A posteriori error estimators for nonconforming finite element methods of the linear elasticity problem

AbstractIn this work we derive and analyze a posteriori error estimators for low-order nonconforming finite element methods of the linear elasticity problem on both triangular and quadrilateral meshes, with hanging nodes allowed for local mesh refinement. First, it is shown that equilibrated Neumann data on interelement boundaries are simply given by the local weak residuals of the numerical solution. The first error estimator is then obtained by applying the equilibrated residual method with this set of Neumann data. From this implicit estimator we also derive two explicit error estimators, one of which is similar to the one proposed by Dörfler and Ainsworth (2005) [24] for the Stokes problem. It is established that all these error estimators are reliable and efficient in a robust way with respect to the Lamé constants. The main advantage of our error estimators is that they yield guaranteed, i.e., constant-free upper bounds for the energy-like error (up to higher order terms due to data oscillation) when a good estimate for the inf–sup constant is available, which is confirmed by some numerical results

주기경계조건을 갖는 P1-비순응유한요소공간과 멀티스케일 문제에 대한 응용

학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 협동과정 계산과학전공, 2018. 2. 신동우.We consider the P1-nonconforming quadrilateral finite space with periodic boundary condition, and investigate characteristics of the finite space and discrete Laplace operators in the first part of this dissertation. We analyze dimension of the finite element spaces in help of concept of minimally essential discrete boundary conditions. Based on the analysis, we classify functions in a basis for the finite space with periodic boundary condition into two types. And we introduce several Krylov iterative schemes to solve second-order elliptic problems, and compare their solutions. Some of the schemes are based on the Drazin inverse, one of generalized inverse operators, since the periodic nature may derive a singular linear system of equations. An application to the Stokes equations with periodic boundary condition is considered. Lastly, we extend our results for elliptic problems to 3-D case. Some numerical results are provided in our discussion. In the second part, we introduce a nonconforming heterogeneous multiscale method for multiscale problems. Its formulation is based on the P1-nonconforming quadrilateral finite element, mainly with periodic boundary condition. We analyze a priori error estimates of the proposed scheme by following general framework for the finite element heterogeneous multiscale method. For numerical implementations, we use one of the proposed iterative schemes for singular linear systems in the previous part. Several numerical examples and results are given.I P1-Nonconforming Quadrilateral Finite Space with Periodic Boundary Condition 1 Chapter 1 Introduction 3 Chapter 2 Preliminaries 7 2.1 P1-nonconforming quadrilateral finite element 7 2.2 Drazin inverse 8 2.3 Notations 9 Chapter 3 Dimension of the Finite Spaces 13 3.1 Induced relation between boundary DoF values 13 3.2 Minimally essential discrete boundary conditions 16 Chapter 4 Deeper Look on the Finite Space with Periodic B.C. 19 4.1 Linear dependence of B 19 4.2 A Basis for V^h_per 21 4.3 Stiffness matrix associated with B 22 4.4 Numerical schemes for elliptic problems with periodic boundary condition 24 4.4.1 Option 1: S = E^♭ for a nonsingular nonsymmetric system 27 4.4.2 Option 2: S = E^♭ for a symmetric positive semi-definite system with rank deficiency 1 28 4.4.3 Option 3: S = E for a symmetric positive semi-definite system with rank deficiency 2 31 4.4.4 Option 4: S = B for a symmetric positive semi-definite system with rank deficiency 2 33 4.5 Numerical results 34 Chapter 5 Application to Stokes Equations 37 5.1 Discrete inf-sup stability 38 5.2 Numerical scheme: Uzawa variant with a semi-definite block 41 5.3 Numerical results 49 Chapter 6 3-D Case 51 6.1 Dimension of finite spaces in 3-D 51 6.2 Linear dependence of B in 3-D 56 6.3 A basis for V^h_per in 3-D 64 6.4 Stiffness matrix associated with B in 3-D 66 6.5 Numerical schemes in 3-D 67 6.6 Numerical results 73 II Nonconforming Heterogeneous Multiscale Method 75 Chapter 1 Introduction 77 Chapter 2 Preliminaries 81 2.1 Homogenization 81 2.2 Notations 83 Chapter 3 FEHMM Based on Nonconforming Spaces 85 Chapter 4 Fundamental Properties of Nonconforming HMM 91 4.1 Existence and uniqueness 91 4.2 Recovered homogenized tensors 93 4.3 The case of periodic coupling 95 4.4 The case of Dirichlet coupling 101 4.5 A priori error estimate 102 4.5.1 Macro error 102 4.5.2 Modeling error 102 4.5.3 Micro error 104 4.6 Main theorem for error estimates 105 Chapter 5 Numerical Results 107 5.1 Periodic diagonal example 108 5.1.1 Comparison between approaches to solve micro problem 110 5.2 Periodic example with off-diagonal terms 112 5.3 Example with noninteger-ε-multiple sampling domain and Dirichlet coupling 112 5.4 Example on mixed domain 115 국문초록 127Docto