The spectral-element method on a glance

To keep the notation simple, we consider the scalar wave equation

with homogenuous initial conditions

and some boundary conditions, which we will ignore for now - more here

The above set of equations are called the strong form of the wave equation. Finite-element methods, however solve the weak or variational form of the PDE.

Weak form

The weak for of the wave equation is obtained by multiplying the equation with a so-called test function $\varphi(\mathbf{x})$ and integrating in space:

Next, we integrate the second term and apply the Gauss-Green theorem to eliminate the second derivative in space:

where the last term is a surface integral along the boundary $\partial\Omega$ of the domain $\Omega$, and $\vec{\mathbf{n}}$ is the unit normal point outward of the domain. Inserting ths into

Putting this back together gives the weak form of the wave equation

A function $u$ that satisfies this weak form for any choice of the test function $\varphi$ at all times $t$ is called a weak solution. Under certain mathematical conditions, $u$ also satisfies the strong form above. We spare the details here, and just assume that both solutions are the same.