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

【摘要】：橢圓方程邊值問題描述了工程應(yīng)用中大量的定常態(tài)問題,例如彈性力學(xué)中平衡問題,導(dǎo)體中的電子密度等。由于問題域及邊值條件的復(fù)雜性,精確解的求解非常困難,因此對(duì)橢圓方程的精確解進(jìn)行數(shù)值近似并且對(duì)數(shù)值近似的方法進(jìn)行收斂性分析具有實(shí)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(cè)等領(lǐng)域出現(xiàn)的一類不適定問題,即測(cè)量數(shù)據(jù)的微小誤差會(huì)引起解的巨大震蕩。因此建立穩(wěn)定的數(shù)值算法并對(duì)其進(jìn)行收斂性分析對(duì)實(shí)際問題具有指導(dǎo)意義。本論文的工作集中于將基于節(jié)點(diǎn)的光滑點(diǎn)插值法和超定Kansa方法應(yīng)用于求解橢圓方程邊值和逆邊值問題并且研究所提出數(shù)值算法的收斂性�；诠�(jié)點(diǎn)的徑向基函數(shù)光滑點(diǎn)插值法被用來求解橢圓方程邊值問題。節(jié)點(diǎn)形函數(shù)通過徑向基函數(shù)的點(diǎn)插值法構(gòu)造。基于三角形和四邊形背景網(wǎng)格,兩種基于節(jié)點(diǎn)的光滑域被構(gòu)造。光滑伽遼金弱形式用來構(gòu)造離散的系統(tǒng)方程。數(shù)值結(jié)果顯示,和有限元相比,在網(wǎng)格變形嚴(yán)重時(shí),該方法可以得到更高精度,更高收斂率的解。對(duì)于能量范數(shù),基于節(jié)點(diǎn)的光滑點(diǎn)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(dāng)精確能量范數(shù)未知時(shí),我們可以結(jié)合這兩種方法對(duì)其進(jìn)行估計(jì)。對(duì)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(shù)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(jì)算誤差,基于三種帶權(quán)重最小二乘公式的自適應(yīng)重構(gòu)算法被提出,這種算法最多只需要三步。自適應(yīng)算法的收斂性定理在對(duì)稱正定徑向基函數(shù)的本核空間內(nèi)離散點(diǎn)近似理論下得到了證明。為了保證數(shù)值穩(wěn)定性的Tikhnov正則化項(xiàng)在算法建立過程中自然出現(xiàn),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(jì)算誤差,橢圓逆邊值問題的優(yōu)化重構(gòu)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(shù)的重構(gòu)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(cè)邊界上配置點(diǎn)處對(duì)半離散解進(jìn)行離散,我們定義了柯西問題離散的數(shù)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?jǐn)?shù)階的抽樣不等式得到證明。二維和三維數(shù)值結(jié)果表明,兩種數(shù)值算法均可以在不同噪聲情況下重構(gòu)出來穩(wěn)定的、高精度的數(shù)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(xué)位授予單位】：太原理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.82
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本文編號(hào)：1736957