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

Author: 

Gatica, G
Gomez-Vargas, B
Ruiz-Baier, R

Publication Date: 

1 March 2019

Journal: 

COMPUTERS & MATHEMATICS WITH APPLICATIONS

Last Updated: 

2019-07-01T07:16:47.34+01:00

Issue: 

5

Volume: 

77

DOI: 

10.1016/j.camwa.2018.11.008

page: 

1312-1330

abstract: 

© 2018 The Author(s) 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.

Symplectic id: 

940492

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Submitted to ORA: 

Submitted

Publication Type: 

Journal Article