本文選題：橢圓方程邊值問題 + 光滑點插值法��； 參考：《太原理工大學》2017年碩士論文

【摘要】：橢圓方程邊值問題描述了工程應用中大量的定常態(tài)問題,例如彈性力學中平衡問題,導體中的電子密度等。由于問題域及邊值條件的復雜性,精確解的求解非常困難,因此對橢圓方程的精確解進行數值近似并且對數值近似的方法進行收斂性分析具有實際意義。而橢圓方程的逆邊值問題是在聲波散射,層析成像及無損檢測等領域出現的一類不適定問題,即測量數據的微小誤差會引起解的巨大震蕩。因此建立穩(wěn)定的數值算法并對其進行收斂性分析對實際問題具有指導意義。本論文的工作集中于將基于節(jié)點的光滑點插值法和超定Kansa方法應用于求解橢圓方程邊值和逆邊值問題并且研究所提出數值算法的收斂性�；诠�(jié)點的徑向基函數光滑點插值法被用來求解橢圓方程邊值問題。節(jié)點形函數通過徑向基函數的點插值法構造�；谌切魏退倪呅伪尘熬W格,兩種基于節(jié)點的光滑域被構造。光滑伽遼金弱形式用來構造離散的系統(tǒng)方程。數值結果顯示,和有限元相比,在網格變形嚴重時,該方法可以得到更高精度,更高收斂率的解。對于能量范數,基于節(jié)點的光滑點插值法和有限元分別得到了精確解的上界解和下界解。這說明當精確能量范數未知時,我們可以結合這兩種方法對其進行估計。對于橢圓方程逆邊值問題,我們基于超定的Kansa方法提出了兩種數值算法并證明了算法的收斂性。通過施加等式約束條件來控制柯西邊界計算誤差,基于三種帶權重最小二乘公式的自適應重構算法被提出,這種算法最多只需要三步。自適應算法的收斂性定理在對稱正定徑向基函數的本核空間內離散點近似理論下得到了證明。為了保證數值穩(wěn)定性的Tikhnov正則化項在算法建立過程中自然出現,且在收斂性分析過程中可以得到其取值公式。通過在柯西邊界上施加二次型約束條件來控制計算誤差,橢圓逆邊值問題的優(yōu)化重構方法被建立。逆邊值問題的半離散解首先被定義為含有二次型約束的優(yōu)化問題。半離散解的收斂性定理基于徑向基函數的重構希爾伯特空間理論和柯西問題的條件穩(wěn)定性得到證明。通過在問題域和可測邊界上配置點處對半離散解進行離散,我們定義了柯西問題離散的數值解。離散的解定義為有二次型約束的最小二乘優(yōu)化問題(LSQI問題)。離散解的收斂性定理通過分數階的抽樣不等式得到證明。二維和三維數值結果表明,兩種數值算法均可以在不同噪聲情況下重構出來穩(wěn)定的、高精度的數值解。
[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.
【學位授予單位】：太原理工大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O241.82
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本文編號：1736957