几类界面问题的非拟合有限元方法分析
发布时间：2018-08-28 11:06
【摘要】：界面问题在材料科学、固体力学以及流体动力学中经常出现。例如,带有不同传导系数的热传导问题、描述不同材料行为的弹性问题以及带有不同粘性系数的两相流问题等。目前,对界面问题数值方法的研究已成为科学计算和工程领域内的研究热点之一。本文主要目的是在非拟合网格(即网格剖分和界面位置无关)下,构造一些新的有限元方法来求解界面问题并对其进行数值分析。首先,基于Nitsche方法和LDG方法的思想,我们提出了一类求解椭圆界面问题的间断Galerkin方法。这个方法的关键是在离散问题的双线性型中,用界面上的加权平均代替经典间断Galerkin方法中的代数平均。我们得到了与界面位置无关的最优误差估计。数值例子验证了我们的理论结果。其次,为了有效解决刚度矩阵病态的问题,我们提出了一类新的间断Galerkin方法。非拟合方法的网格剖分与界面位置无关,导致了在界面附近会出现非常小的单元,从而使得离散问题的刚度矩阵严重病态。为了避免了直接使用这类非常小的单元,我们用与其相邻的较大单元作为它们的延拓单元,这就使得我们证明了经典的逆不等式。从而,我们得到了与界面位置无关的最优误差估计和刚度矩阵的条件数((O(h-2))。然后,我们把这类方法推广到弹性界面问题和Stokes界面问题。针对弹性界面问题,我们提出了一个非拟合非对称间断Galerkin方法,并证明了一个新的延拓定理。利用经典的BDM插值的性质,得到了与界面位置和Lame常数入无关的最优误差估计(Locking-free)。针对Stokes界面问题,我们提出了一个带有惩罚速度跳跃项和应力跳跃项的间断Galerkin方法。利用一些特殊的技巧证明了inf-sup稳定性条件,从而我们得到了在能量范数意义下的最优误差估计。并且,证明了这两类离散问题所对应的刚度矩阵的条件数都是与界面位置无关的。最后,我们提出了一类求解Stokes界面问题的稳定化Nitsche型有限元方法。我们用最低阶的等阶有限元分别来逼近速度和压力空间。结合局部投影方法和ghost罚方法,我们证明了inf-sup稳定性条件,从而得到了在能量范数和L2范数意义下的最优误差估计。同时证明了刚度矩阵的条件数是与界面位置无关的。数值例子验证了我们的理论结果。
[Abstract]:Interface problems often occur in material science, solid mechanics and fluid dynamics. For example, heat conduction problem with different conduction coefficient, elastic problem with different material behavior and two-phase flow problem with different viscosity coefficient, etc. At present, the research on numerical methods of interface problems has become one of the hotspots in the field of scientific calculation and engineering. The main purpose of this paper is to construct some new finite element methods to solve interface problems and analyze them numerically under unfitted meshes (i.e. mesh generation and interface position independence). Firstly, based on the ideas of Nitsche method and LDG method, we propose a class of discontinuous Galerkin methods for solving elliptic interface problems. The key of this method is to replace the algebraic average in the classical discontinuous Galerkin method with the weighted average on the interface in the bilinear form of the discrete problem. We obtain the optimal error estimation which is independent of the interface position. Numerical examples verify our theoretical results. Secondly, in order to solve the problem of stiffness matrix ill-condition effectively, we propose a new class of discontinuous Galerkin method. The mesh generation of the non-fitting method is independent of the interface position, which leads to the appearance of very small elements near the interface, which makes the stiffness matrix of the discrete problem seriously ill-conditioned. In order to avoid the direct use of these very small elements, we use the larger elements adjacent to them as their extension elements, which leads us to prove the classical inverse inequalities. Thus, we obtain the optimal error estimation and the condition number of stiffness matrix (O (h-2)., which are independent of the position of the interface. Then, we extend this method to elastic interface problem and Stokes interface problem. For the elastic interface problem, we propose a nonfitting asymmetric discontinuous Galerkin method and prove a new continuation theorem. By using the properties of classical BDM interpolation, the optimal error estimation (Locking-free) is obtained, which is independent of the interface position and the input of Lame constant. For the Stokes interface problem, we propose a discontinuous Galerkin method with penalty speed jump term and stress jump term. The inf-sup stability conditions are proved by using some special techniques, and the optimal error estimates in the sense of energy norm are obtained. Moreover, it is proved that the condition number of stiffness matrix for these two discrete problems is independent of the interface position. Finally, we propose a stable Nitsche finite element method for solving Stokes interface problems. We use the lowest order equal-order finite element method to approximate the velocity and pressure space respectively. Combining the local projection method and the ghost penalty method, we prove the inf-sup stability condition and obtain the optimal error estimates in the sense of energy norm and L 2 norm. It is also proved that the condition number of stiffness matrix is independent of the interface position. Numerical examples verify our theoretical results.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O241.82
本文编号：2209211
[Abstract]:Interface problems often occur in material science, solid mechanics and fluid dynamics. For example, heat conduction problem with different conduction coefficient, elastic problem with different material behavior and two-phase flow problem with different viscosity coefficient, etc. At present, the research on numerical methods of interface problems has become one of the hotspots in the field of scientific calculation and engineering. The main purpose of this paper is to construct some new finite element methods to solve interface problems and analyze them numerically under unfitted meshes (i.e. mesh generation and interface position independence). Firstly, based on the ideas of Nitsche method and LDG method, we propose a class of discontinuous Galerkin methods for solving elliptic interface problems. The key of this method is to replace the algebraic average in the classical discontinuous Galerkin method with the weighted average on the interface in the bilinear form of the discrete problem. We obtain the optimal error estimation which is independent of the interface position. Numerical examples verify our theoretical results. Secondly, in order to solve the problem of stiffness matrix ill-condition effectively, we propose a new class of discontinuous Galerkin method. The mesh generation of the non-fitting method is independent of the interface position, which leads to the appearance of very small elements near the interface, which makes the stiffness matrix of the discrete problem seriously ill-conditioned. In order to avoid the direct use of these very small elements, we use the larger elements adjacent to them as their extension elements, which leads us to prove the classical inverse inequalities. Thus, we obtain the optimal error estimation and the condition number of stiffness matrix (O (h-2)., which are independent of the position of the interface. Then, we extend this method to elastic interface problem and Stokes interface problem. For the elastic interface problem, we propose a nonfitting asymmetric discontinuous Galerkin method and prove a new continuation theorem. By using the properties of classical BDM interpolation, the optimal error estimation (Locking-free) is obtained, which is independent of the interface position and the input of Lame constant. For the Stokes interface problem, we propose a discontinuous Galerkin method with penalty speed jump term and stress jump term. The inf-sup stability conditions are proved by using some special techniques, and the optimal error estimates in the sense of energy norm are obtained. Moreover, it is proved that the condition number of stiffness matrix for these two discrete problems is independent of the interface position. Finally, we propose a stable Nitsche finite element method for solving Stokes interface problems. We use the lowest order equal-order finite element method to approximate the velocity and pressure space respectively. Combining the local projection method and the ghost penalty method, we prove the inf-sup stability condition and obtain the optimal error estimates in the sense of energy norm and L 2 norm. It is also proved that the condition number of stiffness matrix is independent of the interface position. Numerical examples verify our theoretical results.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O241.82
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