The spectral-element method on a glance

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

c1(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:

Ωc1(x)t2u(x,t)φ(x)dxΩΔu(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=Ωu(x,t)φ(x)dx+Ω(u(x,t)n(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 n{`\\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

Ωc1(x)t2u(x,t)φ(x)dx+Ωu(x,t)φ(x)dxΩ(u(x,t)n)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 u{`u`} that satisfies this weak form for any choice of the test function φ{`\\varphi`} at all times t{`t`} is called a weak solution. Under certain mathematical conditions, u{`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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