Synthesizing realistic images is a fundamental challenge for computer graphics. One of the most difficult aspects of this task is the simulation of global illumination: the interreflection of light between surfaces. This research applies finite element methods to the simulation of global illumination. The work is divided into three subprojects: (1) error metrics for global illumination, (2) adaptive meshing for radiosity, and (3) generalization to scenes with arbitrary reflectance. Rather than attempting to measure ``realism'', which is a rather nebulous perceptual quality, simpler, quantitative accuracy measures are being developed. Using these error measures, existing and future image synthesis algorithms can be evaluated and quantitatively ranked. Adaptive meshes for radiosity mesh generation is only partially automated in many radiosity systems, and little is known about the errors introduced by various meshes. Recent experiments on simplified, two-dimensional scenes show that the use of adaptive meshes and higher degree approximations can lead to much faster and/or more accurate solutions than those of existing algorithms. More accurate, better looking pictures result when the radiosity mesh follows discontinuities in the solution function such as those at shadow boundaries. These ``discontinuity meshing'' techniques have been embraced by researchers. In this research, meshes for radiosity are compared and evaluated using error metrics as an objective criterion. This rigorous analysis of radiosity functions suggests new mesh generation techniques allowing more accurate simulations. Generalization to scenes with arbitrary reflectance, the final sub-project, is the generalization of these methods to scenes with realistic light scattering. The reflectance of real surfaces is not purely diffuse or specular, but is described by a complex bidirectional reflectance distribution function (BRDF). Recent work suggests that the use of spherical harmonics or other appropriate basis functions greatly facilitates the approximation of direction-dependent light functions. Such representations for functions of both position and direction are being analyzed and developed. The ultimate goal is the development of accurate, efficient algorithms for simulating global illumination on complex scenes with arbitrary reflectance and transmittance.