introduce a set of $m$ degrees of freedom (DoF). These DoF are, by definition, linear functionals, $l_\beta(u)$ ($\beta = 1,2,\dots, m$), each mapping any admissible field $u$ to a number (real or complex, depending on the problem). The simplest example of DoF for a scalar field $u$ is as set of nodal values $l_\beta(u) \equiv u(\mathbf{r}_\beta)$, where $\mathbf{r}_1,\dots,\mathbf{r}_m$ are a set of grid nodes in $\Omega_h$. In the case of vector fields, one may also consider fluxes, circulations, etc. as other examples of DoF .

Locally, within $\Omega_h$, the solution $u$ is approximated by a linear combination of Trefftz functions $\psi_\alpha$ ($\alpha = 1,2,\dots, n$) : $$\label{eqn:uh-eq-c-psi}
   u(\mathbf{r}) \,\approx\, u_h(\mathbf{r}) \equiv 
   \sum\nolimits_{\alpha=1}^n c_\alpha \psi_\alpha (\mathbf{r}) \,=\,
   \underline{c}^T \underline{\psi}(\mathbf{r}),$$ where $\underline{c} \in \mathbb{C}^n$ is a coefficient vector and $\underline{\psi}$ is a vector of basis functions (both generally complex). In $\Omega_h$, we seek an FD equation of the form $$\label{eqn:s-beta-l-beta-eq-0}
    \sum\nolimits_{\beta=1}^m s_\beta l_\beta (u) = 0,$$ where $\underline{s} = (s_1, s_2,\dots, s_m)^T$ is a vector of complex coefficients (a “scheme”) to be determined. In the simplest version of FLAME, the scheme is required to be exact for any linear combination of basis functions. Then, after straightforward algebra, one obtains [@Tsukerman05; @Tsukerman06] $$\label{eqn:s-in-null-Nt}
