Form of the solution

An analytical solution for simple geometries can be obtained by many methods; one of them — the so-called Fourier method — we have already used in the section “General analytical solution of the boundary value problem”. We assume in advance that the solution, even before it is obtained, is already expanded into a Fourier series — the expansions in the individual coordinates are considered mutually independent — and all that remains is to determine the specific values of the general solution (2.17$$\begin{split} &T(M, t) =\\ &\sum_{n=1}^{\infty} \frac{\displaystyle \Psi_n(M)}{\displaystyle \|\Psi_n(M)\|^2} \cdot e^{- a^2 \cdot \gamma_n^2 \cdot t} \cdot \iiint_G \rho(M) \cdot T_0(M) \cdot \Psi_n(M) \,dG +\\ &\sum_{n=1}^{\infty} \frac{\displaystyle \Psi_n(M)}{\displaystyle \|\Psi_n(M)\|^2} \cdot \int_0^t e^{- a^2 \cdot \gamma_n^2 \cdot (t - \tau)} \cdot \iiint_G \rho(M) \cdot f(M, \tau) \cdot \Psi_n(M) \,dG \,d\tau. \end{split}$$). The difference is that in the Fourier method we partition the solution but not the geometry — which is why an analytical solution is possible only for a limited set of geometries — while in FEM we partition the geometry itself into simple elements, usually called simplices. The solution on these elements is approximated by so-called trial functions.

So, let us partition the space $M$ into simplex subspaces $\sigma_h$, which completely cover the original space. In each subspace there will exist its own solution $\upsilon_h$ $= T(\sigma_h)$ — the restriction of the sought solution to the simplex, and the collection of such solutions will be the solution of the boundary value problem. Next, let us talk about simplices.