Time-dependent Stokes Equations Research Articles

Error Analysis of the Vector Penalty-Projection Methods for the Time-Dependent Stokes Equations with Open Boundary Conditions

Abstract We present in this paper a rigorous error analysis of the vector penalty-projection method for solving the time-dependent incompressible Stokes equations with open boundary conditions on part of the boundary. First, we prove the stability of the scheme. Then we provide an error analysis for the second-order vector penalty-projection method which shows that the convergence rate of the error on the velocity and the pressure is of order 2 in l ∞ ⁢ ( L 2 ⁢ ( Ω ) ) l^{\infty}(\mathbf{L}^{2}(\Omega)) and l 2 ⁢ ( L 2 ⁢ ( Ω ) ) l^{2}(L^{2}(\Omega)) respectively. In addition, it is shown that the splitting errors of the method varies as O ⁢ ( ε ) \mathcal{O}(\varepsilon) , where 𝜀 is a penalty parameter chosen as small as desired. Several numerical tests in agreement with the theoretical results are presented. To the best of our knowledge, this paper provides the first rigorous proof of optimal error estimates for second-order splitting schemes with open boundary conditions.

A computational algorithm for optimal design of a bioartificial organ scaffold architecture.

We develop a computational algorithm based on a diffuse interface approach to study the design of bioartificial organ scaffold architectures. These scaffolds, composed of poroelastic hydrogels housing transplanted cells, are linked to the patient's blood circulation via an anastomosis graft. Before entering the scaffold, the blood flow passes through a filter, and the resulting filtered blood plasma transports oxygen and nutrients to sustain the viability of transplanted cells over the long term. A key issue in maintaining cell viability is the design of ultrafiltrate channels within the hydrogel scaffold to facilitate advection-enhanced oxygen supply ensuring oxygen levels remain above a critical threshold to prevent hypoxia. In this manuscript, we develop a computational algorithm to analyze the plasma flow and oxygen concentration within hydrogels featuring various channel geometries. Our objective is to identify the optimal hydrogel channel architecture that sustains oxygen concentration throughout the scaffold above the critical hypoxic threshold. The computational algorithm we introduce here employs a diffuse interface approach to solve a multi-physics problem. The corresponding model couples the time-dependent Stokes equations, governing blood plasma flow through the channel network, with the time-dependent Biot equations, characterizing Darcy velocity, pressure, and displacement within the poroelastic hydrogel containing the transplanted cells. Subsequently, the calculated plasma velocity is utilized to determine oxygen concentration within the scaffold using a diffuse interface advection-reaction-diffusion model. Our investigation yields a scaffold architecture featuring a hexagonal network geometry that meets the desired oxygen concentration criteria. Unlike classical sharp interface approaches, the diffuse interface approach we employ is particularly adept at addressing problems with intricate interface geometries, such as those encountered in bioartificial organ scaffold design. This study is significant because recent developments in hydrogel fabrication make it now possible to control hydrogel rheology and utilize computational results to generate optimized scaffold architectures.

A coupled model of the fluid flow with nonlinear slip Tresca boundary condition

In this article, we consider the finite element discretization of the time-dependent Stokes equations written under the Boussinesq approximation, coupled with the convection-diffusion equation and provided with Tresca's Friction Law. Existence and uniqueness of a solution are established. To solve Stokes variational inequality, we suggest an alternating direction method of multiplier. We derive some a priori estimates and validate the obtained theoretical results by numerical simulations.

Analysis of a diffuse interface method for the Stokes-Darcy coupled problem

We consider the interaction between a free flowing fluid and a porous medium flow, where the free flowing fluid is described using the time dependent Stokes equations, and the porous medium flow is described using Darcy’s law in the primal formulation. To solve this problem numerically, we use a diffuse interface approach, where the weak form of the coupled problem is written on an extended domain which contains both Stokes and Darcy regions. This is achieved using a phase-field function which equals one in the Stokes region and zero in the Darcy region, and smoothly transitions between these two values on a diffuse region of width (ϵ) around the Stokes-Darcy interface. We prove convergence of the diffuse interface formulation to the standard, sharp interface formulation, and derive rates of convergence. This is performed by deriving a priori error estimates for discretizations of the diffuse interface method, and by analyzing the modeling error of the diffuse interface approach at the continuous level. The convergence rates are also shown computationally in a numerical example.

Modal Representation of Inertial Effects in Fluid–Particle Interactions and the Regularity of the Memory Kernels

This article develops a modal expansion (in terms of functions exponentially decaying with time) of the force acting on a micrometric particle and stemming from fluid inertial effects (usually referred to as the Basset force) deriving from the application of the time-dependent Stokes equation to model fluid–particle interactions. One of the main results is that viscoelastic effects induce the regularization of the inertial memory kernels at t=0, eliminating the 1/t-singularity characterizing Newtonian fluids. The physical origin of this regularization stems from the finite propagation velocity of the internal shear stresses characterizing viscoelastic constitutive equations. The analytical expression for the fluid inertial kernel is derived for a Maxwell fluid, and a general method is proposed to obtain accurate approximations of it for generic complex viscoelastic fluids, characterized by a spectrum of relaxation times.

Error analysis of a fully discrete consistent splitting MAC scheme for time dependent Stokes equations

We construct and analyze a numerical scheme based on the truly consistent splitting approach in time and the MAC discretization in space for the time dependent Stokes equations. The scheme only requires solving several Poisson type equations for the velocity and pressure at each time step. We establish the equivalence between two different formulations of the fully discrete consistent splitting schemes, prove unconditional stability, and establish first-order in time and second-order in space error estimates for velocity and pressure in different discrete norms. We also provide numerical experiments to verify our theoretical results.

A novel approach of superconvergence analysis of the bilinear-constant scheme for time-dependent Stokes equations

The superconvergent properties for time-dependent Stokes equations are investigated in this paper. Firstly, based on a new approach and the special characters of the bilinear-constant finite element space pair on the rectangular mesh, the superclose error estimations in L∞(H1)-norm for velocity and in L∞(L2)-norm for pressure are obtained. Then, the corresponding superconvergence results for velocity and pressure are derived in terms of a suitable postprocessing approach. Finally, some numerical experiments are provided to verify the theoretical analysis.

An unfitted Eulerian finite element method for the time-dependent Stokes problem on moving domains

Abstract We analyse a Eulerian finite element method, combining a Eulerian time-stepping scheme applied to the time-dependent Stokes equations with the CutFEM approach using inf-sup stable Taylor–Hood elements for the spatial discretization. This is based on the method introduced by Lehrenfeld & Olshanskii (2019, A Eulerian finite element method for PDEs in time-dependent domains. ESAIM: M2AN, 53, 585–614) in the context of a scalar convection–diffusion problems on moving domains, and extended to the nonstationary Stokes problem on moving domains by Burman et al. (2019, arXiv:1910.03054 [math.NA]) using stabilized equal-order elements. The analysis includes the geometrical error made by integrating over approximated level set domains in the discrete CutFEM setting. The method is implemented and the theoretical results are illustrated using numerical examples.

Analysis of Fully Discrete Mixed Finite Element Methods for Time-dependent Stochastic Stokes Equations with Multiplicative Noise

This paper is concerned with fully discrete mixed finite element approximations of the time-dependent stochastic Stokes equations with multiplicative noise. A prototypical method, which comprises of the Euler–Maruyama scheme for time discretization and the Taylor-Hood mixed element for spatial discretization is studied in detail. Strong convergence with rates is established not only for the velocity approximation but also for the pressure approximation (in a time-averaged fashion). A stochastic inf-sup condition is established and used in a nonstandard way to obtain the error estimate for the pressure approximation in the time-averaged fashion. Numerical results are also provided to validate the theoretical results and to gauge the performance of the proposed fully discrete mixed finite element methods.

Optimally convergent mixed finite element methods for the stochastic Stokes equations

Abstract We propose some new mixed finite element methods for the time-dependent stochastic Stokes equations with multiplicative noise, which use the Helmholtz decomposition of the driving multiplicative noise. It is known (Langa, J. A., Real, J. & Simon, J. (2003) Existence and regularity of the pressure for the stochastic Navier--Stokes equations. Appl. Math. Optim., 48, 195--210) that the pressure solution has low regularity, which manifests in suboptimal convergence rates for well-known inf-sup stable mixed finite element methods in numerical simulations; see Feng X., & Qiu, H. (Analysis of fully discrete mixed finite element methods for time-dependent stochastic Stokes equations with multiplicative noise. arXiv:1905.03289v2 [math.NA]). We show that eliminating this gradient part from the noise in the numerical scheme leads to optimally convergent mixed finite element methods and that this conceptual idea may be used to retool numerical methods that are well known in the deterministic setting, including pressure stabilization methods, so that their optimal convergence properties can still be maintained in the stochastic setting. Computational experiments are also provided to validate the theoretical results and to illustrate the conceptual usefulness of the proposed numerical approach.

Error analysis of a fully discrete finite element method for variable density incompressible flows in two dimensions

An error estimate is presented for a fully discrete, linearized and stabilized finite element method for solving the coupled system of nonlinear hyperbolic and parabolic equations describing incompressible flow with variable density in a two-dimensional convex polygon. In particular, the error of the numerical solution is split into the temporal and spatial components, separately. The temporal error is estimated by applying discrete maximal Lp-regularity of time-dependent Stokes equations, and the spatial error is estimated by using energy techniques based on the uniform regularity of the solutions given by semi-discretization in time.

POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation

In this work we deal with parametrized time dependent optimal control problems governed by partial differential equations. We aim at extending the standard saddle point framework of steady constraints to time dependent cases. We provide an analysis of the well-posedness of this formulation both for parametrized scalar parabolic constraint and Stokes governing equations and we propose reduced order methods as an effective strategy to solve them. Indeed, on one hand, parametrized time dependent optimal control is a very powerful mathematical model which is able to describe several physical phenomena, on the other, it requires a huge computational effort. Reduced order methods are a suitable approach to have rapid and accurate simulations. We rely on POD–Galerkin reduction over the physical and geometrical parameters of the optimality system in a space-time formulation. Our theoretical results and our methodology are tested on two examples: a boundary time dependent optimal control for a Graetz flow and a distributed optimal control governed by time dependent Stokes equations. With these two test cases the convenience of the reduced order modelling is further extended to the field of time dependent optimal control.

Divergence-free radial kernel for surface Stokes equations based on the surface Helmholtz decomposition

In this paper, we develop and analyze a new divergence-free kernel approximation method for the time-dependent incompressible Stokes equations on surfaces. The novelty of our proposed method comes from the surface Helmholtz decomposition, which can convert the surface Stokes equations into a coupled equations in which the velocity is deadly divergence-free so that no inf–sup conditions have to be satisfied. Spatial discretization of velocity is implemented by the radial kernel divergence-free approximation spaces, which only need scatter nodes on surfaces. Using the radial basis function collocation method in space, we derive the rigorous stability and convergence result. Numerical examples are presented, demonstrating the efficiency of some model problems on more general surfaces.

Discontinuous Galerkin time discretization methods for parabolic problems with linear constraints

Abstract We consider time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. Prototype examples that fit into the general framework are the heat equation with inhomogeneous (time-dependent) Dirichlet boundary conditions and the time-dependent Stokes equation with an inhomogeneous divergence constraint. Two common ways of treating such linear constraints, namely explicit or implicit (via Lagrange multipliers) are studied. These different treatments lead to different variational formulations of the parabolic problem. For these formulations we introduce a modification of the standard discontinuous Galerkin (DG) time discretization method in which an appropriate projection is used in the discretization of the constraint. For these discretizations (optimal) error bounds, including superconvergence results, are derived. Discretization error bounds for the Lagrange multiplier are presented. Results of experiments confirm the theoretically predicted optimal convergence rates and show that without the modification the (standard) DG method has sub-optimal convergence behavior.

Analysis of a linear 3D fluid–mesh–shell interaction problem

We study a linear fluid–structure interaction problem between an incompressible, viscous 3D fluid flow, a 2D linearly elastic Koiter shell, and an elastic 1D net of curved rods. This problem is motivated by studying fluid–structure interaction between blood flow through coronary arteries treated with metallic mesh-like devices called stents. The flow is assumed to be laminar, modeled by the time-dependent Stokes equations, and the structure displacement is assumed to be small, modeled by a system of linear Koiter shell equations allowing displacement in all three spatial directions. The fluid and the mesh-supported structure are coupled via the kinematic and dynamic coupling conditions describing continuity of velocity and balance of contact forces. The coupling conditions are evaluated along a linearized fluid–structure interface, which coincides with the fixed fluid domain boundary. No smallness on the structure velocity is assumed. We prove the existence of a weak solution to this linear fluid–composite structure interaction problem. This is the first result in the area of fluid–structure interaction that includes a 1D elastic mesh and takes into account structural displacements in all three spatial directions. Numerical simulations based on the finite element discretization of the coupled FSI problem are presented.

Stability and superconvergence of MAC schemes for time dependent Stokes equations on nonuniform grids

In this paper, two MAC schemes are introduced and analyzed to solve the time dependent Stokes equations on nonuniform grids. One scheme is the Euler backward scheme with first order accuracy in time increment while the other one is the Crank Nicolson scheme with second order accuracy in time increment. By constructing an auxiliary function depending on the velocity and discretizing parameters, we obtain the second order superconvergence in L2 norm for both velocity and pressure. Besides, second order superconvergence for some terms of H1 norm of the velocity on nonuniform grids is obtained. Finally, some numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis.

A posteriori error analysis of the fully discretized time-dependent coupled Darcy and Stokes equations

We present an a posteriori error analysis of the fully discretized time-dependent Darcy and Stokes equations, that models laminar fluid flow over a porous medium in two- or three-dimensional connected open domains which are coupled via appropriate matching conditions on the interface. The problem is discretized by the backward Euler scheme in time and finite elements in space.

A posteriori estimates for the two-step backward differentiation formula and discrete regularity for the time-dependent Stokes equations

A posteriori estimates for the two-step backward differentiation formula and discrete regularity for the time-dependent Stokes equations

Analysis of a velocity–stress–pressure formulation for a fluid–structure interaction problem

We consider a fluid–structure interaction problem consisting of the time-dependent Stokes equations in the fluid domain coupled with the equations of linear elastodynamics in the solid domain. For simplicity, all changes of geometry are neglected. We propose a new method in terms of the fluid velocity, the fluid pressure, the structural velocity and the Cauchy stress tensor. We show that the new weak formulation is well-posed. Then, we propose a new semidiscrete problem where the velocities and the fluid pressure are approximated using a stable pair for the Stokes problem in the fluid domain and compatible finite elements in the solid domain. We obtain a priori estimates for the solution of the semidiscrete problem, prove the convergence of these solutions to the solution of the weak formulation and obtain error estimates. A time discretization based on the backward Euler method leads to a fully discrete scheme in which the computation of the approximated Cauchy stress tensor can be decoupled from that of the remaining unknowns at each time step. The displacements in the structure (if needed) can be recovered by quadrature. Finally, some numerical experiments showing the performance of the method are provided.

A parallel partition of unity method for the unsteady Stokes equations

PurposeThe purpose of this paper is to propose a parallel partition of unity method to solve the time-dependent Stokes problems.Design/methodology/approachThis paper solved the time-dependent Stokes equations using the finite element method and the partition of unity method.FindingsThe proposed method in this paper obtained the same accuracy as the standard Galerkin method, but it, in general, saves time.Originality/valueBased on a combination of the partition of unity method and the finite element method, the authors, in this paper, propose a new parallel partition of unity method to solve the unsteady Stokes equations. The idea is that, at each time step, one need to only solve a series of independent local sub-problems in parallel instead of one global problem. Numerical tests show that the proposed method not only reaches the same convergence orders as the fully discrete standard Galerkin method but also saves ample computing time.