The Finite Element Method in 1D

Abstract

In this chapter we shall introduce the finite element method as a general tool for the numerical solution of two-point boundary value problems. In doing so, the basic idea is to first rewrite the boundary value problem as a variational equation, and then seek a solution approximation to this equation from the space of continuous piecewise linears. We prove basic error estimates and show how to use these to formulate adaptive algorithms that can be used to automatically improve the accuracy of the computed solution. The derivation and areas of application of the studied boundary value problems are also discussed.