Post-processing for spatial accuracy-enhancement of pure Lagrange-Galerkin schemes applied to convection-diffusion equations

Marta Benítez, Bernardo Cockburn

Research output: Contribution to journalArticlepeer-review

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Abstract

We analyze a technique to improve the spatial accuracy, by the single application at the end of the simulation of a local post-processing, for pure Lagrange-Galerkin (PLG) methods applied to evolutionary convection-diffusion (possibly pure convection/diffusion) equations with time-dependent domains. The post-processing technique is based on a simple convolution that extracts the 'hidden accuracy' of Galerkin schemes, and it is used and rigorously analyzed in a fully discrete context.We prove that, when applied to the numerical solution of PLG schemes, it improves the spatial accuracy in the l∞ (L2(ω0))-norm from order k+1 to at least order 2k, where k is the degree of the polynomials defining the finite element space and ω0 any interior region of the computational domain meshed with translation-invariant meshes. For pure convection, a spatial accuracy enhancement in the l∞ (L2(ω0))-norm from order k+1 to order 2k+2 is obtained by post-processing the numerical solution of PLG schemes. Numerical tests are presented that confirm these theoretical results.

Original languageEnglish (US)
Pages (from-to)54-77
Number of pages24
JournalIMA Journal of Numerical Analysis
Volume42
Issue number1
DOIs
StatePublished - Jan 1 2022

Bibliographical note

Funding Information:
Fondo Europeo de Desarrollo Regional (FEDER) and Xunta de Galicia (Spain) (2017 GRC GI-1563 and ED431G/01 to M.B.); FEDER and the Spanish Ministry of Economy and Competitiveness (MTM2017-86459-R to M.B.); National Science Foundation (DMS 1522657 to B.C.).

Publisher Copyright:
© 2020 The Author(s).

Keywords

  • Lagrange-Galerkin methods
  • convection-diffusion equations
  • negative-order norms error estimates
  • post-processing
  • spatial accuracy enhancement
  • superconvergence error estimates

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