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Adaptive anisotropic Petrov-Galerkin methods for first order transport equations

: Dahmen, W.; Kutyniok, G.; Lim, W.-Q.; Schwab, C.; Welper, G.


Journal of computational and applied mathematics 340 (2018), pp.191-220
ISSN: 0377-0427
ISSN: 0771-050X
Journal Article
Fraunhofer HHI ()

This paper builds on recent developments of adaptive methods for linear transport equations based on certain stable variational formulations of Petrov–Galerkin type. The key issues can be summarized as follows. The variational formulations allow us to employ meshes with cells of arbitrary aspect ratios. We develop a refinement scheme generating highly anisotropic partitions that is inspired by shearlet systems. We establish L2 approximation rates for N-term approximations from corresponding piecewise polynomials for certain compact cartoon classes of functions. In contrast to earlier results in a curvelet or shearlet context the cartoon classes are concisely defined through certain characteristic parameters and the dependence of the approximation rates on these parameters is made explicit here. The approximation rate results serve then as a benchmark for subsequent applications to adaptive Galerkin solvers for transport equations. We outline a new class of directionally adaptive, Petrov–Galerkin discretizations for such equations. In numerical experiments, the new algorithms track C2 discontinuity curves stably and accurately, and realize essentially optimal rates. Finally, we treat parameter dependent transport problems, which arise in kinetic models as well as in radiative transfer. In heterogeneous media these problems feature propagation of singularities along curved characteristics precluding, in particular, fast marching methods based on ray-tracing. Since now the solutions are functions of spatial variables and parameters one has to address the curse of dimensionality. As for non-adaptive schemes considered in Grella and Schwab (2011) and Grella (2014), we show computationally, for a model parametric transport problem in heterogeneous media in 2+1
dimension, that sparse tensorization of the presently proposed directionally adaptive variational discretization scheme in physical space combined with hierarchic collocation in ordinate space can overcome the curse of dimensionality when approximating averaged bulk quantities.