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Adaptive Petrov-Galerkin methods for first order transport equations. (English) Zbl 1260.65091

The authors propose stable variational formulations for certain linear, unsymmetric operators with first order transport equations in bounded domains. 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 \(L_2\), special emphasis is placed on \(L_2\)-stable formulations. The proposed stability concept is based on perturbations of certain “ideal” test spaces in Petrov-Galerkin formulations. 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 is proposed. Moreover, based on this iteration, the authors develop adaptive solution concepts with provable error reduction. Some numerical examples are presented to support the theoretical results.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L02 First-order hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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