分数阶稳态方程的弱有限元方法

发布时间：2018-07-06 21:36

本文选题：弱有限元方法 + 分数阶微分方程　；参考：《兰州大学》2017年硕士论文

【摘要】：一种解决整数阶微分方程的弱有限元方法已经被介绍和分析了,举个列子,用弱有限元方法求解二次椭圆方程,在求解二次椭圆方程近似解的基础就是离散弱梯度算子,用弱有限元空间代替传统的有限元空间,让离散弱梯度算子代替经典梯度算子.并且用不同的多项式空间的多项式组合来逼近弱函数和弱梯度算子,不同的多项式空间的组合产生不同的弱有限元方法,这使得弱有限元方法拥有很多的优点,可以应用到不连续函数上面,在数值计算上面有很高的灵活性和有效性,在有些文献中,作者是通过加稳定子,来保持单元之间的弱连续性.在这篇论文中,我们第一次尝试在分数阶微分方程上运用弱有限元方法.看看用弱有限元方法方法是否能够解决分数阶微分方程,以简单地分数阶稳态方程为例,我们也用弱有限元空间代替传统的有限元空间,在分数阶方程中定义了弱梯度算子的形式,将基函数分为内部和边界,并且加上稳定子s(ν,ω).在稳态方程中,我们探讨了该方程解的存在唯一性和误差估计,最终都让弱梯度算子可以被分片多项式来逼近,最后会有一些数值实验结果.这篇文章主要工作是第一次在简单的二维分数阶稳态方程中运用加稳定子的弱有限元方法,并研究解的存在唯一性和误差估计。
[Abstract]:A weak finite element method for solving integer-order differential equations has been introduced and analyzed. For example, a weak finite element method is used to solve quadratic elliptic equations. The basis of solving the approximate solutions of quadratic elliptic equations is discrete weak gradient operator. The weak finite element space is used to replace the traditional finite element space and the discrete weak gradient operator is used to replace the classical gradient operator. And the weak function and the weak gradient operator are approximated by the polynomial combination of different polynomial space. The combination of different polynomial space produces different weak finite element method, which makes the weak finite element method have many advantages. It can be applied to discontinuous functions and has high flexibility and effectiveness in numerical computation. In some literatures, the authors maintain the weak continuity between elements by adding stabilizers. In this paper, we first try to apply the weak finite element method to fractional differential equations. See if the weak finite element method can solve the fractional differential equation, take the simple fractional steady state equation as an example, we also use the weak finite element space instead of the traditional finite element space. In the fractional order equation, the form of weak gradient operator is defined, the basis function is divided into internal and boundary, and the stable subs (谓, 蠅) is added. In the steady-state equation, we discuss the existence and uniqueness of the solution of the equation and the error estimation. Finally, the weak gradient operator can be approximated by piecewise polynomials. Finally, some numerical results are obtained. The main work of this paper is to study the existence and uniqueness of the solution and the error estimation for the first time in a simple two-dimensional fractional steady-state equation by using a weak finite element method with a stabilizer.
【学位授予单位】：兰州大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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