 # 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. PAGE CONTENTS