2022/10/24 by Paula Egging, Egging, Paula, George Avalos +1
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2210.12895
openalex publication_date 2022/10/24 · openalex created_date 2022/10/31 · openalex updated_date 2026/07/28
This work presents qualitative and numerical results on a system of partial differential equations (PDEs) which models certain fluid-fluid interaction dynamics. This system models a compressible fluid in a domain Ω+ ⊂ ℝ2, coupled to an incompressible fluid modeled by Stokes flow in domain Ω- ⊂ ℝ2, with the strong coupling implemented through certain boundary conditions on the shared interface, Γ. The wellposedness of this system is established by means of constructing for it a semigroup generator representation. This representation is accomplished by eliminating one of the pressure variables via identifying it as the solution of a certain boundary value problem, while the wellposedness is established via a nonstandard usage of the Babuska-Brezzi Theorem. In later sections, we demonstrate how the earlier constructive proof of wellposedness naturally lends itself to a certain finite element method (FEM), by which to numerically approximate solutions of the given coupled PDE system. This FEM is provided with error estimates and associated rates of convergence.