To keep the notation simple, we consider the scalar wave equation
c−1(x)∂t2u(x,t)−Δu(x,t)=f(x,t).
with homogenuous initial conditions
u(x,0)=0,∂tu(x,0)=0,
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.
where the last term is a surface integral along the boundary ∂Ω of the domain Ω, and 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 φ 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.