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Well-posed variational formulations of Friedrichs-type systems

Authors
  • Berggren, Martin
  • Hägg, Linus
Publication Date
Jan 01, 2021
Identifiers
DOI: 10.1016/j.jde.2021.05.002
OAI: oai:DiVA.org:umu-175317
Source
DiVA - Academic Archive On-line
Keywords
Language
English
License
Green
External links

Abstract

All finite element methods, as well as much of the Hilbert-space theory for partial differential equations, rely on variational formulations, that is, problems of the type: find u∈V such that a(v,u)=l(v) for each v∈L, where V,L are Sobolev spaces. However, for systems of Friedrichs type, there is a sharp disparity between established well-posedness theories, which are not variational, and the very successful discontinuous Galerkin methods that have been developed for such systems, which are variational. In an attempt to override this dichotomy, we present, through three specific examples of increasing complexity, well-posed variational formulations of boundary and initial–boundary-value problems of Friedrichs type. The variational forms we introduce are generalizations of those used for discontinuous Galerkin methods, in the sense that inhomogeneous boundary and initial conditions are enforced weakly through integrals in the variational forms. In the variational forms we introduce, the solution space is defined as a subspace V of the graph space associated with the differential operator in question, whereas the test function space L is a tuple of L2 spaces that separately enforce the equation, boundary conditions of characteristic type, and initial conditions. / <p>Previously included in thesis in manuscript form.</p>

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