This chapter is devoted to some fundamentals of linear vector spaces. First, we define the linear vector space.

Model PDEs and Basic Numerical Methods Model PDEs and Basic Numerical Methods Model PDEs and Basic Numerical Methods

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.

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.

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.

H(grad), H(div) and H(curl) Finite element spaces

A Posteriori Error Estimates and Adaptive Finite Element Methods

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.

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.

Some examples

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.

Mixed finite element methods for second order elliptic problem

Linear elasticity and finite elements

Mixed Methods for Stokes equations

In this chapter, we will introduce discontinuous Galerkin (DG) Methods.

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.

Finite Volume Methods

Appendix: Di↵erential forms and a unified theory of exact sequences

Di↵erential Form

Given an activation function

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.

Introduction to Finite Element Packages

Exercises

References