Divergence-free cut finite element methods for Stokes flow

We develop two unfitted finite element methods for the Stokes equations based on $\mathbf{H}^{\text{div}}$-conforming finite elements. The first method is a cut finite element discretization of the Stokes equations based on the Brezzi-Douglas-Marini elements and involves interior penalty terms to enforce tangential continuity of the velocity at interior edges in the mesh. The second method is a cut finite element discretization of a three-field formulation of the Stokes problem involving the vorticity, velocity, and pressure and uses the Raviart-Thomas space for the velocity. We present mixed ghost penalty stabilization terms for both methods so that the resulting discrete problems are stable and the divergence-free property of the $\mathbf{H}^{\text{div}}$-conforming elements is preserved also for unfitted meshes. We compare the two methods numerically. Both methods exhibit robust discrete problems, optimal convergence order for the velocity, and pointwise divergence-free velocity fields, independently of the position of the boundary relative to the computational mesh

Cut finite element methods for interface problems

Interface problems modeled by Partial Differential Equations (PDEs) appear in a wide range of fields such as biology, fluid dynamics, micro and nano-technologies. In computer simulations of such problems a fundamental task is to numerically solve PDEs in/on domains defined by deformable interfaces/geometries. Those interfaces may for example model fractures, cell membranes, ice sheets or an airplane wing. This thesis is devoted to the development of Cut Finite Element Methods (CutFEM) that can accurately discretize PDEs with weak or strong discontinuities in parameters and solutions across stationary or evolving interfaces and conserve physical quantities. On evolving geometries, spatial discretization based on the cut finite element method is combined with a finite element method for the time discretization using discontinuous piecewise polynomials. Different strategies have been investigated throughout this thesis in order to achieve optimal approximation properties, well-posed resulting linear systems, and conservation of physical quantities. In Paper I and III, we consider two-phase flow problems where the interface separates immiscible incompressible fluids with different densities and viscosities. In Paper I we develop a numerical algorithm for obtaining high order approximations of the mean curvature vector and hence the surface tension force. Coupling this strategy with a space-time cut finite element discretization of the Navier-Stokes equations gives us a method that can accurately capture discontinuities across evolving interfaces and be used to accurately simulate the dynamics of two-phase flow problems, see Paper I. In Paper II, we develop a cut finite element discretization for linear hyperbolic conservation laws with an interface and show that stability and conservation can be obtained when using appropriate penalty parameters. In Paper III, discrete conservation of the total mass of surfactant is obtained by introducing a new weak formulation of the convectiondiffusion interface problem modeling the evolution of insoluble surfactants, with the help of the Reynold’s transport theorem. In Paper IV we propose a new stabilization for the discretization of the Darcy interface problem using cut finite element methods. The new stabilized cut finite element method preserves the divergencefree property of Hdiv conforming elements also in an unfitted setting. Thus, with the new scheme optimal approximation can be obtained for the velocity, pressure, and the divergence with control on the condition number of the resulting system matrix

Cut finite element methods for interface problems

Interface problems modeled by Partial Differential Equations (PDEs) appear in a wide range of fields such as biology, fluid dynamics, micro and nano-technologies. In computer simulations of such problems a fundamental task is to numerically solve PDEs in/on domains defined by deformable interfaces/geometries. Those interfaces may for example model fractures, cell membranes, ice sheets or an airplane wing. This thesis is devoted to the development of Cut Finite Element Methods (CutFEM) that can accurately discretize PDEs with weak or strong discontinuities in parameters and solutions across stationary or evolving interfaces and conserve physical quantities. On evolving geometries, spatial discretization based on the cut finite element method is combined with a finite element method for the time discretization using discontinuous piecewise polynomials. Different strategies have been investigated throughout this thesis in order to achieve optimal approximation properties, well-posed resulting linear systems, and conservation of physical quantities. In Paper I and III, we consider two-phase flow problems where the interface separates immiscible incompressible fluids with different densities and viscosities. In Paper I we develop a numerical algorithm for obtaining high order approximations of the mean curvature vector and hence the surface tension force. Coupling this strategy with a space-time cut finite element discretization of the Navier-Stokes equations gives us a method that can accurately capture discontinuities across evolving interfaces and be used to accurately simulate the dynamics of two-phase flow problems, see Paper I. In Paper II, we develop a cut finite element discretization for linear hyperbolic conservation laws with an interface and show that stability and conservation can be obtained when using appropriate penalty parameters. In Paper III, discrete conservation of the total mass of surfactant is obtained by introducing a new weak formulation of the convectiondiffusion interface problem modeling the evolution of insoluble surfactants, with the help of the Reynold’s transport theorem. In Paper IV we propose a new stabilization for the discretization of the Darcy interface problem using cut finite element methods. The new stabilized cut finite element method preserves the divergencefree property of Hdiv conforming elements also in an unfitted setting. Thus, with the new scheme optimal approximation can be obtained for the velocity, pressure, and the divergence with control on the condition number of the resulting system matrix

A Cut Finite Element Method for two-phase flows with insoluble surfactants

We propose a new unfitted finite element method for simulation of two-phase flows in presence of insoluble surfactant. The key features of the method are 1) discrete conservation of surfactant mass; 2) the possibility of having meshes that do not conform to the evolving interface separating the immiscible fluids; 3) accurate approximation of quantities with weak or strong discontinuities across evolving geometries such as the velocity field and the pressure. The new discretization of the incompressible Navier--Stokes equations coupled to the convection-diffusion equation modeling the surfactant transport on evolving surfaces is based on a space-time cut finite element formulation with quadrature in time and a stabilization term in the weak formulation that provides function extension. The proposed strategy utilize the same computational mesh for the discretization of the surface Partial Differential Equation (PDE) and the bulk PDEs and can be combined with different techniques for representing and evolving the interface, here the level set method is used. Numerical simulations in both two and three space dimensions are presented including simulations showing the role of surfactant in the interaction between two drops

A Cut Finite Element Method for two-phase flows with insoluble surfactants

QC 20220520</p

Prise en charge des patients sous agents antiplaquettaires en chirurgie buccale

RENNES1-BU Santé (352382103) / SudocPARIS-BIUM (751062103) / SudocSudocFranceF

A Cut Finite Element Method for two-phase flows with insoluble surfactants

QC 20220520</p

High Order Discontinuous Cut Finite Element Methods for Linear Hyperbolic Conservation Laws with an Interface

We develop a family of cut finite element methods of different orders based on the discontinuous Galerkin framework, for hyperbolic conservation laws with stationary interfaces in both one and two space dimensions, and for moving interfaces in one space dimension. Interface conditions are imposed weakly and so that both conservation and stability are ensured. A CutFEM with discontinuous elements in space is developed and coupled to standard explicit time stepping schemes for linear advection problems and the acoustic wave problem with stationary interfaces. In the case of moving interfaces, we propose a space-time CutFEM based on discontinuous elements both in space and time for linear advection problems. We show that the proposed CutFEM are conservative and energy stable. For the stationary interface case an a priori error estimate is proven. Numerical computations in both one and two space dimensions support the analysis, and in addition demonstrate that the proposed methods have the expected accuracy

A divergence preserving cut finite element method for Darcy flow

We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs $\textbf{RT}_0\times\text{P}_0$, $\textbf{BDM}_1\times \text{P}_0$, and $\textbf{RT}_1\times \text{P}_1$. We show that the standard ghost penalty stabilization, often added in the weak forms of cut finite element methods for stability and control of the condition number of the resulting linear system matrix, pollutes the computed velocity field so the divergence-free property of the considered elements is lost. Therefore, we propose two corrections to the standard stabilization strategy; using macro-elements and new stabilization terms for the pressure. By decomposing the computational mesh into macro-elements and applying ghost penalty terms only on interior edges of macro-elements, stabilization is active only where needed. By modifying the standard stabilization terms for the pressure we recover the optimal approximation of the divergence without losing control of the condition number of the linear system matrix. Numerical experiments indicate that with the new stabilization terms the unfitted finite element discretization, for the given element pairs, results in 1) optimal rates of convergence of the approximate velocity and pressure; 2) well-posed linear systems where the condition number of the system matrix scales as for fitted finite element discretizations; 3) optimal rates of convergence of the approximate divergence with pointwise divergence-free approximations of solenoidal velocity fields. All three properties hold independently of how the interface is positioned relative the computational mesh

Cut finite element methods for Darcy flow in fractured porous media

We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs RT0 × P0, BDM1 × P0, and RT1 × P1. We show that the standard ghost penalty stabilization, often added in the weak forms of Cut Finite Element Methods (CutFEM) for stability and control of the condition number of the resulting linear system matrix, pollutes the computed velocity field so the optimal approximation of the divergence is lost. Therefore, we propose two corrections to the standard stabilization strategy; using macro-elements and new stabilization terms for the pressure. By decomposing the computational mesh into macro-elements and applying ghost penalty terms only on interior edges of macro-elements stabilization is active only where needed. By modifying the standard stabilization terms for the pressure we recover the optimal approximation of the divergence without losing control of the condition number of the linear system matrix. Numerical experiments indicate that with the new stabilization terms the unfitted finite element discretization results in 1) optimal rates of convergence of the approximate velocity and pressure; 2) well-posed linear systems where the condition number of the system matrix scales as for fitted finite element discretizations; 3) optimal approximation of the divergence with pointwise divergence-free approximations of solenoidal velocity fields. All three properties hold independently of how the interface is positioned relative the computational mesh.QC 20220520</p