求解界面问题的扩展杂交间断有限元方法研究

发布时间：2018-07-10 01:00

  本文选题：杂交间断伽略金方法 + 复杂界面　； 参考：《湖南师范大学》2016年博士论文

【摘要】：本学位论文提出并分析了一种求解界面问题的一致网格方法。首先,本文以泊松界面问题为例给出了该算法的基本思想。通过在界面附近构造一个新的分片多项式函数,得到一个界面与网格的边界一致的扩展界面问题。然后,采用杂交间断伽略金有限元方法(HDG)来求解该扩展问题,通过合理地选择数值通量,解在单元边界上的跳跃被自然地引入到了数值格式当中。与现有的方法相比,该方法构造的分片多项式函数是通过一个巧妙的二次Hermite多项式插值,外加一个标准的拉格朗日多项式插值的后处理得到。上述显式构造的多项式能够准确地捕捉到界面上的跳跃信息,而且存在唯一,并以3阶精度逼近原问题精确解的间断部分,更重要的是它与界面的形状及位置无关。另外,它使得扩展界面问题的解具有较高的正则性,从而保证了我们得以用HDG方法高精度求解。最后证明了该方法在L2范数意义下,其精确解及梯度均具有二阶收敛精度。文中涉及各种复杂界面椭圆问题的数值例子也验证了该算法的稳定性及收敛性。基于求解泊松界面问题的成功经验,本文随后研究了抛物界面问题,并重点考虑了移动界面情形。由于界面的位置和形状随着时间不断变化,因此需要在每一个时刻构造逼近解的间断部分的高精度分片多项式,在此基础上,它将原问题转化成界面与网格的边界重合的扩展界面问题。之后,利用HDG方法对扩展界面问题进行空间离散。通过合理设计数值通量,解在单元边界上的跳跃被自然地引入到了数值格式当中,从而保证了离散格式的二阶收敛精度。在时间方向,本文采用经典的向后欧拉格式进行离散,以保证全离散格式的数值稳定性。值得指出的是,每一时刻构造解的间断部分的分片多项式逼近的方法是不变的,只是界面的位置以及界面上的跳跃条件发生了变化,而这种变化只会对每一步要求解的线性方程组的右端产生影响,并不会改变线性方程组的系数矩阵。因此,在计算时只需要在第一个时间步完成对线性方程组系数矩阵的计算和组装,在后面所有时间步,只需要反复使用已经组装好的系数矩阵,从而大幅度提高了算法的计算效率。大量数值实验表明,在笛卡尔网格下,随着界面的移动,该方法不仅稳定,而且能够保证解及其梯度在L2范数意义下具有二阶收敛精度。在对泊松界面问题的研究中,出于理论分析的考虑,只讨论了带有形如[[%絬·n]]仿射跳跃条件的界面问题。为了处理更一般的带间断系数的界面问题,本文引进一种迭代技巧。通过该迭代技巧,一般的带有间断系数的界面问题的解,可以被一系列带有简单仿射跳跃条件的界面问题的解逼近,并且只要选取适当的收敛因子即可保证这种迭代法的收敛性。因此我们只需对这一系列逼近问题采用本文所提出的数值方法就可以实现对一般界面问题的高精度求解。与移动界面问题的求解类似,每一个逼近的界面问题,需要根据其解所满足的跳跃条件构造相应的解的间断部分的分片多项式逼近。为了验证算法的有效性,我们考察了圆环区域上带有一阶吸收边界条件的Helmholtz界面问题。数值实验表明,在拟一致网格下,该数值方法不仅稳定,而且解及其梯度在L2范数的意义下皆具有二阶收敛精度。
[Abstract]:In this thesis, a uniform grid method for solving interface problems is proposed and analyzed. First, the basic idea of this algorithm is given as an example of the Poisson interface problem. By constructing a new piecewise polynomial function near the interface, an extended interface problem with the boundary of the interface to the grid is obtained. The discontinuous Galerkin finite element method (HDG) is used to solve the extension problem. By selecting the numerical flux reasonably, the jump is naturally introduced into the numerical scheme in the element boundary. Compared with the existing method, the piecewise polynomial function constructed by this method is interpolated by a clever two times Hermite polynomial, plus a standard. The postprocessing of the quasi Lagrange polynomial interpolation is obtained. The polynomial of the above explicit construction can accurately capture the jumping information on the interface, and there is a unique, and the 3 order accuracy approximates the discontinuous part of the exact solution of the original problem, and more importantly, it is independent of the shape and position of the interface. In addition, it makes the solution of the extended interface problem. It has high regularity, which ensures that we can solve the high precision by HDG method. Finally, it is proved that the method has two order convergence precision in the sense of L2 norm. The numerical examples of various complex interface elliptic problems in this paper also verify the stability and convergence of the method. After the successful experience of the surface problem, this paper studies the problem of the parabolic interface and focuses on the mobile interface. Because the position and shape of the interface vary with time, the high precision piecewise polynomial of the discontinuous part of the solution is constructed at every moment. On this basis, it transforms the original problem into the interface and the net. The extended interface problem of the boundary coincidence of the lattice is solved. After that, the HDG method is used to discrete the extended interface problem. By reasonably designing the numerical flux, the jump is naturally introduced into the numerical scheme, thus ensuring the two order convergence accuracy of the discrete scheme. In the time direction, the classical backward direction is used in this paper. The Euler scheme is discrete to ensure the numerical stability of the full discrete scheme. It is worth noting that the piecewise polynomial approximation of the discontinuous part of the structural solution at every moment is invariable, only the position of the interface and the jumping conditions on the interface change, and this change will only be a linear equation set for each step. The right end has an influence and does not change the coefficient matrix of the linear equations. Therefore, the calculation and assembly of the linear equation group coefficient matrix is only needed in the first time step. In all the time steps, only the coefficient matrix which has been assembled is used repeatedly, which greatly improves the computational efficiency of the algorithm. Numerical experiments show that in Cartesian grid, with the movement of the interface, the method is not only stable, but also can guarantee the two order convergence precision of the solution and its gradient in the sense of L2 norm. In the study of the Poisson interface problem, only the interface problem with the shape like [%] n]] affine jumping condition is discussed for the consideration of the theoretical analysis. In order to deal with more general interface problems with discontinuous coefficients, an iterative technique is introduced in this paper. Through this iterative technique, the general solution of interface problems with discontinuous coefficients can be approximated by a series of solutions with simple affine jumping conditions, and the iterative method can be guaranteed by selecting the appropriate convergence factor. Therefore, we only need to use the numerical method proposed in this series of approximation problems to achieve a high precision solution to the general interface problem. It is similar to the solution of the mobile interface problem. The interface problem of each approximation needs to be divided into the discontinuous parts of the corresponding solution according to the jump conditions satisfied by the solution. In order to verify the validity of the algorithm, in order to verify the effectiveness of the algorithm, we examine the Helmholtz interface problem with the first order absorption boundary conditions in the ring region. The numerical experiment shows that the numerical method is not only stable but also has two order convergence precision in the sense of L2 norm.
【学位授予单位】：湖南师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82

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