# Finite Element Method

The finite element method is a general discretization technique that can utilize unstructured grids to approximate the solutions of many partial differential equations (PDEs).

There is a large body of literature on finite elements, including the following excellent books:

The MFEM library is designed to be lightweight, general and highly scalable finite element toolkit that provides the building blocks for developing finite element algorithms in a manner similar to that of MATLAB for linear algebra methods.

Some of the C++ classes for the finite element realizations of these PDE-level concepts in MFEM are described below.

### Primal and Dual Vectors

The finite element method uses vectors of data in a variety of ways and the differences can be subtle. MFEM defines `GridFunction`, `LinearForm`, and `Vector` classes which help to distinguish the different roles that vectors of data can play.

### Bilinear Form Integrators

Bilinear form integrators are at the heart of any finite element method, they are used to compute the integrals of products of basis functions over individual mesh elements (or sometimes over edges or faces). The `BilinearForm` class adds several `BilinearFormIntegrator`s together to build the global sparse finite element matrix.

### Linear Form Integrators

Linear form integrators are used to compute the integrals of products of a basis function with a given source function over individual mesh elements (or sometimes over edges or faces). The `LinearForm` class adds several `LinearFormIntegrator`s together to build the global right-hand side for the finite element linear system.

### Linear Interpolators

Unlike Bilinear and Linear forms, Linear Interpolators do not perform integrations, but project one basis function (or a linear function of a basis function) onto another basis function. The `DiscreteLinearOperator` class adds one or more `LinearInterpolators` together to build a global sparse matrix representation of the linear operator.