两类界面问题的有限体积元方法

发布时间：2018-06-24 18:03

  本文选题：有限体积元方法 + 界面问题　； 参考：《南京师范大学》2015年博士论文

【摘要】：本文致力于两类界面问题的有限体积元方法的研究,全文共分为三个部分.第一章首先我们介绍了关于界面问题的一些浸入方法,阐述了发展浸入有限体积元方法的目的。然后我们介绍了一维带界面的双相延迟方程、高阶紧有限体积元方法和Pade型紧有限体积方法。第二章第一部分讨论了带界面的泊松方程的浸入有限体积元方法。通过源项移去技巧,将带非齐次跳跃条件的界面问题转化为带齐次跳跃条件的界面问题,与跳跃条件相关的项被转移到了方程的右端。此时,在四边形网格下,其双线性基函数是通常的有限元基函数。两个数值算例验证了格式的可行性和有效性。第二部分进一步讨论了带变系数的二维椭圆界面问题的浸入有限体积元方法,其变系数在通过界面时有一个有限的跳跃。由此导致其解和通量在通过界面时也会产生一个有限的跳跃,增加了数值计算上的困难。我们仍先使用源项移去技巧,得到一个等价的带齐次跳跃条件的椭圆界面问题。由于变系数的存在,在界面附近的节点基函数是分片多项式函数,其构造需满足齐次跳跃条件。若远离界面,我们使用通常的有限元节点基函数。四边形网格对分片多项式的构造在某些情况下会产生奇异性,故我们使用了三角形网格。由此产生的线性问题简单并且容易求解。我们对此进行了在能量范数意义下的误差估计。并给出了数值实验。两个数值实验进一步验证了我们的结论：在L2范数意义下,界面附近的误差和整体误差均有O(h2)阶精度；在H1范数意义下,均有O(h)阶精度。第三章给出了一维带界面的双相延迟热传导方程的高阶紧有限体积元方法。除界点格式外所得到的系数矩阵是三对角的,具有较好的对称与对角性质,且易于求解。该高阶方法有助于对这个方程在相对疏松的网格上研究纳米级别的热传导现象,有重要的实际应用价值。我们应用离散能量方法在L2和L∞范数意义下给出误差估计,其收敛阶是O(△t~2+h~(3.5))。数值例子验证了该方法的有效性和可行性。该高阶有限体积元方法的构造涉及到对原方程的回代。一旦我们遇到含多个变量的复杂方程时,该方法就不是很实用了。进一步我们考虑了一个四阶Pade型紧有限体积方法更简便的高阶方法,可以处理多维的界面问题。方程的解及其导数都达到了四阶精度。第四章给出了本文的主要结论和有待进一步解决的问题。
[Abstract]:This paper is devoted to the study of finite volume element method for two kinds of interface problems, which is divided into three parts. In the first chapter, we introduce some immersion methods about interface problems and expound the purpose of developing immersion finite volume element method. Then we introduce the biphase delay equation with interface, the high order compact finite volume element method and the Pade type compact finite volume method. In the second chapter, the immersion finite volume element method for Poisson equation with interface is discussed. By means of the source term removal technique, the interface problem with non-homogeneous jump condition is transformed into the interface problem with homogeneous jump condition, and the term related to the jump condition is transferred to the right end of the equation. In this case, the bilinear basis function of the quadrilateral mesh is the usual finite element basis function. Two numerical examples show that the scheme is feasible and effective. In the second part, we further discuss the finite volume element method for two-dimensional elliptic interface problem with variable coefficients, which has a finite jump when passing through the interface. As a result, the solution and flux also lead to a finite jump when passing through the interface, which increases the difficulty of numerical calculation. We still use the source term removal technique to obtain an equivalent elliptic interface problem with homogeneous jump conditions. Because of the existence of variable coefficients, the nodal basis function near the interface is a piecewise polynomial function, and its construction needs to satisfy the homogeneous jump condition. Far from the interface, we use the usual finite element node basis function. The construction of quadrilateral meshes to piecewise polynomials may result in singularity in some cases, so we use triangular meshes. The resulting linear problem is simple and easy to solve. We estimate the error in the sense of energy norm. Numerical experiments are also given. Two numerical experiments further verify our conclusion: in the sense of L _ 2 norm, the errors near the interface and the global errors have O (H2) order accuracy, and in the sense of H _ 1 norm, both have O (h) order accuracy. In chapter 3, the high order compact finite volume element method for the biphase delay heat conduction equation with interface is given. The coefficient matrix obtained in addition to the bounded point scheme is tridiagonal with good symmetry and diagonal properties and is easy to solve. The high-order method is helpful to the study of the nano-scale heat conduction phenomenon on the relatively loose grid, and has important practical application value. In the sense of L _ 2 and L _ 鈭，

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