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# The spectral-element method on a glance

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

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

with homogenuous initial conditions

$u(\mathbf{x},0) = 0, \quad \partial_t u(\mathbf{x},0) = 0,$

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 $\varphi(\mathbf{x})$ and integrating in space:

$\int_\Omega c^{-1}(\mathbf{x}) \partial_t^2 u(\mathbf{x},t)\, \varphi(\mathbf{x})\,d\mathbf{x} - \int_\Omega \Delta u(\mathbf{x},t)\,\varphi(\mathbf{x})\,d\mathbf{x} =\int_\Omega f(\mathbf{x},t)\,\varphi(\mathbf{x})\,d\mathbf{x}.$

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

$\int_\Omega \Delta u(\mathbf{x},t)\,\varphi(\mathbf{x})\,d\mathbf{x} = - \int_\Omega \nabla u(\mathbf{x},t)\cdot \nabla \varphi(\mathbf{x})\,d\mathbf{x} + \int_{\partial\Omega} \left(\nabla u(\mathbf{x},t)\cdot \vec{\mathbf{n}}(\mathbf{x})\right) \, \varphi(\mathbf{x})\,d\mathbf{x},$

where the last term is a surface integral along the boundary $\partial\Omega$ 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

$\int_\Omega c^{-1}(\mathbf{x}) \partial_t^2 u(\mathbf{x},t)\, \varphi(\mathbf{x})\,d\mathbf{x} + \int_\Omega \nabla u(\mathbf{x},t)\cdot \nabla \varphi(\mathbf{x})\,d\mathbf{x} - \int_{\partial\Omega} \left(\nabla u(\mathbf{x},t)\cdot \vec{\mathbf{n}}\right) \, \varphi(\mathbf{x})\,d\mathbf{x} = \int_\Omega f(\mathbf{x},t)\,\varphi(\mathbf{x})\,d\mathbf{x}.$

A function $u$ 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.

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