The Finite Element Method (FEM)

FEM is a versatile numerical method for finding approximate solutions to partial differential equations (PDEs). In this section, we introduce the fundamental concepts of FEM through a classical, second-order elliptic PDE: the Poisson equation.

The model problem: Poisson equation

Consider a bounded domain  ,  , with a sufficiently smooth boundary . We seek a function defined on that satisfies the Poisson equation with homogeneous Dirichlet boundary conditions:

where is a given source function, and    is the Laplace operator.

Strong form vs. weak form

The equation above represents the strong form of the problem. It requires the solution to have continuous second derivatives, which is often a very restrictive condition, especially when is not smooth.

To relax this regularity requirement, we derive the weak (or variational) form. We multiply the strong form by an arbitrary test function that vanishes on the boundary (  on ) and integrate over the domain :

Applying integration by parts (Green's first identity) to the left-hand side, we get:

Since the test function vanishes on the boundary , the boundary integral is zero. The resulting problem can then be stated in the following form: find   such that

where

•   is the Sobolev space of functions with square-integrable weak derivatives that vanish on the boundary;

•     is a symmetric bilinear form;

•    is a linear form.

This formulation requires and to only have (square-integrable) first derivatives and thus describes a much larger class of situations than the strong form.

Bubnov-Galerkin approximation

The weak formulation is infinite-dimensional because the space is infinite-dimensional. The (Bubnov-)Galerkin method consists of replacing the continuous space with a finite-dimensional subspace  . (The subscript here contains a positive parameter which characterises the discrete nature of the problem, with   denoting that we are approaching the infinite-dimensional setting.)

The corresponding discrete problem can then be written down as: find   such that

To compute the solution, we will take to be a finite element space consisting of piecewise-defined functions on a suitable partition of the domain called a mesh.

For instance, in 1D, an -element mesh for   can be defined by choosing mesh vertices ,  :

The -th mesh element is defined to be  ,  . Finally, an example finite element space on this mesh could be the space of -smooth piecewise-quadratic functions satisfying homogeneous boundary conditions:

The parameter characterizing the discrete nature of this space could be defined in different ways; for instance, a couple of popular choices are   and   .

In the general setting, once we have chosen such a finite element space, let   and let be a basis for . Then, we can express the approximate solution as a linear combination of these basis functions:

where   are the unknown coefficients called degrees of freedom.

Testing the discrete problem with each basis function   for  , we obtain a system of linear equations:

This can be written in matrix-vector form as:

where

• is the matrix with entries     .

• is the source vector with entries    .

•   is the vector of unknowns.