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We consider the numerical approximation of boundary conditions in radiativetransfer problems by a perfectly matched layer approach. The main idea is toextend the computational domain by an absorbing layer and to use an appropriatereflection boundary condition at the boundary of the extended domain. A carefulanalysis shows that the consistency error introduced by this approach can bemade arbitrarily small by increasing the size of the extension domain or themagnitude of the artificial absorption in the surrounding layer. A particularchoice of the reflection boundary condition allows us to circumvent thehalf-space integrals that arise in the variational treatment of the originalvacuum boundary conditions and which destroy the sparse coupling observed innumerical approximation schemes based on truncated spherical harmonicsexpansions. A combination of the perfectly matched layer approach with a mixedvariational formulation and a PN-finite element approximation leads todiscretization schemes with optimal sparsity pattern and provable quasi-optimalconvergence properties. As demonstrated in numerical tests these methods areaccurate and very efficient for radiative transfer in the scattering regime.

A perfectly matched layer approach for radiative transfer in highly scattering regimes