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.
The weak for of the wave equation is obtained by multiplying the equation with a so-called test function 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 of the domain , and 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 that satisfies this weak form for any choice of the test function at all times is called a weak solution. Under certain mathematical conditions, also satisfies the strong form above. We spare the details here, and just assume that both solutions are the same.