Abstract
The linear radiative transfer equation, a partial differential equation for the radiation intensity u ( x , s ) , with independent variables x ∈ D ⊂ R n in the physical domain D of dimension n = 2 , 3 , and angular variable s ∈ S 2 : = { y ∈ R 3 : | y | = 1 } , is solved in the n + 2 -dimensional computational domain D × S 2 . We propose an adaptive multilevel Galerkin finite element method (FEM) for its numerical solution. Our approach is based on (a) a stabilized variational formulation of the transport operator, (b) on so-called sparse tensor products of two hierarchic families of finite element spaces in H 1 ( D ) and in L 2 ( S 2 ) , respectively, and (c) on wavelet thresholding techniques to adapt the discretization to the underlying problem. An a priori error analysis shows, under strong regularity assumptions on the solution, that the sparse tensor product method is clearly superior to a discrete ordinates method, as it converges with essentially optimal asymptotic rates while its complexity grows essentially only as that for a linear transport problem in R n . Numerical experiments for n = 2 on a set of example problems agree with the convergence and complexity analysis of the method and show that introducing adaptivity can improve performance in terms of accuracy vs. number of degrees even further.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.