Irregular Regions and Grids

Abstract

Throughout ten chapters of this text we have studied many algorithms, and we hope that we solved many problems. Most of these problems were defined on the interval [0,1], the square [0,1] × [0,1] or a cube. (There were probably a couple of times when we got adventuresome and used [−2, 1] instead of [0, 1].) We all know that the world and the objects in the world are not intervals, squares and cubes. It is convenient to treat these easier problems first while we are learning about numerical schemes. It is now time to face some reality and discuss more complicated situations. We will discuss both situations where the geometry is more complex than a square or a cube and the situation where it is best to use a nonuniform grid. We will not discuss these topics in depth. We will try to introduce a variety of the methods that are used to confront problems caused by very irregular solutions and irregular geometries. We hope to provide an overview of some of the difficulties and some of the solutions.