The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach Spaces

IGNACIO PATRICIO PEDRO MUGA URQUIZA, Matthew J.W. Tyler, Kristoffer G. Van Der Zee

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

We propose and analyze a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak formulation allows for the direct approximation of solutions in the Lebesgue Lp, 1 < p < ∞. The greater generality of this weak setting is natural when dealing with rough data and highly irregular solutions, and when enhanced qualitative features of the approximations are needed. We first present a rigorous analysis of the well-posedness of the underlying continuous weak formulation, under natural assumptions on the advection-reaction coefficients. The main contribution is the study of several discrete subspace pairs guaranteeing the discrete stability of the method and quasi-optimality in L p {L^{p}}, and providing numerical illustrations of these findings, including the elimination of Gibbs phenomena, computation of optimal test spaces, and application to 2-D advection.

Original languageEnglish
Pages (from-to)557-579
Number of pages23
JournalComputational Methods in Applied Mathematics
Volume19
Issue number3
DOIs
StatePublished - 1 Jul 2019

Keywords

  • Advection-Reaction
  • Banach Spaces
  • Compatible FE Pairs
  • DDMRes
  • Discrete Dual Norms
  • Fortin Condition
  • Petrov-Galerkin Method
  • Residual Minimization

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