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The finite‐element method (FEM) has been successfully applied for finding approximate solutions to the radiosity equation which plays an important role in the field of computer graphics. The author shows how these methods can quite similarly be utilised for the calculation of the transfer function of the optical wireless communication channel. The analysis is performed in the frequency domain and is not restricted to a finite order of light reflections as is the case for the time‐domain approaches known from the literature. The numerical results show that the Galerkin FEM approach increases the accuracy obtained for the received optical power and for the channel transfer function in the frequency region that is relevant to data transmission or, vice versa, it reduces the computational complexity for a given required accuracy. [ABSTRACT FROM AUTHOR]

While it is well-known that the standard integral operator K of (stationary) diffuse-gray radiation, as it occurs in the radiosity equation, is compact if the domain of radiative interaction is sufficiently regular, we show noncompactness of the operator if the domain is polyhedral. We also show that a stationary operator is never compact when reinterpreted in a transient setting. Moreover, we provide new proofs, which do not use the compactness of K, for 1 being a simple eigenvalue of K for connected enclosures, and for $${I-(1-\epsilon)K}$$ being invertible, provided the emissivity $${\epsilon}$$ does not vanish identically. [ABSTRACT FROM AUTHOR]

We study the mapping properties of the reflection operator on a conical surface. This allows us to derive regularity results for the solution of the radiosity equation on conical surfaces in a scale of weighted Sobolev spaces. To motivate the calculations we first study the operator on a cylinder. Here we estimate the asymptotic behavior of the spectrum of the reflection operator by partial integration. This method works also for the conical case, but first we have to find a simple representation for some hypergeometric functions. [ABSTRACT FROM AUTHOR]

This paper investigates the stability of the collocation method for the radiosity equation. We introduce graded meshes and trial spaces of piecewise polynomials and prove stability for a modified collocation method. Graded meshes are necessary for higher convergence rates. The generation of triangulations which allow higher‐order approximations leads to geometrical problems which are interesting in themselves, but do not affect the stability of the collocation method. [ABSTRACT FROM PUBLISHER]

A “fast matrix–vector multiplication method” is proposed for iteratively solving discretizations of the radiosity equation (I — К)u = E. The method is illustrated by applying it to a discretization based on the centroid collocation method. A convergence analysis is given for this discretization, yielding a discretized linear system (I — K n ) u n = E n. The main contribution of the paper is the presentation of a fast method for evaluating multiplications Kn v, avoiding the need to evaluate Kn explicitly and using fewer than O( n 2) operations. A detailed numerical example concludes the paper, and it illustrates that there is a large speedup when compared to a direct approach to discretization and solution of the radiosity equation. The paper is restricted to the surface S being unoccluded, a restriction to be removed in a later paper. [ABSTRACT FROM AUTHOR]