Adaptive Petrov--Galerkin Methods for First Order Transport Equations

We propose stable variational formulations for certain linear, unsymmetric operators with first order transport equations in bounded domains serving as the primary focus of this paper. The central objective is to develop for such classes adaptive solution concepts with provable error reduction. To adaptively resolve anisotropic solution features such as propagating singularities, the presently proposed variational formulations allow, in particular, the employment of trial spaces spanned by directional representation systems. Since such systems, typically given as frames, are known to be stable only in L₂, special emphasis is placed on L₂-stable formulations. The proposed stability concept is based on perturbations of certain “ideal” test spaces in Petrov--Galerkin formulations; see also L. F. Demkowicz and J. Gopalakrishnan, Comput. Methods Appl. Mech. Engrg. , 199 (2010), pp. 1558--1572, L. Demkowicz and J. Gopalakrishnan, Numer. Methods Partial Differential Equations, 27 (2011), pp. 70--105, J. Zitelli, I. Muga, L. Demkowicz, J. Gopalakrishnan, D. Pardo, and V. Calo, J. Comput. Phys. , 230 (2011), pp. 2406--2432. We propose a general strategy for realizing the resulting Petrov--Galerkin schemes based on an Uzawa iteration circumventing an excessively expensive computation of corresponding test basis functions. Moreover, based on this iteration, we develop adaptive solution concepts with provable error reduction. The results are illustrated by numerical experiments.