321 IGPM321.pdf        January 2011
TITLE Adaptive Petrov-Galerkin Methods for First Order Transport Equations
AUTHORS Wolfgang Dahmen, Chunyan Huang, Christoph Schwab, Gerrit Welper
ABSTRACT We propose a general framework for well posed variational formulations of linear unsymmetric operators, taking first order transport and evolution equations in bounded domains as primary orientation. We outline a general variational framework for stable discretizations of boundary value problems for these operators. To adaptively resolve anisotropic solution features such as propagating singularities the variational formulations should allow one, in particular, to employ as trial spaces directional representation systems. Since such systems are known to be stable 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. We develop a general strategy for realizing corresponding schemes without actually computing excessively expensive test basis functions. Moreover, we develop adaptive solution concepts with provable error reduction. The results are illustrated by first numerical experiments.
KEYWORDS Linear transport problems, L2-stable Petrov-Galerkin formulations, trace theorems, δ-proximality, adaptive refinement schemes, residual approximation, error reduction
DOI 10.1137/110823158
PUBLICATION SIAM journal on numerical analysis
50(5), 2420-2445 (2012)