This page here has been created for the latest stable release of Salvus. You have chosen to view the documentation for another Salvus version. Please be aware there might be some small differences you have to account for.
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.