幾類界面問題的非擬合有限元方法分析
發(fā)布時間：2018-08-28 11:06
【摘要】：界面問題在材料科學、固體力學以及流體動力學中經常出現(xiàn)。例如,帶有不同傳導系數(shù)的熱傳導問題、描述不同材料行為的彈性問題以及帶有不同粘性系數(shù)的兩相流問題等。目前,對界面問題數(shù)值方法的研究已成為科學計算和工程領域內的研究熱點之一。本文主要目的是在非擬合網格(即網格剖分和界面位置無關)下,構造一些新的有限元方法來求解界面問題并對其進行數(shù)值分析。首先,基于Nitsche方法和LDG方法的思想,我們提出了一類求解橢圓界面問題的間斷Galerkin方法。這個方法的關鍵是在離散問題的雙線性型中,用界面上的加權平均代替經典間斷Galerkin方法中的代數(shù)平均。我們得到了與界面位置無關的最優(yōu)誤差估計。數(shù)值例子驗證了我們的理論結果。其次,為了有效解決剛度矩陣病態(tài)的問題,我們提出了一類新的間斷Galerkin方法。非擬合方法的網格剖分與界面位置無關,導致了在界面附近會出現(xiàn)非常小的單元,從而使得離散問題的剛度矩陣嚴重病態(tài)。為了避免了直接使用這類非常小的單元,我們用與其相鄰的較大單元作為它們的延拓單元,這就使得我們證明了經典的逆不等式。從而,我們得到了與界面位置無關的最優(yōu)誤差估計和剛度矩陣的條件數(shù)((O(h-2))。然后,我們把這類方法推廣到彈性界面問題和Stokes界面問題。針對彈性界面問題,我們提出了一個非擬合非對稱間斷Galerkin方法,并證明了一個新的延拓定理。利用經典的BDM插值的性質,得到了與界面位置和Lame常數(shù)入無關的最優(yōu)誤差估計(Locking-free)。針對Stokes界面問題,我們提出了一個帶有懲罰速度跳躍項和應力跳躍項的間斷Galerkin方法。利用一些特殊的技巧證明了inf-sup穩(wěn)定性條件,從而我們得到了在能量范數(shù)意義下的最優(yōu)誤差估計。并且,證明了這兩類離散問題所對應的剛度矩陣的條件數(shù)都是與界面位置無關的。最后,我們提出了一類求解Stokes界面問題的穩(wěn)定化Nitsche型有限元方法。我們用最低階的等階有限元分別來逼近速度和壓力空間。結合局部投影方法和ghost罰方法,我們證明了inf-sup穩(wěn)定性條件,從而得到了在能量范數(shù)和L2范數(shù)意義下的最優(yōu)誤差估計。同時證明了剛度矩陣的條件數(shù)是與界面位置無關的。數(shù)值例子驗證了我們的理論結果。
[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.
【學位授予單位】：南京師范大學
【學位級別】：博士
【學位授予年份】：2015
【分類號】：O241.82
本文編號：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.
【學位授予單位】：南京師范大學
【學位級別】：博士
【學位授予年份】：2015
【分類號】：O241.82
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