The Radiosity Equation on Certain Spaces of Continuous Functions and Its Numerical Solution

Abstract

In this article we study the radiosity equation on a polyhedral surface S in R 3 . We construct a special space of continuous functions on S where we can prove the existence of a unique solution of the radiosity equation. These results enable us to construct grids for the numerical approximation of the solution which guarantee convergence in the maximum norm for the collocation method. In a last section we present numerical results which confirm our theoretical prediction and these results also show that graded meshes will increase the order of convergence.

The mapping properties of the radiosity operator along an edge

“…This corresponds to results which were obtained in Reference [11]. Our ÿnal Corollary 3.7 brings us to the usual situation for integral equations of the second kind (on smooth domains), where the right-hand side determines exactly the smoothness of the solution.…”

“…Therefore, we cannot expect smoothing properties along the edge. This corresponds to the results in Reference [11]. There we considered two triangles 1 and 2 , which may be part of two half-planes with a common edge.…”

On the Numerical Treatment of Heat Conduction Problem by Boundary Element and Multigrid Methods

“…The entries of the matrix A as well as the entries of the vector f have been computed numerically. To keep the numerical integration error small, we handle the singularity of the integral kernels by employing double partial derivatives, see (Hansen, 2003). Note that the step size h k is associated with the dimension parameter n k , where h k = 1 n k with n k = 2 k and k is called the level number.…”