Math professor publishes new book on surveys and tutorials
This year, Springer has published the new book, "Graded Finite Element Methods for Elliptic Problems in Nonsmooth Domains," by Professor Hengguang Li, chair of Wayne State University's Department of Mathematics. Professor Li's book is part of Springer's series "Surveys and Tutorials in the Applied Mathematical Sciences."
The idea of the finite element method is as follows. Suppose you are trying to solve a PDE (partial differential equation). For concreteness' sake, focus on the Poisson equation: you start with some function f, and you ask to solve the equation, Laplacian(u) = f, for the variable u.
Recall that the Laplacian of u is the sum of the pure second partial derivatives of u. For example, if u is a function of variables x and y and z, then the Laplacian of u is: (d^2 u)/dx^2 + (d^2 u)/dy^2 + (d^2 u)/dz^2.
Presumably the functions f and u are defined on some domain X. The Dirichlet initial condition for that PDE then consists of letting u be zero on the boundary of the domain X. We start with f, and we aim to find the function u which satisfies the Poisson equation (i.e., the Laplacian of u is equal to f) and also the Dirichlet boundary condition (i.e., u is zero on the boundary of the domain X).
The idea of the finite element method is this: choose a positive integer d, and choose a decomposition of the domain X into a union of topological simplices (i.e., dots, edges, triangles, tetrahedra, and higher-dimensional generalizations of tetrahedra). This decomposition is called a "mesh." Consider all the functions on the domain X which are, on each simplex in the mesh, given by a polynomial of total degree at most d, and whose values agree on the centroids of each of the shared faces of the simplices in the mesh. These functions comprise a finite-dimensional vector space. We will search for approximate solutions to the original PDEs by restricting our attention to only these piecewise-polynomial functions.
To find the closest approximate solution to the Poisson equation with respect to a given mesh and a given degree d, this boils down to a problem of linear algebra! The solution to that problem of linear algebra is not generally an exact solution to the original PDE, but it is at least an approximation to a solution of that original PDE, and by taking the degree d to be larger and larger, and taking the simplicial "mesh" to be finer and finer (i.e., with smaller and smaller topological simplices), one hopes that these approximate solutions converge to an actual solution to the PDE. In good cases, one proves that indeed this is so.
Professor Li's book is about the techniques used to carry out the finite-element method when one wants to find approximate solutions to an elliptic partial differential equation in a domain whose boundary has a singularity. For example, a square (including its interior) is a perfectly reasonable domain in which to solve a PDE, but the boundary of a square has corners. It is easy to imagine that the singularities along the boundary of the domain can cause problems with the convergence of the approximate solutions, as the number of cells and the degree of the polynomials goes to infinity. Typically, one deals with these convergence problems by having smaller cells in the mesh near the singular points along the boundary, in order to better approximate the behavior of solutions to the PDE. It is, however, not usually clear how best to design the geometry of these cells, and how one ought to introduce new cells to form a finer mesh so that the approximations converge to an actual solution of the PDE at a reasonable rate.
The focus of Professor Li's book is a priori methods (as opposed to "adaptive" methods which rely on first trying a mesh that works badly, and then modifying it to compensate for its bad behavior) for the design of meshes for the finite element method on domains with singularities. Some examples of mesh designs are in the figure included in this article. Consequently, the ideas involve some combinatorial topology, as well as PDEs and numerical analysis. Rather than aiming for the broadest generality, Professor Li's book focuses on concrete cases, with worked examples and useful illustrations. The book is quite clear and readable and very accessible to students. It is designed to be understandable after a first course in analysis and a first course in the finite element method.