In this chapter, we discuss the finite element method for the solution of partial differential equations. It is an important solution approach when the shape of the domain (including possible holes inside the domains) cannot be conveniently transformed to a single rectangular domain. This includes domains whose boundaries cannot be formulated easily under existing coordinate systems.

In contrast to finite difference methods which are based on replacing derivatives with discrete approximations, finite element (FE) methods approach the problem by piecewise interpolation methods. Thus the FE method first partitions the whole domain Ω into several small pieces Ωn, which are known as the finite elements represented by a set of nodes in the domain. The sizes and shapes of the finite elements do not have to be uniform, and often the sizes may need to be varied to balance accuracy with computational efficiency.

Instead of tackling the differential equations directly, the problem is to first recast it as a set of integral equations known as the weak form of the partial differential equation. There are several ways in which this integral is formulated, including least squares, collocation, and weighted residual. We focus on a particular weighted residual method known as the Galerkin method. These integrals are then imposed on each of the finite elements. The finite elements that are attached to the boundaries of Ω will have the additional requirements of satisfying the boundary conditions.