一类抛物型界面问题的浸入有限元方法
发布时间:2018-04-13 01:20
本文选题:二阶抛物界型面问题 + 部分惩罚方法 ; 参考:《山东师范大学》2015年硕士论文
【摘要】:界面问题刻画了诸如由复杂地质结构或多向流导致的具有问断扩散系数的混溶驱替等实际渗流过程,建立其准确高效的数值模拟方法和完整的数值分析理论体系,对深刻揭示实际渗流的运动机理、指导科学工程实践具有重要的理论价值和应用前景.本文旨在对可描述各向异性渗流问题的一类二阶抛物型界面问题提出相应的有限元数值模拟格式,并建立严格的数值分析理论.主要内容分为两个部分: 1.基于线性拉格朗日插值的部分惩罚浸入界面有限元方法 在这一部分中,我们运用线性拉格朗日插值构造浸入界面有限元空问,对于问中函数的存在唯一性.进一步,将上述空问构造中这一条件弱化至若sl0,则证明了浸入界面线性有限元空问中的函数可由三角形单元顶点值唯一确定,从而运用较弱的条件构造出相应的分片线性有限元空问,拓广了该方法的应用范围.基于由上述条件构造出的有限元空问,我们对二维抛物型界面问题建立了对称、非对称及不完全的部分惩罚浸入界面有限元(PIFE)半离散和全离散格式.利用椭圆投影、Gronwall不等式等数值分析技术,证明了格式的可解性、稳定性和最优能量模与L2模误差估计. 2.基于旋转Q1元的部分惩罚浸入界面有限元方法 在这一部分中,我们运用旋转Q1元构造浸入界面有限元空问,对于扩散系元空问中函数的存在唯一性.证明了浸入界面双线性有限元空问中的函数可由矩形单元四边中点处的函数值唯一确定.从而,构造出了相应的分片线性有限元空间.基于由上述条件构造的有限元空间,我们对二维抛物型界面问题建立了矩形剖分下对称、非对称及不完全的浸入界面双线性有限元半离散和全离散格式.利用椭圆投影、Gronwall不等式等数值分析技术,证明了格式的可解性与稳定性,最终给出了次最优能量模误差估计.
[Abstract]:The interface problem depicts the actual seepage process such as mixed displacement with fault diffusion coefficient caused by complex geological structure or multidirectional flow, and establishes its accurate and efficient numerical simulation method and complete numerical analysis theory system.It has important theoretical value and application prospect for revealing the movement mechanism of actual seepage and guiding scientific engineering practice.The purpose of this paper is to propose a finite element numerical simulation scheme for a class of second-order parabolic interface problems which can describe anisotropic seepage problems, and to establish a strict numerical analysis theory.The main content is divided into two parts:1.Partial penalty Immersion Interface finite element method based on Linear Lagrange interpolationIn this part, we use the linear Lagrange interpolation to construct the space question of the finite element immersed in the interface, and the existence and uniqueness of the function in the question is obtained.Furthermore, by weakening this condition to sl0, it is proved that the function immersed in the linear finite element space problem of the interface can be uniquely determined by the vertex value of the triangular element.The corresponding piecewise linear finite element space problems are constructed by using the weaker conditions, and the application range of the method is extended.Based on the finite element space problems constructed from the above conditions, the symmetric, asymmetric and incomplete partial penalty immersion finite element (PIFE) semi-discrete and fully discrete schemes are established for two-dimensional parabolic interface problems.By using the elliptic projection Gronwall inequality and other numerical analysis techniques, the solvability and stability of the scheme and the error estimates of the optimal energy norm and L 2 norm are proved.2.Partial penalty Immersion Interface finite element method based on rotating Q1 elementIn this part, we use the rotating Q1 element to construct the space question of the immersion interface finite element, and the existence and uniqueness of the function in the diffusion system element space problem are obtained.It is proved that the function in the space problem of the bilinear finite element immersed in the interface can be uniquely determined by the function value at the midpoint of the four edges of the rectangular element.Thus, the corresponding piecewise linear finite element space is constructed.Based on the finite element space constructed by the above conditions, we establish the semi-discrete and fully discrete schemes of bilinear finite element for two-dimensional parabolic interface problems with symmetric, asymmetric and incomplete immersion interfaces under rectangular subdivision.By using the elliptic projection Gronwall inequality and other numerical analysis techniques, the solvability and stability of the scheme are proved. Finally, the suboptimal energy modulus error estimates are given.
【学位授予单位】:山东师范大学
【学位级别】:硕士
【学位授予年份】:2015
【分类号】:O241.82
【参考文献】
相关期刊论文 前1条
1 王淑燕;陈焕贞;;基于Crouzeix-Raviart元的界面浸入有限元方法及其收敛性分析[J];计算数学;2012年02期
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