution are obtained based on the nodal smooth point interpolation method and the finite element method, respectively.This shows that when the exact energy norm is unknown, we can estimate it with these two methods.For the inverse boundary value problem of elliptic equations, we propose two numerical algorithms based on the overdetermined Kansa method and prove the convergence of the algorithm.By applying equality constraints to control the Cauchy boundary calculation error, an adaptive reconstruction algorithm based on three weighted least squares formulas is proposed, which requires only three steps at most.The convergence theorem of the adaptive algorithm is proved by the discrete point approximation theory in the kernel space of the symmetric positive definite radial basis function.In order to ensure the numerical stability of the Tikhnov regularization term in the establishment of the algorithm naturally appear in the process of convergence analysis can be obtained in the process of its value formula.By applying quadratic constraints on the Cauchy boundary to control the calculation error, the optimal reconstruction method for the elliptic inverse boundary value problem is established.The semi-discrete solution of inverse boundary value problem is firstly defined as an optimization problem with quadratic constraints.The convergence theorem of semi-discrete solutions is proved based on the reconstruction of Hilbert space theory of radial basis function and the conditional stability of Cauchy problem.The numerical solution of Cauchy problem is defined by discretization of the semi-discrete solution at the collocation point on the problem domain and measurable boundary.The discrete solution is defined as the LSQI problem with quadratic constraints.The convergence theorem of discrete solutions is proved by fractional sampling inequality.Two-dimensional and three-dimensional numerical results show that both algorithms can reconstruct stable and high-precision numerical solutions under different noise conditions.
【学位授予单位】：太原理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
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本文编号：1736957