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Zbl 1005.65143
Hansen, Olaf
On the stability of the collocation method for the radiosity equation on polyhedral domains. (English)
[J] IMA J. Numer. Anal. 22, No.3, 463-479 (2002). ISSN 0272-4979; ISSN 1464-3642/e

The radiosity equation is a second-kind integral equation of the form $$(I-\Cal K)u(x) = E(x),\quad x\in S,\tag 1$$ where $$(\Cal K)u(x):=\frac{\rho(x)}{\pi} \int_S\beta(x,y)\frac{{\overset\rightarrow\to n}(x)\cdot(y - x) {\overset\rightarrow\to n}(y)\cdot(x-y)}{|x-y|^4}u(y) dy\tag 2$$ is the radiosity operator, $S$ is a surface in $\Bbb R^3$ and the emissivity function $E : S\to\Bbb R$ describes the emission of light at point $x$.\par The unknown function $u$ characterizes the brightness at every point $x$ and (1) implies that this brightness is due to the emissivity $E(x)$ and the reflection of incoming light $\Cal Ku$. Formula (2) describes the physics which is assumed to be valid. The author defines a graded triangulation of the polyhedral surface $S$, constructs the trial space of piecewise polynomials and introduces some modification ($i^\ast$-trick) near the edges.\par The mapping properties of the operator $\Cal K$ are studied. The first result is the contractivity of $\Cal K$ in $L^\infty(S)$. This is sufficient for the invertibility of $I-\Cal K$ in $L^\infty(S)$. A face $\Delta_j$ of $S$ is fixed and the properties of $\Cal Ku$ on $\Delta_j$ are investigated. Denoting by $\Cal N_j$ the faces of $S$ which are neighbours of $\Delta_j$, it is shown that $\Cal K(u|_{S\setminus\Cal N_j})$ is Lipschitz continuous. The Lipschitz continuity of $\Cal K(u|_{\Cal N_j})$ on $\Delta_j$ is also proved.\par With these results, it is proved that the collocation projector $P_N\Cal K$ is also a contraction in $L^\infty(S)$ if $N$ is sufficiently large and some $i^\ast$-modification is used. This is enough to prove the stability of the collocation method, used for the convergence proof of the method. The only assumption needed for the convergence proof is the piecewise continuity of the right-hand side $E$ of (1).
[Nikolay Yakovlevich Tikhonenko (Odessa)]

MSC 2000:: *65R20 Integral equations (numerical methods)
45E10 Integral equations of the convolution type
78A40 Waves and radiation

Keywords: radiosity equation; radiosity operator; contractivity; mapping properties; collocation method; stability; convergence; graded triangulation of polyhedral surface; trial space of piecewise polynomials; $i^\ast$-trick; second-kind integral equation

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