Numerical Methods for the Radiosity Equation and Related Problems

In this work we present numerical methods for the solution of Fredholm integral equations of the second type for smooth and piecewise smooth surfaces We use a collocation method based on piecewise polynomial interpolation of the solution We consider only collocation methods for which the collocation nodes are interior to each triangular face In Chapter II we give the general framework for collocation methods based on interpolation We show that interpolation of degree r of the solution leads to an error in the collocation method of O h where h is the mesh size of the triangulation and so collocation methods of any given order can be developed In Chapter III we discuss superconvergent methods as particular cases of the methods introduced in Chapter II The radiosity equation is introduced along with some of its properties Next we discuss two superconvergent collocation methods based on piecewise quadratic interpolation for the radiosity equation followed by numerical examples We conclude this chapter with giving generalized superconver gent methods based on interpolation of any degree r considering separately the case where r is odd and the case where r is even In the following chapter the ideas described earlier are used for nding numerical solutions of the exterior Neumann problem since in solving this problem we encounter integral equations whose properties are very similar to the ones of the radiosity equa tion Considering collocation methods that use only interior nodes is especially useful in solving this problem We describe a collocation method based on interpolation of the solution for solving the integral equation derived from the exterior Neumann problem giving numerical examples for the case of piecewise constant interpolation of the solution centroid rule In the concluding chapter we draw some important and interesting conclusions as well as discuss some possible ideas for future work in this area Abstract approved Thesis supervisorapproved Thesis supervisor Title and department