This article deals with a finite dimensional reduced order state observer for a class of nonlinear partial differential equations (PDEs) described by a radiative transfer equation (RTE) coupled with a nonlinear heat equation (NHE) in two-dimensional domains. First, the original plant is approximated by an N-dimensional ordinary differential equation (ODE) system using both discontinuous and continuous Galerkin methods. Thanks to the differential mean value theorem (DMVT), both high order and reduced order state observers are provided. The convergence of the discretized observer to the state of the original system is established. The error dynamic system was written as a linear parameter varying (LPV) system, and a linear matrix inequality (LMI) methodology is used to prove sufficient convex conditions for global convergence. Furthermore, we show how to construct the observer gains to ensure exponential convergence. Finally, an extension to H-infinity performance analysis, in the presence of disturbances and/or discretization errors, is also developed. In order to show the high accuracy of the proposed technique, in terms of precision and low computational requirements, numerical examples are provided.