Coupling nonlinear Stokes and Darcy flow using mortar finite elements

Ervin, Vincent J.; Jenkins, Eleanor W.; Sun, Shuyu

Coupling nonlinear Stokes and Darcy flow using mortar finite elements

Type

Article

Authors

Ervin, Vincent J.
Jenkins, Eleanor W.
Sun, Shuyu

KAUST Department

Computational Transport Phenomena Lab
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Earth Science and Engineering Program
Environmental Science and Engineering Program
Physical Science and Engineering (PSE) Division

Date

2011-11

Abstract

We study a system composed of a nonlinear Stokes flow in one subdomain coupled with a nonlinear porous medium flow in another subdomain. Special attention is paid to the mathematical consequence of the shear-dependent fluid viscosity for the Stokes flow and the velocity-dependent effective viscosity for the Darcy flow. Motivated by the physical setting, we consider the case where only flow rates are specified on the inflow and outflow boundaries in both subdomains. We recast the coupled Stokes-Darcy system as a reduced matching problem on the interface using a mortar space approach. We prove a number of properties of the nonlinear interface operator associated with the reduced problem, which directly yield the existence, uniqueness and regularity of a variational solution to the system. We further propose and analyze a numerical algorithm based on mortar finite elements for the interface problem and conforming finite elements for the subdomain problems. Optimal a priori error estimates are established for the interface and subdomain problems, and a number of compatibility conditions for the finite element spaces used are discussed. Numerical simulations are presented to illustrate the algorithm and to compare two treatments of the defective boundary conditions. © 2010 Published by Elsevier B.V. on behalf of IMACS.

Citation

Ervin, V. J., Jenkins, E. W., & Sun, S. (2011). Coupling nonlinear Stokes and Darcy flow using mortar finite elements. Applied Numerical Mathematics, 61(11), 1198–1222. doi:10.1016/j.apnum.2011.08.002