Mondaic

The spectral-element method on a glance

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

c−1(x)∂t2u(x,t)−Δu(x,t)=f(x,t). c^{-1}(\mathbf{x}) \partial_t^2 u(\mathbf{x},t) - \Delta u(\mathbf{x},t) = f(\mathbf{x},t).

with homogenuous initial conditions

u(x,0)=0,∂tu(x,0)=0,u(\mathbf{x},0) = 0, \quad \partial_t u(\mathbf{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.

Weak form

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

∫Ωc−1(x)∂t2u(x,t) φ(x) dx−∫ΩΔu(x,t) φ(x) dx=∫Ωf(x,t) φ(x) dx. \int_\Omega c^{-1}(\mathbf{x}) \partial_t^2 u(\mathbf{x},t)\, \varphi(\mathbf{x})\,d\mathbf{x} - \int_\Omega \Delta u(\mathbf{x},t)\,\varphi(\mathbf{x})\,d\mathbf{x} =\int_\Omega f(\mathbf{x},t)\,\varphi(\mathbf{x})\,d\mathbf{x}.

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

∫ΩΔu(x,t) φ(x) dx=−∫Ω∇u(x,t)⋅∇φ(x) dx+∫∂Ω(∇u(x,t)⋅n⃗(x)) φ(x) dx,\int_\Omega \Delta u(\mathbf{x},t)\,\varphi(\mathbf{x})\,d\mathbf{x} = - \int_\Omega \nabla u(\mathbf{x},t)\cdot \nabla \varphi(\mathbf{x})\,d\mathbf{x} + \int_{\partial\Omega} \left(\nabla u(\mathbf{x},t)\cdot \vec{\mathbf{n}}(\mathbf{x})\right) \, \varphi(\mathbf{x})\,d\mathbf{x},

where the last term is a surface integral along the boundary ∂Ω\partial\Omega of the domain Ω\Omega, and n⃗\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

∫Ωc−1(x)∂t2u(x,t) φ(x) dx+∫Ω∇u(x,t)⋅∇φ(x) dx−∫∂Ω(∇u(x,t)⋅n⃗) φ(x) dx=∫Ωf(x,t) φ(x) dx. \int_\Omega c^{-1}(\mathbf{x}) \partial_t^2 u(\mathbf{x},t)\, \varphi(\mathbf{x})\,d\mathbf{x} + \int_\Omega \nabla u(\mathbf{x},t)\cdot \nabla \varphi(\mathbf{x})\,d\mathbf{x} - \int_{\partial\Omega} \left(\nabla u(\mathbf{x},t)\cdot \vec{\mathbf{n}}\right) \, \varphi(\mathbf{x})\,d\mathbf{x} = \int_\Omega f(\mathbf{x},t)\,\varphi(\mathbf{x})\,d\mathbf{x}.

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