Radiative transfer through absorbing and scattering media carries a large part of the energy balance in high-temperature reactor concepts, yet open, independently verified three-dimensional solvers for the radiative transport equation (RTE) remain scarce. Three solvers are developed and verified here on an isothermal spherical cavity at reactor-relevant conditions (R = 0.25 m, medium at 1200 K, black wall at 600 K, optical radius βR = 1.5, albedo 1/3): a continuous Galerkin FEM (CG-FEM) on linear tetrahedra with analytically derived elemental matrices, a discontinuous Galerkin FEM (DG-FEM) with an upwind numerical flux, and a mesh-free physics-informed neural network (RTE-PINN). The reference is not another code but a semi-analytical solution obtained by integrating the RTE along its characteristics; it is exact in angle and reproduces its own closed form in the purely absorbing limit to 1.1 × 10⁻¹⁵. On a common mesh and a common 20 000-point sample the relative L₂ errors in the incident radiation are 3.71 % (CG-FEM), 1.19 % (DG-FEM) and 1.33 % (RTE-PINN). Both FEM schemes converge at O(h^1.3)–O(h^1.5), while the angular error saturates completely between S₄ and S₈: the residual at βR = 1.5 is spatial, not a ray effect, and concentrates in the near-wall layer where CG-FEM suffers from strong nodal enforcement of the inflow condition. Over a sweep of βR from 0.15 to 5 the unstabilised CG-FEM error rises from 1.56 % to 6.83 %, while DG-FEM (0.55–2.02 %) and the PINN (1.74–2.83 %) are far less sensitive.

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