Formulation and analysis of fully-mixed methods for stress-assisted diffusion problems

Gabriel N. Gatica, Bryan Gomez-Vargas, Ricardo Ruiz-Baier

Research output: Contribution to journalArticleResearchpeer-review

Abstract

This paper is devoted to the mathematical and numerical analysis of a mixed-mixed PDE system describing the stress-assisted diffusion of a solute into an elastic material. The equations of elastostatics are written in mixed form using stress, rotation and displacements, whereas the diffusion equation is also set in a mixed three-field form, solving for the solute concentration, for its gradient, and for the diffusive flux. This setting simplifies the treatment of the nonlinearity in the stress-assisted diffusion term. The analysis of existence and uniqueness of weak solutions to the coupled problem follows as combination of Schauder and Banach fixed-point theorems together with the Babuška–Brezzi and Lax–Milgram theories. Concerning numerical discretization, we propose two families of finite element methods, based on either PEERS or Arnold–Falk–Winther elements for elasticity, and a Raviart–Thomas and piecewise polynomial triplet approximating the mixed diffusion equation. We prove the well-posedness of the discrete problems, and derive optimal error bounds using a Strang inequality. We further confirm the accuracy and performance of our methods through computational tests.

Original languageEnglish
Pages (from-to)1312-1330
Number of pages19
JournalComputers and Mathematics with Applications
Volume77
Issue number5
DOIs
Publication statusPublished - 1 Mar 2019
Externally publishedYes

Keywords

  • A priori error analysis
  • Augmented fully-mixed formulation
  • Finite element methods
  • Fixed-point theory
  • Stress–diffusion coupling

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