## UNLV Theses, Dissertations, Professional Papers, and Capstones

5-1-2020

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematical Sciences

Pengtao Sun

Hongtao Yang

Jichun Li

Monika Neda

Hui Zhao

196

#### Abstract

In this dissertation, two kinds of arbitrary Lagrangian-Eulerian (ALE)-finite element methods (FEM) within the monolithic approach are studied for unsteady multiphysics coupling problems involving the moving interfaces/boundaries. For the classical affine-type ALE mapping that is studied in the first part of this dissertation, we develop the monolithic ALE-FEM and conduct stability and optimal convergence analyses in the energy norm for the transient Stokes/parabolic interface problem with jump coefficients, and more realistically, for the dynamic fluid-structure interaction (FSI) problems by taking the discrete ALE mapping and the discrete mesh velocity into a careful consideration of our numerical analyses and computations, where the affine-type ALE mapping preserves $H^1$-invariance for both the Stokes (fluid) equations and the parabolic (structure) equation in their moving subdomains all the time. In particular, we analyze the ALE-FEM for Stokes/parabolic interface problem by introducing a novel mixed-type $H^1$-projection with a moving interface and the discrete mesh velocity. We first obtain the well-posedness and convergence properties for this new $H^1$-projection and its ALE time derivative, by means of which we then derive the optimal error estimate in the energy norm and the sup-optimal error estimate in $L^2$ norm for both semi- and fully discrete mixed finite element approximations to the Stokes/parabolic interface problem. As for the realistic FSI problem, we build the classical affine-type ALE mapping into our novel mixed-type $H^1$-projection that couples the Eulerian fluid equation and the Lagrangian structure equation through a moving interface, and study its well-posedness and optimal convergence properties. Then we are able to analyze the (nearly) optimal error estimate in various norms for the ALE-finite element approximation to FSI problem as well.

In the second part of this dissertation, a novel Piola-type ALE mapping and the associated ALE-FEM are developed and are well analyzed for two types of moving interface problems whose weak forms are associated with $H(\text{div})$ space: the mixed parabolic problem in a moving domain, and the mixed parabolic/parabolic moving interface problem. In practice, the multiphysics problems involving the pore (Darcy's) fluid equation, or more sophisticatedly, the poroelasticity (Biot's) model, which may stay alone in a moving domain or interact with other field models through a moving interface, essentially belong to these two types of problems that we study in this part. The key idea of the developed Piola-type ALE mapping is to preserve $H(\text{div})$-invariance with time for the moving interfaces/boundaries problems that are associated with $H(\text{div})$ space in moving (sub)domains. Utilizing a specific stabilization technique, we apply the stable Stokes-pair to the mixed ALE-finite element discretization of both problems, design their semi- and fully discrete Piola-type ALE-finite element schemes, and analyze their stability and optimal convergence results using the MINI mixed element. All theoretical results obtained in this dissertation are appropriately validated by our numerical experiments using various numerical examples.

#### Keywords

arbitrary Lagrangian Eulerian; Aubin Nitsche Trick; Finite element method; Fluid struture interaction; H^1-projection; Piola mapping

Mathematics

pdf

3.2 MB

English