The conforming virtual element method for polyharmonic and elastodynamics problems: a review

In this paper, we review recent results on the conforming virtual element approximation of polyharmonic and elastodynamics problems. The structure and the content of this review is motivated by three paradigmatic examples of applications: classical and anisotropic Cahn-Hilliard equation and phase field models for brittle fracture, that are briefly discussed in the first part of the paper. We present and discuss the mathematical details of the conforming virtual element approximation of linear polyharmonic problems, the classical Cahn-Hilliard equation and linear elastodynamics problems.Comment: 30 pages, 7 figures. arXiv admin note: text overlap with arXiv:1912.0712

A Posteriori Error Analysis of hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic Problems

We develop the a-posteriori error analysis of hp-version interior-penalty discontinuous Galerkin finite element methods for a class of second-order quasilinear elliptic partial differential equations. Computable upper and lower bounds on the error are derived in terms of a natural (mesh-dependent) energy norm. The bounds are explicit in the local mesh size and the local degree of the approximating polynomial. The performance of the proposed estimators within an automatic hp-adaptive refinement procedure is studied through numerical experiments

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

Review of Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/90641/1/AIAA-53965-537.pd