On monotonicity-preserving stabilized finite element approximations of transport problems

Santiago Badia, Alba Hierro

Research output: Contribution to journalArticleResearchpeer-review

20 Citations (Scopus)

Abstract

The aim of this work is to design monotonicity-preserving stabilized finite element techniques for transport problems as a blend of linear and nonlinear (shock-capturing) stabilization. As linear stabilization, we consider and analyze a novel symmetric projection stabilization technique based on a local Scott-Zhang projector. Next, we design a weighting of the aforementioned linear stabilization such that, when combined with a finite element discretization enjoying a discrete maximum principle (usually attained via nonlinear stabilization), it does not spoil these monotonicity properties. Then, we propose novel nonlinear stabilization schemes in the form of an artificial viscosity method where the amount of viscosity is proportional to gradient jumps at either finite element boundaries or nodes. For the nodal scheme, we prove a discrete maximum principle for time-dependent multidimensional transport problems. Numerical experiments support the numerical analysis and we show that the resulting methods provide excellent results. In particular, we observe that the proposed nonlinear stabilization techniques do an excellent job eliminating oscillations around shocks.

Original languageEnglish
Pages (from-to)A2673-A2697
Number of pages25
JournalSIAM Journal on Scientific Computing
Volume36
Issue number6
DOIs
Publication statusPublished - 1 Jan 2014
Externally publishedYes

Keywords

  • Convection-dominated flows
  • Nonlinear stabilization
  • Shock-capturing
  • Stabilized finite elements
  • Transport problem

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