二维间断扩散系数界面问题的间断有限元方法研究

发布时间：2018-02-01 12:11

  本文关键词： 间断扩散系数 界面问题 三角形网格 内部惩罚 浸入有限元 局部间断Galerkin方法 隐式积分因子方法 Krylov子空间逼近　出处：《中国工程物理研究院》2016年博士论文　论文类型：学位论文

【摘要】：本文主要针对二维区域上间断扩散系数界面问题,研究了在非贴体三角形网格上基于Crouzeix-Raviart元部分惩罚浸入有限元(PIFE)方法和贴体三角形网格上局部间断Galerkin (LDG)方法.同时,为了保持间断Galerkin(DG)方法可分单元计算的优势,减轻显式时间离散对时间步长的严格限制(Δt=O(h(h2min)),我们研究了隐式积分因子(IIF)方法,显著提高了计算效率.本文内容主要分为两大部分.第一部分,在非贴体三角形网格上,给出了求解二阶椭圆界面问题的PIFE方法.首先,在有界面线穿过的界面单元上,构造了满足界面跳跃条件的浸入有限元(IFE)空间,并研究了空间性质；然后,给出了基于对称,非对称以及不完全内部惩罚间断Galerikin (IPDG)的三种PIFE格式,证明了格式解的存在唯一性,并给出了最优能量模误差估计；最后,分别计算了扩散系数是分片常数和对称正定矩阵的算例,验证了以上三种PIFE格式的有效性及最优收敛性.第二部分,在贴体三角形网格上,针对齐次与非齐次抛物界面问题采用LDG方法进行空间离散,不仅得到数值解,还得到流的数值逼近.这一部分我们分两步进行研究.第一步,对齐次和非齐次抛物界面问题进行LDG空间离散,分析了LDG半离散格式的稳定性,证明其先验误差估计.借助显式时间离散,数值验证了LDG方法求解界面问题的空间收敛精度,解和流分别是最优阶和次优阶.这里,可以看到LDG方法求解非齐次界面问题十分自然,只需要将非齐次界面跳跃条件强加在数值通量中,格式本身形式和齐次情形相同,无论分析还是程序都可归于齐次框架,无需进行特殊处理.第二步,考虑更为有效的时间离散方法.我们将二阶IIF方法与LDG方法相结合(ⅡF-LDG),应用于求解不含界面的反应扩散系统,以验证方法的有效性及其优势.方法在得到数值解的同时,也得到了流的数值逼近；发挥了DG方法分单元计算的优势,不必求解大型代数方程组,且可使用较大时间步长(At=O(hmin)),从而节约了计算时间.之后,将这种ⅡF-LDG方法应用于求解二阶抛物界面问题,给出其全离散格式,并分别针对间断常量扩散系数与非线性系数两种情形,数值验证了其有效性和空间收敛性.另外,将计算所用CPU时间与二阶显式Runge-Kuttta时间离散下的结果对比,表明IIF-LDG方法确实缩短了计算时间,提高了计算效率.
[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，

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