Spectral method

Definition

Let $L$ be a linear differential operator on an interval or a box $\Omega$, and consider $L u = f$ with boundary or periodicity conditions. Fix a family of smooth functions $\varphi_k$ on all of $\Omega$, each nonzero on most of $\Omega$. Typical choices are $e^{ikx}$ for periodic problems, Chebyshev polynomials or Legendre polynomials otherwise. Let $I_N$ be a finite set of $n$ indices. A spectral method seeks $u_N = \sum_{k \in I_N} a_k \varphi_k$ and fixes the $a_k$ by one of three conditions:

1. Galerkin: $\ip{L u_N - f}{\varphi_j} = 0$ for all $j \in I_N$, with the $\varphi_k$ satisfying the boundary conditions;
2. tau: the same equations for all but $m$ indices $j$, plus the $m$ boundary conditions written as equations on the $a_k$;
3. collocation: $L u_N(x_j) = f(x_j)$ at chosen points $x_j$, together with the boundary conditions, with $n$ equations in all.

Here $\ip{\cdot}{\cdot}$ is the $L^2$ inner product, with the weight of the chosen polynomial family.