TY - JOUR
T1 - Analysis of an Augmented Fully-mixed Finite Element Method for a Three-dimensional Fluid-solid Interaction Problem
AU - G. Gatica, A. Marquez & S. Meddahi
JO - International Journal of Numerical Analysis and Modeling
VL - 3
SP - 624
EP - 656
PY - 2014
DA - 2014/11
SN - 11
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/545.html
KW - mixed finite elements
KW - Arnold-Falk-Winther elements
KW - Helmholtz
KW - elastodynamic
AB - We introduce and analyze an augmented fully-mixed finite element method for a fluid-solid interaction problem in 3D. The media are governed by the acoustic and elastodynamic
equations in time-harmonic regime, and the transmission conditions are given by the equilibrium
of forces and the equality of the corresponding normal displacements. We first employ dualmixed
variational formulations in both domains, which yields the Cauchy stress tensor and the
rotation of the solid, together with the gradient of the pressure of the fluid, as the preliminary
unknowns. This approach allows us to extend an idea from a recent own work in such a way that
both transmission conditions are incorporated now into the definitions of the continuous spaces,
and therefore no unknowns on the coupling boundary appear. As a consequence, the pressure
of the fluid and the displacement of the solid become explicit unknowns of the coupled problem,
and hence two redundant variational terms arising from the constitutive equations, both of them
multiplied by stabilization parameters, need to be added for well-posedness. In fact, we show that
explicit choices of the above mentioned parameters and a suitable decomposition of the spaces
allow the application of the Babuška-Brezzi theory and the Fredholm alternative for concluding
the solvability of the resulting augmented formulation. The unknowns of the fluid and the solid
are then approximated by a conforming Galerkin scheme defined in terms of Arnold-Falk-Winther
and Lagrange finite element subspaces of order 1. The analysis of the discrete method relies on
a stable decomposition of the finite element spaces and also on a classical result on projection
methods for Fredholm operators of index zero. Finally, numerical results illustrating the theory
are also presented.