Abstract
This chapter explores the finite element method for elliptic differential equations. The finite element method is a special method for the numerical solution of partial differential equations. The name was coined by engineers who used the method in structural mechanics. The finite element method became a very widely used method in practice. The theoretical investigation of different aspects began a few years ago. Nevertheless, many fundamental results are known at present. There is a variety of methods called the finite element method. The chapter presents one special approach aimed especially at solving elliptical equations. To avoid incidental technical difficulties, the chapter presents the results and numerical experiments for simple model problems. The chapter discusses approximation theory, which plays a very important role in the investigation of the finite element method.
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