Techniques are presented for applying more advanced finite element techniques to the global illumination problem in order to yield more accurate results. First, piecewise-linear, piecewise-quadratic, and higher order elements are discussed as a superior alternative to current piecewise-constant radiosity assumptions. Second, Galerkin techniques are a more robust alternative to current point collocation (point sampling) techniques. Finally, occlusions in a scene give rise to discontinuities such as shadow edges in the solution function. Discontinuity meshing is introduced as a technique for resolving these discontinuities by adaptive placement of element boundaries. Illustrations, algorithms, and results are given for two-dimensional radiosity in flatland problems.