Elements of Numerical Analysis

Introduction

We discussed boundary value problems (bvp's) in one-dimension in Chapter 7 and in two dimensions (elliptic equations) in Chapter 11. In case of ordinary differential equation the domain of the problem was a finite interval along, say x-axis. Some nodal points or nodes were selected over the interval by subdividing it into uniform (evenly-spaced) subintervals/ subdomains. Similarly an elliptic equation was defined over a domain in the x-y plane which was subdivided into rectangular/square meshes. In both cases the sizes of the subdomains were of uniform size except perhaps near the boundary. The solution was obtained by approximating the derivatives by finite differences (forward, backward or central). These methods are called Finite Difference Methods (FDMs).

The Finite Element Method (FEM) in this chapter, will be discussed mainly in respect of boundary value problems in one and two dimensions only. It can be used for solving transient (time-dependent) problems also and in that case, as one of the methods, the time derivative will be approximated by the finite difference techniques. They are also discussed briefly. The FEM approach for solving a bvp differs from FDM in mainly two aspects. First, in the FEM, the domain may be subdivided in an arbitrary manner. For example, in two dimension it is usually subdivided into triangles which are not necessarily of equal sizes – on the other hand they are often of unequal sizes. This gives us the freedom to choose larger triangles in the part of the domain where variation in the solution is expected to be not too large and choose smaller triangles where solution may vary too sharply or it may be too sensitive over some part of the domain. These triangles (or polygons in general) are called ‘elements’ and vertices of the triangles (or polygons) as ‘nodes’. We are required to find the solution at these nodes. The second difference is with regard to the mathematical strategy adopted in FEM which is totally different from FDM.