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How would you describe the Galerkin method to solving the 3D wave equation $$u_{tt}= c^2\Delta u$$ to someone who wants to implement it immediately?

More precisely, we want to solve the Cauchy problem $$u_{tt}= c^2\Delta u , \\ u_t(x,y,z,0)= g(x,y,z) , \\ u(x,y,z,0)= f(x,y,z)$$ where $f, g$ are given compactly supported functions.

Say in a bounded region of the space we are interested in, say a ball of radius $R$, note that the wave propagates towards the boundary of the region, the sphere, thus we can't impose the condition at the boundary.

How would you implement this method ? I am looking for a description of the finite elements, how to decompose this region? (you can use any other shape, not necessarily a ball, take a cube for example), then describing the numerical algorithm necessary to find the values of the solution $u$ at certain element or node $(x,y,z,T)$ at time $T$

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  • $\begingroup$ This is one of the three standard equations discussed in nearly every book about the finite element method. Have you taken a look at the typical finite element books? $\endgroup$
    – Wolfgang Bangerth
    May 19, 2022 at 17:24
  • $\begingroup$ What book would you recommend please? $\endgroup$
    – NotaChoice
    May 20, 2022 at 2:03
  • $\begingroup$ I doubt this is a Cauchy problem since you don't have boundary conditions. $\endgroup$
    – nicoguaro ♦
    May 20, 2022 at 17:21

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