Adaptive Anisotropic Petrov-Galerkin Methods for First Order Transport Equations

TL;DR

This paper introduces adaptive anisotropic Petrov-Galerkin methods for first-order transport equations, enabling stable, accurate, and efficient approximation of solutions with complex features and parameter dependencies, surpassing previous approaches.

Contribution

It develops a novel anisotropic refinement scheme inspired by shearlet systems and provides explicit approximation rates for cartoon classes, advancing adaptive Galerkin methods for transport equations.

Findings

Algorithms effectively track curved shear layers and discontinuities.

Achieves near-optimal approximation rates in numerical experiments.

Sparse tensorization mitigates the curse of dimensionality in parametric problems.

Abstract

This paper builds on recent developments of adaptive methods for linear transport equations based on certain stable variational formulations of Petrov-Galerkin type. 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 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. In numerical experiments, the new algorithms track C^2-curved shear layers and discontinuities 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. We show computationally, for a model parametric transport problem in heterogeneous media in 2 + 1 dimension, that sparse tensorization of the presently proposed spatial directionally adaptive scheme with hierarchic collocation in ordinate space based on a stable variational formulation high-dimensional phase space, the curse of dimensionality can be removed when approximating averaged bulk quantities.