The Finite Element Method
2019-12-04 Browse times：540
This set of notes are prepared by Jinchao Xu for the class MATH/CSE 556 at Penn State in Fall 2018. All rights reserved. Not to be disseminated without explicit permission of the author.
Chapter 1: Linear Vector Spaces and Dualsdownload
This chapter is devoted to some fundamentals of linear vector spaces. First, we define the linear vector space.
Chapter 2: Model PDEs and Basic Numerical Methodsdownload
Model PDEs and Basic Numerical Methods Model PDEs and Basic Numerical Methods Model PDEs and Basic Numerical Methods
Chapter 3: Finite Difrence Methoddownload
In this chapter, we will consider finite di↵erence method for solving the following model problem on ⌦ ⇢ R d (d = 1,2): (3.1) ( ??u = f, in ⌦, u = 0, on @⌦, The idea is to use this example in order to determine how the linear algebraic systems are related to partial di↵erential equations (PDEs). Then we consider the error estimates for the finite di↵erence schemes by using maximum principle.
Chapter 4: An overview of the finite element methoddownload
In this chapter, we give an overview of the finite element method for the discretization of partial di↵er- ential equations. We will mainly consider the linear finite element discretization of the Poisson equations in one, two and three dimensions. We will walk through the main issues related to the finite element method, including the basic setup, theory, basic data structures and some solution methods. The presentation in this chapter is meant to be concise and informative, rather than rigorous.
Chapter 5: H(grad), H(curl) and H(div) spacesdownload
The Laplacian operator is the most important example of partial di↵erential operator of elliptic type. In this chapter, we will present some main results on this operator.
Chapter 6: H(grad), H(div) and H(curl) Finite element spacesdownload
H(grad), H(div) and H(curl) Finite element spaces
Chapter 7: A Posteriori Error Estimates and Adaptive Finite Element Methodsdownload
A Posteriori Error Estimates and Adaptive Finite Element Methods
Chapter 8: The sparse grid methoddownload
It is well-known that a smooth function defined on [0,1] d can be pointwisely approximated with O(h 2 ) accuracy by a piecewise bilinear function in a subspace V h of dimension O(h ?d ). In the socalled sparse grid method proposed by Zenger, an O(h 2 |logh| d?1 ) pointwise accuracy can be achieved by using a substantially smaller subspace S h ⇢ V h of dimension O(h ?1 |logh| d?1 ). As a result, a function u on a general domain in R d can be approximated with O(h|logh| d?1 ) in H 1 norm with only O(h ?1 |logh| d?1 ) number of operations.
Chapter 9: A Generalized Lax Equivalence Theorem: Consistency, Stability and Convergencedownload
Consistency, stability and convergence are basic concepts for almost all discretization methods. These concepts are mostly commonly encountered in numerical methods for partial di↵erential equations. In the finite di↵erence methods, the best known result is the theory of P. Lax [?] on the equivalence of stability and convergence for certain class of consistent finite di↵erence schemes. In the discretization methods based on variational principle such as Petrov-Galerkin methods (including finite element and finite volume methods), there are the fundamental theories by Babuˇ ska [?] for the standard Galerkin method and Brezzi [?, ?] for the mixed Galerkin method.
Chapter 10: Some examplesdownload
Chapter 11: Drawbacks of conforming finite element methodsdownload
It is evident that the conforming finite element is a natural way to construct an approximation space for a given Sobolev space. However, because of some essential character of the conforming element, it is far from perfect in practical use. For example, we take the two aspects below.
Chapter 12: Mixed finite element methods for second order elliptic problemdownload
Mixed finite element methods for second order elliptic problem
Chapter 13: Linear elasticity and finite elementsdownload
Linear elasticity and finite elements
Chapter 14: Mixed Methods for Stokes equationsdownload
Mixed Methods for Stokes equations
Chapter 15: Discontinuous Galerkin Methodsdownload
In this chapter, we will introduce discontinuous Galerkin (DG) Methods.
Chapter 16: Extended Finite Element Methodsdownload
The interface problems which involve partial di↵erential equations having discontinuous coe?cients across certain interfaces are often encountered in fluid dynamics, electromagnetics, and materials science. Because of the low global regularity and the irregular geometry of the interface, the standard numerical methods which are e?cient for smooth solutions usually lead to loss in accuracy across the interface.
Chapter 17: Finite Volume Methodsdownload
Finite Volume Methods
Chapter 18: Appendix: Di↵erential forms and a unified theory of exact sequencesdownload
Appendix: Di↵erential forms and a unified theory of exact sequences
Chapter 19: Di↵erential Formdownload
Chapter 20: Approximation Properties of Neural Network Function Classdownload
Given an activation function
Chapter 21: Deep Neural Networks and Finite Elementsdownload
In this chapter, we will give a brief introduction to a special function class related to deep neural networks (DNN) used in machine learning. We then explore the relationship between DNN (with ReLU as activation function) and linear finite element methods.
Chapter 22: Introduction to Finite Element Packagesdownload
Introduction to Finite Element Packages
Chapter 23: Exercisesdownload
Chapter 24: Referencesdownload