Adaptive Strategies for Transport Equations

TL;DR

This paper develops efficient a posteriori error bounds and adaptive strategies for linear transport equations, analyzing their reliability, computational cost, and error reduction properties in both one and multiple dimensions.

Contribution

It introduces new a posteriori error bounds for transport equations and investigates adaptive refinement criteria, including global error reduction analysis in multiple dimensions.

Findings

Error estimators are efficient and reliable under mild conditions.

Adaptive strategies achieve fixed error reduction in 1D, partial results in multidimensional cases.

Global arguments are crucial for error reduction in transport equations.

Abstract

This paper is concerned with a posteriori error bounds for linear transport equations and related questions of contriving corresponding adaptive solution strategies in the context of Discontinuous-Petrov-Galerkin schemes. After indicating our motivation for this investigation in a wider context the first major part of the paper is devoted to the derivation and analysis of a posteriori error bounds that, under mild conditions on variable convection fields, are efficient and, modulo a data-oscillation term, reliable. In particular, it is shown that these error estimators are computed at a cost that stays uniformly proportional to the problem size. The remaining part of the paper is then concerned with the question whether typical bulk criteria known from adaptive strategies for elliptic problems entail a fixed error reduction rate also in the context of transport equations. This turns out to be significantly more difficult than for elliptic problems and at this point we can give a complete affirmative answer for a single spatial dimension. For the general multidimensional case we provide partial results which we find of interest in their own right. An essential distinction from known concepts is that global arguments enter the issue of error reduction. An important ingredient of the underlying analysis, which is perhaps interesting in its own right, is to relate the derived error indicators to the residuals that naturally arise in related least squares formulations. This reveals a close interrelation between both settings regarding error reduction in the context of adaptive refinements.