Graded Finite Element Methods for Elliptic Problems in Nonsmooth Domains

✍ Scribed by Hengguang Li

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 186
• Series: Surveys and Tutorials in the Applied Mathematical Sciences, 10
• Edition: 1
• Category: Library

✦ Synopsis

This book develops a class of graded finite element methods to solve singular elliptic boundary value problems in two- and three-dimensional domains. It provides an approachable and self-contained presentation of the topic, including both the mathematical theory and numerical tools necessary to address the major challenges imposed by the singular solution. Moreover, by focusing upon second-order equations with constant coefficients, it manages to derive explicit results that are accessible to the broader computation community. Although written with mathematics graduate students and researchers in mind, this book is also relevant to applied and computational mathematicians, scientists, and engineers in numerical methods who may encounter singular problems.

✦ Table of Contents

Preface
Contents
1 The Finite Element Method
1.1 The Finite Element Algorithm
1.1.1 The Variational Formulation
1.1.2 The Finite Element Space
1.2 Examples
1.2.1 A One-Dimensional Example
1.2.2 A Two-Dimensional Example
1.2.3 A Three-Dimensional Example
2 The Function Space
2.1 Vector Spaces
2.2 Sobolev Spaces
2.2.1 Domains and Sobolev Spaces
2.2.2 Extension and Embedding Theorems
2.2.3 Trace Theorems
2.3 Regularity Theorems
2.4 Basic Finite Element Error Estimates
3 Singularities and Graded Mesh Algorithms
3.1 A Numerical Example
3.2 Singularities in Polygonal Domains
3.3 Singularities in Polyhedral Domains
3.3.1 The 3D Edge Singularity
3.3.2 The 3D Vertex Singularity
3.4 The Graded Mesh Algorithm
4 Error Estimates in Polygonal Domains
4.1 Regularity Analysis in Weighted Sobolev Spaces
4.2 2D Graded Meshes and Mesh Layers
4.3 Interpolation Error Estimates
4.4 Analysis for Neumann Boundary Conditions
4.5 Graded Quadrilateral Meshes
5 Regularity Estimates and Graded Meshes in Polyhedral Domains
5.1 Regularity Analysis in Anisotropic Weighted Sobolev Spaces
5.1.1 Dirichlet Boundary Conditions and Weighted Spaces
5.1.2 Dirichlet Boundary Conditions and Rough Given Data
5.1.3 Mixed Boundary Conditions
5.1.3.1 Regularity Estimates in a Dihedron
5.1.3.2 Regularity Results in a Dihedral Cone
5.1.3.3 Regularity Estimates for (5.1) with Mixed Boundary Conditions
5.2 3D Graded Meshes and Mesh Layers
6 Anisotropic Error Estimates in Polyhedral Domains
6.1 Interpolation Error Estimates for uMm+1μ+1(Ω)
6.1.1 Estimates on Initial o-, v-, and ve-Tetrahedra in T0
6.1.2 Estimates on Initial e-Tetrahedra in T0
6.1.3 Estimates on Initial ev-Tetrahedra in T0
6.2 Error Estimates for uH2γ(Ω) (Rough Given Data)
6.3 Error Estimates for Mixed Boundary Conditions
6.4 Numerical Illustrations
6.4.1 The Case of uM2μ+1(Ω)
6.4.2 The Case of Rough Given Data
6.4.3 The Case of Mixed Boundary Conditions
References
Index

✦ Subjects

Graded Finite Element Method, Elliptic Boundary Value Problem, Nonsmooth Domain, Singular Solution, Weighted Sobolev Space