In solving the Ion Cyclotron Resonance Heating wave equation, plane wave Fourier mode $\exp[ikx]$ base functions are often exploited to produce expressions for the dielectric response when including the effect of finite temperature, requiring locally integrating the equation of motion of charged particles by hand. Standard Fourier analysis notation is used: $x$ is the position, $k$ is the wave vector component in the $x$-direction and $i$ is the imaginary number. In contrast, finite element techniques adopt base functions – typically low - order polynomials – that individually are only non-zero in a small domain. They are a natural and easy approach to capture inhomogeneity effects and readily allow grid refinement to zoom in on regions where this is required. To enable ample realism of kinetic effects while profiting from the rich pool of numerical tools available for solving differential and integro-differential equations relying on finite elements, it is desirable to have a procedure allowing us to profit from both approaches: the detailed physics brought by finite temperature effects – commonly described in terms of Fourier modes – as well as the simplicity from a local polynomial representation. A novel technique is offered to achieve that. It consists of finding the Fourier representation of the localised base functions exploited in finite elements so that the richness of the dielectric response in $k$-space can be accounted for. The resulting equation is assembled just like the finite element method prescribes but the coefficients of the linear local system are assembled differently. The technique allows us to capture finite temperature corrections in both the parallel and perpendicular directions for a dielectric tensor model of choice. The focus in this paper is on the numerical technique while the expressions for the dielectric response are assumed to be known. A few first examples are briefly discussed: the wave equation solutions for a typical minority heating (H)-D JET plasma are provided for (i) an all-FLR model, (ii) the FLR-0 ‘tepid’ equivalent and (iii) a cold plasma model; FLR refers to the Finite Larmor Radius expansion exploited to account for finite temperature corrections.

The two-plasmon decay instability of high-power microwaves used for electron cyclotron resonance heating has been observed experimentally in multiple fusion devices. This type of instability is a nonlinear three-wave interaction that can transfer energy away from the cyclotron harmonics. Analytical models quantify exponential growth rates and power thresholds, but typically evaluate the growth as a spatially averaged gain over the interaction region of the instability. This description effectively excludes field inhomogeneities and noise, leaving their impact on the instability growth uncertain. We assess this assumption by solving the full nonlinear system on a spatial grid. Across all cases considered, we find that the instabilities drive the wave fields toward the spatially averaged behaviour. After a transient period that scales inversely with the instability growth rate, the simulated growth converges to the averaged prediction, indicating that the established models are asymptotically valid.