本文關(guān)鍵詞： 間斷擴(kuò)散系數(shù) 界面問(wèn)題 三角形網(wǎng)格 內(nèi)部懲罰 浸入有限元 局部間斷Galerkin方法 隱式積分因子方法 Krylov子空間逼近　出處：《中國(guó)工程物理研究院》2016年博士論文　論文類型：學(xué)位論文

【摘要】：本文主要針對(duì)二維區(qū)域上間斷擴(kuò)散系數(shù)界面問(wèn)題,研究了在非貼體三角形網(wǎng)格上基于Crouzeix-Raviart元部分懲罰浸入有限元(PIFE)方法和貼體三角形網(wǎng)格上局部間斷Galerkin (LDG)方法.同時(shí),為了保持間斷Galerkin(DG)方法可分單元計(jì)算的優(yōu)勢(shì),減輕顯式時(shí)間離散對(duì)時(shí)間步長(zhǎng)的嚴(yán)格限制(Δt=O(h(h2min)),我們研究了隱式積分因子(IIF)方法,顯著提高了計(jì)算效率.本文內(nèi)容主要分為兩大部分.第一部分,在非貼體三角形網(wǎng)格上,給出了求解二階橢圓界面問(wèn)題的PIFE方法.首先,在有界面線穿過(guò)的界面單元上,構(gòu)造了滿足界面跳躍條件的浸入有限元(IFE)空間,并研究了空間性質(zhì)；然后,給出了基于對(duì)稱,非對(duì)稱以及不完全內(nèi)部懲罰間斷Galerikin (IPDG)的三種PIFE格式,證明了格式解的存在唯一性,并給出了最優(yōu)能量模誤差估計(jì)；最后,分別計(jì)算了擴(kuò)散系數(shù)是分片常數(shù)和對(duì)稱正定矩陣的算例,驗(yàn)證了以上三種PIFE格式的有效性及最優(yōu)收斂性.第二部分,在貼體三角形網(wǎng)格上,針對(duì)齊次與非齊次拋物界面問(wèn)題采用LDG方法進(jìn)行空間離散,不僅得到數(shù)值解,還得到流的數(shù)值逼近.這一部分我們分兩步進(jìn)行研究.第一步,對(duì)齊次和非齊次拋物界面問(wèn)題進(jìn)行LDG空間離散,分析了LDG半離散格式的穩(wěn)定性,證明其先驗(yàn)誤差估計(jì).借助顯式時(shí)間離散,數(shù)值驗(yàn)證了LDG方法求解界面問(wèn)題的空間收斂精度,解和流分別是最優(yōu)階和次優(yōu)階.這里,可以看到LDG方法求解非齊次界面問(wèn)題十分自然,只需要將非齊次界面跳躍條件強(qiáng)加在數(shù)值通量中,格式本身形式和齊次情形相同,無(wú)論分析還是程序都可歸于齊次框架,無(wú)需進(jìn)行特殊處理.第二步,考慮更為有效的時(shí)間離散方法.我們將二階IIF方法與LDG方法相結(jié)合(ⅡF-LDG),應(yīng)用于求解不含界面的反應(yīng)擴(kuò)散系統(tǒng),以驗(yàn)證方法的有效性及其優(yōu)勢(shì).方法在得到數(shù)值解的同時(shí),也得到了流的數(shù)值逼近；發(fā)揮了DG方法分單元計(jì)算的優(yōu)勢(shì),不必求解大型代數(shù)方程組,且可使用較大時(shí)間步長(zhǎng)(At=O(hmin)),從而節(jié)約了計(jì)算時(shí)間.之后,將這種ⅡF-LDG方法應(yīng)用于求解二階拋物界面問(wèn)題,給出其全離散格式,并分別針對(duì)間斷常量擴(kuò)散系數(shù)與非線性系數(shù)兩種情形,數(shù)值驗(yàn)證了其有效性和空間收斂性.另外,將計(jì)算所用CPU時(shí)間與二階顯式Runge-Kuttta時(shí)間離散下的結(jié)果對(duì)比,表明IIF-LDG方法確實(shí)縮短了計(jì)算時(shí)間,提高了計(jì)算效率.
[Abstract]:This paper focuses on the interfacial problem of discontinuous diffusion coefficient in two-dimensional region. The partial penalty immersion in finite element based on Crouzeix-Raviart element on non-body-fitted triangular meshes is studied. Methods and local discontinuous Galerkin Galerkin method on body-fitted triangular meshes. In order to maintain the advantage of the discontinuous Galerkin DG method, the strict limitation of explicit time discretization on the time step size (螖 t ~ (t)) is reduced. We study the implicit integral factor (IIFs) method and improve the computational efficiency significantly. The content of this paper is divided into two parts. The first part is on the non-body-fitted triangular meshes. The PIFE method for solving the second order elliptic interface problem is presented. Firstly, the immersion finite element space satisfying the jumping condition of the interface is constructed on the interface element with interfacial line crossing. The properties of space are also studied. Then, three kinds of PIFE schemes based on symmetric, asymmetric and incomplete internal penalty discontinuous Galerikin schemes are given, and the existence and uniqueness of the solutions are proved. The error estimation of the optimal energy mode is given. Finally, the numerical examples of the diffusion coefficient which are piecewise constant and symmetric positive definite matrix are calculated, and the validity and optimal convergence of the above three PIFE schemes are verified. The second part, on the body-fitted triangular meshes. For the homogeneous and non-homogeneous parabolic interface problem, the LDG method is used to discretize the space. Not only the numerical solution is obtained, but also the numerical approximation of the flow is obtained. In this part, we study the problem in two steps. The LDG space discretization of homogeneous and nonhomogeneous parabolic interface problems is carried out. The stability of LDG semi-discrete scheme is analyzed and its prior error estimate is proved. Numerical results verify the spatial convergence accuracy of the LDG method for solving interface problems. The solutions and flows are the optimal order and the sub-optimal order, respectively. Here, we can see that the LDG method is very natural to solve the non-homogeneous interface problems. Only the non-homogeneous interface jump condition is imposed on the numerical flux, the format itself is the same as the homogeneous case, both the analysis and the program can be attributed to the homogeneous framework, no special treatment is required. The second step. The second order IIF method is combined with the LDG method to solve the reaction-diffusion system without interface. In order to verify the validity of the method and its advantages, the numerical solution is obtained, and the numerical approximation of the flow is also obtained. The advantage of DG method in unit calculation is given full play to, the large algebraic equations need not be solved, and the large time step can be used to save the calculation time. The 鈪，

本文編號(hào)：1481828