Journal of Computational Physics
The solution of multidimensional real Helmholtz equations on Sparse Grids
SIAM Journal on Scientific Computing
Least-Squares Finite-Element Solution of the Neutron Transport Equation in Diffusive Regimes
SIAM Journal on Numerical Analysis
A Boundary Functional for the Least-Squares Finite- Element Solution of Neutron Transport Problems
SIAM Journal on Numerical Analysis
Adaptive wavelet methods for elliptic operator equations: convergence rates
Mathematics of Computation
Sparse grid spaces for the numerical solution of the electronic Schrödinger equation
Numerische Mathematik
An accurate solver for forward and inverse transport
Journal of Computational Physics
Sparse tensor spherical harmonics approximation in radiative transfer
Journal of Computational Physics
Multiple point evaluation on combined tensor product supports
Numerical Algorithms
Hi-index | 31.46 |
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 DxS^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.