### Mathematics Colloquium

# Ralf Hiptmair

### (ETH Zürich)

## "Sparse adaptive finite elements for radiative transfer"

** Date: ** |
Mon, September 6, 2010 |

** Time: ** |
17:15 |

** Place: ** |
Research II Lecture Hall |

**Abstract:** We propose an adaptive multilevel Galerkin FEM
for the numerical solution of the linear radiative transfer equation.
Our approach is based

- on a stabilized variational formulation of the transport operator,
- on so-called sparse tensor products of two hierarchic families of
Finite Element spaces in
*H*^{1} and in *L*^{2}, respectively,
- on the local use of full tensor product spaces for the resolution of influx boundary conditions,
- on thresholding techniques to adapt the discretization to the underlying problem,
- on subspace correction preconditioning of the resulting large
linear system of equations.

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.