A Lagrange multiplier method for a Stokes-Biot fluid-poroelastic structure interaction model

We study a finite element computational model for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium. The free fluid is governed by the Stokes equations, while the flow in the poroelastic medium is modeled using the Biot poroelasticity system. Equilibrium and kinematic conditions are imposed on the interface. A mixed Darcy formulation is employed, resulting in continuity of flux condition of essential type. A Lagrange multiplier method is employed to impose weakly this condition. A stability and error analysis is performed for the semi-discrete continuous-in-time and the fully discrete formulations. A series of numerical experiments is presented to confirm the theoretical convergence rates and to study the applicability of the method to modeling physical phenomena and the sensitivity of the model with respect to its parameters

Robust Discretization of Flow in Fractured Porous Media

Flow in fractured porous media represents a challenge for discretization methods due to the disparate scales and complex geometry. Herein we propose a new discretization, based on the mixed finite element method and mortar methods. Our formulation is novel in that it employs the normal fluxes as the mortar variable within the mixed finite element framework, resulting in a formulation that couples the flow in the fractures with the surrounding domain with a strong notion of mass conservation. The proposed discretization handles complex, non-matching grids, and allows for fracture intersections and termination in a natural way, as well as spatially varying apertures. The discretization is applicable to both two and three spatial dimensions. A priori analysis shows the method to be optimally convergent with respect to the chosen mixed finite element spaces, which is sustained by numerical examples

Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche's coupling approach

We develop a computational model to study the interaction of a fluid with a poroelastic material. The coupling of Stokes and Biot equations represents a prototype problem for these phenomena, which feature multiple facets. On one hand it shares common traits with fluid-structure interaction. On the other hand it resembles the Stokes-Darcy coupling. For these reasons, the numerical simulation of the Stokes-Biot coupled system is a challenging task. The need of large memory storage and the difficulty to characterize appropriate solvers and related preconditioners are typical shortcomings of classical discretization methods applied to this problem. The application of loosely coupled time advancing schemes mitigates these issues because it allows to solve each equation of the system independently with respect to the others. In this work we develop and thoroughly analyze a loosely coupled scheme for Stokes-Biot equations. The scheme is based on Nitsche's method for enforcing interface conditions. Once the interface operators corresponding to the interface conditions have been defined, time lagging allows us to build up a loosely coupled scheme with good stability properties. The stability of the scheme is guaranteed provided that appropriate stabilization operators are introduced into the variational formulation of each subproblem. The error of the resulting method is also analyzed, showing that splitting the equations pollutes the optimal approximation properties of the underlying discretization schemes. In order to restore good approximation properties, while maintaining the computational efficiency of the loosely coupled approach, we consider the application of the loosely coupled scheme as a preconditioner for the monolithic approach. Both theoretical insight and numerical results confirm that this is a promising way to develop efficient solvers for the problem at hand

Mathematical and Numerical Modeling of Inflammation

When the body is attacked by a bacterial infection, it initiates a series of events designed to eradicate the infection while causing minimal damage to the body. Our goal is to investigate the defenses of the organ walls to the spread of infection. To do this we have chosen to model a volume of the body that includes the organ wall, the lumen outside of it and the blood and tissue within it. We have also taken into account the varied responses of the body, and our model includes many interacting agents that are part of the infection and defense processes, including the agents that attempt to prevent the infection from breaching the organ wall. The mathematical model is based on a system of nonlinear transient partial differential equations. The numerical model is based on cell-centered finite differences in space and implicit Euler in time. The model is implemented in MATLAB, and has many visualization options to better see the progression of the infection. It is hoped that this model will help in better understanding the failure of the body’s defenses in such situations as Necrotizing Enterocolitis (NEC), and eventually lead to the development of a method of prevention

Multiscale mortar mixed finite element methods for the Biot system of poroelasticity

We develop a mixed finite element domain decomposition method on non-matching grids for the Biot system of poroelasticity. A displacement-pressure vector mortar function is introduced on the interfaces and utilized as a Lagrange multiplier to impose weakly continuity of normal stress and normal velocity. The mortar space can be on a coarse scale, resulting in a multiscale approximation. We establish existence, uniqueness, stability, and error estimates for the semidiscrete continuous-in-time formulation under a suitable condition on the richness of the mortar space. We further consider a fully-discrete method based on the backward Euler time discretization and show that the solution of the algebraic system at each time step can be reduced to solving a positive definite interface problem for the composite mortar variable. A multiscale stress-flux basis is constructed, which makes the number of subdomain solves independent of the number of iterations required for the interface problem, as well as the number of time steps. We present numerical experiments verifying the theoretical results and illustrating the multiscale capabilities of the method for a heterogeneous benchmark problem

Mathematical and Numerical Modeling of Inflammation

When the body is attacked by a bacterial infection, it initiates a series of events designed to eradicate the infection while causing minimal damage to the body. Our goal is to investigate the defenses of the organ walls to the spread of infection. To do this we have chosen to model a volume of the body that includes the organ wall, the lumen outside of it and the blood and tissue within it. We have also taken into account the varied responses of the body, and our model includes many interacting agents that are part of the infection and defense processes, including the agents that attempt to prevent the infection from breaching the organ wall. The mathematical model is based on a system of nonlinear transient partial differential equations. The numerical model is based on cell-centered finite differences in space and implicit Euler in time. The model is implemented in MATLAB, and has many visualization options to better see the progression of the infection. It is hoped that this model will help in better understanding the failure of the body’s defenses in such situations as Necrotizing Enterocolitis (NEC), and eventually lead to the development of a method of prevention