On the stability of the Collocation method for the Radiosity equation on polyhedral domains

Hansen, Olaf

doi:10.1093/imanum/22.3.463

Cited by 4 publications

(10 citation statements)

References 7 publications

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“…Although there is a huge literature on the radiosity equation in computer graphics, see the references in [7, 20 25], there are not so many mathematical investigations on the structure, for example regularity, of the solution ū of (1.1). Existence theory in the L p (S) spaces, p ∈ [1, ∞], or in spaces of continuous functions, under certain assumptions on S and E, can be found in [1,3,4,10,16,17]. In [17] the properties of ū along an edge were studied, if the function E is differentiable and in [3] one finds results for the regularity of ū near corners in the two-dimensional cases.…”

Section: Introductionmentioning

confidence: 99%

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

Hansen¹

2003

J. Integral Equations Applications

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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.

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Section: Introductionmentioning

confidence: 99%

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

Hansen¹

2003

J. Integral Equations Applications

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“…in order to have (28) be satisfied. A less restrictive analysis is given in Hansen [21] to obtain (29).…”

Section: General Framework Of the Projection Methodsmentioning

confidence: 99%

Analytic Evaluation of Collocation Integrals for the Radiosity Equation

Seol

2005

Appl. Num. Anal. Comp. Math.

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In this work, we consider solving the radiosity equation using the collocation method. We develop analytic evaluation of the integrations which are needed to setup the linear system in solving the radiosity equation using the collocation method. These integrations are over triangular elements in R 3 . Our approach is to use affine transformations to convert integrations over elements in R 3 to integrations over elements in R 2 and then to use a change of variables. For this, we introduce functions H m,n,k for m, n, k ∈ N0 and use these to give our analytic formulas. The analytic evaluations of Hm,n,4 and other relevant integrations are given in detail for some values of m and n. Finally, a performance comparison of the analytic evaluation integration with that of other well-known numerical integration schemes is given.

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“…This can be generalized to the case of a piecewise C 1 -boundary (see for example [3], [15], [2]). The second constitutive relation between R and J is then given by (0.8)…”

Section: Problem Descriptionmentioning

confidence: 99%

“…Thanks to (4.23) and (4.24), we can elementarily pass to the limit in (4.4) and get The following lemma has been proved in [3] for polyhedral surfaces, in [14] for piecewise C 1 −boundaries.…”

Section: P R O O F (1)mentioning

confidence: 99%

Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary p-summable right-hand side

Druet

2010

Appl Math

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On the stability of the Collocation method for the Radiosity equation on polyhedral domains

Cited by 4 publications

References 7 publications

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

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

Analytic Evaluation of Collocation Integrals for the Radiosity Equation

Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary p-summable right-hand side

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