This monograph focuses on the mathematical and numerical analysis of simplicial partitions and the finite element method. This active area of research has become an essential part of physics and engineering, for example in the study of problems involving heat conduction, linear elasticity, semiconductors, Maxwell's equations, Einstein's equations and magnetic and gravitational fields.
These problems require the simulation of various phenomena and physical fields over complicated structures in three (and higher) dimensions. Since not all structures can be decomposed into simpler objects like d-dimensional rectangular blocks, simplicial partitions are important. In this book an emphasis is placed on angle conditions guaranteeing the convergence of the finite element method for elliptic PDEs with given boundary conditions.
It is aimed at a general mathematical audience who is assumed to be familiar with only a few basic results from linear algebra, geometry, and mathematical and numerical analysis.
Author(s): Jan Brandts, Sergey Korotov, Michal Křížek
Series: Springer Monographs in Mathematics
Publisher: Springer
Year: 2020
Language: English
Pages: 203
City: Cham
Preface
Contents
Glossary of Symbols
1 Introduction
1.1 Motivation
1.2 Preliminaries
2 Simplices: Definitions and Properties
2.1 The Triangle
2.2 The Tetrahedron
2.3 Higher-Dimensional Simplices
3 Simplicial Partitions
3.1 Triangulations
3.2 Tetrahedral and Simplicial Partitions
3.3 Coloring Simplicial Partitions
3.4 The Basic Idea of the Finite Element Method
4 Angle Conditions
4.1 Zlámal's Minimum Angle Condition
4.2 Volumic Regularity Conditions
4.3 Minimum Angle Conditions in Higher Dimensions
4.4 The Maximum Angle Conditions
4.5 The FEM on Highly Distorted Partitions
4.6 Superconvergence and Post-processing
5 Nonobtuse Simplicial Partitions
5.1 Preliminaries
5.2 Acute Partitions
5.3 Nonobtuse Partitions and Path-Simplices
5.4 Applications in Numerical Mathematics
5.5 Further Applications
6 Nonexistence of Acute Simplicial Partitions in R5
6.1 Preliminaries
6.2 Auxiliary Lemmas
6.3 The Proposed Proof Technique for d=3
6.4 The Proposed Proof Technique for d=4
6.5 The Nonexistence of Acute Partitions in R5
6.6 Extension to Higher Dimensions
7 Tight Bounds on Angle Sums of Simplices
7.1 Preliminaries
7.2 Discussion of Gaddum's Dihedral Angle Bounds
7.3 Dihedral Angle Bounds for Nonobtuse Simplices
8 Refinement Techniques
8.1 Refinements of Unstructured Partitions
8.2 Red and Green Refinements of Tetrahedra
8.3 Properties of Red Refinement Techniques
8.4 Refinements into Path-Tetrahedra
8.5 Partitions into Well-Centered Simplices
8.6 Local Nonobtuse Refinements
8.7 The Longest-Edge Bisection Algorithm
8.8 Refinements of Triangular Prisms
9 The Discrete Maximum Principle
9.1 The Maximum Principle for an Elliptic Problem
9.2 Finite Element Discretization
9.3 Sufficient Algebraic Conditions
9.4 Associated Geometrical Conditions
9.5 Typical Problems with Standard Conditions
9.6 Less Severe Conditions Based on Stieltjes Matrices
10 Variational Crimes
10.1 What Are Variational Crimes?
10.2 Efficient Quadrature Formulae on Simplices
10.3 Isoparametric Quadratic Elements
10.4 Setting the Problem
10.5 Approximate Solution
10.6 Slice and Hat Elements
10.7 A Convergence Result
11 0/1-Simplices and 0/1-Triangulations
11.1 Congruence Versus 0/1-Equivalence
11.2 Representation as 0/1-Matrices
11.3 Counting and Enumeration of 0/1-Equivalence Classes
11.4 Orthogonal 0/1-Simplices
11.5 Acute 0/1-Simplices
11.6 Neighbor Theorems for Nonobtuse Simplices
11.7 The Hadamard Conjecture
11.8 Triangulations of Id Using 0/1-Simplices
12 Tessellations of Maximally Symmetric Manifolds
12.1 Regular Polytopes
12.2 Regular Triangular Tessellations
12.3 Regular Tetrahedral Tessellations
12.4 Regular Simplicial Tessellations
Appendix References
Index
Author Index
