Mixed Finite Element Approximation Research Articles

Numerical Analysis of A Mixed Finite Element Approximation of a Coupled System Modeling Biofilm Growth in Porous Media with Simulations

Numerical Analysis of A Mixed Finite Element Approximation of a Coupled System Modeling Biofilm Growth in Porous Media with Simulations

A Mixed Finite Element Approximation for Time-Dependent Navier–Stokes Equations with a General Boundary Condition

In this paper, we present a numerical scheme for addressing the unsteady asymmetric flows governed by the incompressible Navier–Stokes equations under a general boundary condition. We utilized the Finite Element Method (FEM) for spatial discretization and the fully implicit Euler scheme for time discretization. In addition to the theoretical analysis of the error in our numerical scheme, we introduced two types of a posteriori error indicators: one for time discretization and another for spatial discretization, aimed at effectively controlling the error. We established the equivalence between these estimators and the actual error. Furthermore, we conducted numerical simulations in two dimensions to assess the accuracy and effectiveness of our scheme.

An unconditionally energy stable finite element scheme for a nonlinear fluid–fluid interaction model

Abstract In this paper, we design a decoupled scheme for solving a fluid–fluid interaction problem, which includes two Navier–Stokes equations coupled by some nonlinear interface conditions. Compared with two decoupled schemes proposed by Connors et al. (2012, Decoupled time stepping methods for fluid–fluid interaction. SIAM J. Numer. Anal., 50, 1297–1319) for the fluid–fluid interaction problem, we deal with these nonlinear interface conditions by applying explicit scheme. The presented fully discrete scheme is a combination of a mixed finite element approximation for spatial discretization, the first-order backward Euler scheme for temporal discretization and explicit treatment for the interface conditions and the nonlinear terms. Moreover, the unconditional energy stability is established and error estimate for the fully discrete scheme is also showed. Finally, some numerical experiments are provided to verify the theoretical results, which illustrate the accuracy and efficiency of the presented scheme.

A second-order modular grad-div stabilized scheme for the Darcy–Brinkman model

In this article, a fully discrete modular grad-div stabilized finite element scheme for the Darcy–Brinkman equations are considered. This fully discrete scheme includes two steps: the first step is a combination of a mixed finite element approximation for space discretization, the second-order backward differentiation formula for temporal discretization, and extrapolated treatments in linearization for the nonlinear terms. Then, in the second step, a modular grad-div stabilized technique is applied, which can improve solution accuracy without increasing computational time for large stabilized parameters. Moreover, we show that the scheme is unconditionally stable and convergent with second-order accuracy with respect to time step. Finally, several numerical examples are provided to support the derived theoretical results, and demonstrate the efficiency of the presented scheme.

On mixed finite element approximations of shape gradients in shape optimization with the Navier–Stokes equation

AbstractFor shape optimization of fluid flows governed by the Navier–Stokes equation, we investigate effectiveness of shape gradient algorithms by analyzing convergence and accuracy of mixed finite element approximations to both the distributed and boundary types of shape gradients. We present convergence analysis with a priori error estimates for the two approximate shape gradients. The theoretical analysis shows that the distributed formulation has superconvergence property. Numerical results with comparisons are presented to verify theory and show that the shape gradient algorithm based on the distributed formulation is highly effective and robust for shape optimization.

Mixed finite elements for Bingham flow in a pipe

We consider mixed finite element approximations of viscous, plastic Bingham flow in a cylindrical pipe. A novel a priori and a posteriori error analysis is introduced which is based on a discrete mesh dependent norm for the normalized Lagrange multiplier. This allows proving stability for various conforming finite elements. Numerical examples are presented to support the theory and to demonstrate adaptive mesh refinement.

Shape optimization of Navier–Stokes flows by a two-grid method

We consider the energy dissipation minimization constrained by steady Navier–Stokes flows. The nonlinearity of the Navier–Stokes equation causes its numerical solver computationally expensive and thus leads to inefficiency of shape optimization algorithms. We propose a new efficient shape gradient algorithm of distributed type by using a two-grid solver for the Navier–Stokes equation. Asymptotically optimal convergence rate is shown for mixed finite-element approximation of the two-grid distributed shape gradient. Moreover, a Uzawa iterative scheme for Stokes-type systems is adopted in the two-grid solver and a new scalar-type shape gradient flow is proposed to improve significantly the computational efficiency especially for 3D. Numerical experiments are presented to verify theoretical analysis and show effectiveness of two-grid shape gradient optimization algorithms.

A vector penalty-projection approach for the time-dependent incompressible magnetohydrodynamics flows

In this paper, we study a fully discrete vector penalty-projection method for the time-dependent incompressible magnetohydrodynamics flows. This fully discrete scheme is a combination of a mixed finite element approximation for spatial discretization and first-order backward Euler for temporal discretization. Moreover, unconditionally energy stable is established, and error estimates for the fully discrete scheme is also derived. Finally, some numerical experiments are provided to verify the theoretical results and illustrate the accuracy and efficiency of the proposed scheme.

An optimal error estimate of the BDF-Galerkin FEM for the incompressible MHD system

In this article, a linearized backward differential formula type scheme for the numerical approximations of the magnetohydrodynamics equations is analyzed. The proposed fully discretized numerical scheme, which is comprised of a third order backward differential formula for time derivative terms, extrapolated treatments in linearization for nonlinear terms and mixed finite element approximation for spatial discretization. Based on the temporal-spatial error splitting technique, we demonstrate the almost unconditional optimal error estimate in L2 norm for the fluid velocity and the magnetic by getting boundedness of numerical solutions in certain norms. Finally, several numerical tests are carried out to verify the theoretical results.

Flux-Mortar Mixed Finite Element Methods on NonMatching Grids

We investigate a mortar technique for mixed finite element approximations of Darcy flow on non-matching grids in which the normal flux is chosen as the coupling variable. It plays the role of a Lagrange multiplier to impose weakly continuity of pressure. In the mixed formulation of the problem, the normal flux is an essential boundary condition and it is incorporated with the use of suitable extension operators. Two such extension operators are considered and we analyze the resulting formulations with respect to stability and convergence. We further generalize the theoretical results, showing that the same domain decomposition technique is applicable to a class of saddle point problems satisfying mild assumptions. An example of coupled Stokes-Darcy flows is presented.

High order approximation of Hodge Laplace problems with local coderivatives on cubical meshes

In mixed finite element approximations of Hodge Laplace problems associated with the de Rham complex, the exterior derivative operators are computed exactly, so the spatial locality is preserved. However, the numerical approximations of the associated coderivatives are nonlocal and can be regarded as an undesired effect of standard mixed methods. For numerical methods with local coderivatives, a perturbation of low order mixed methods in the sense of variational crimes has been developed for simplicial and cubical meshes. In this paper we extend the low order method to all high orders on cubical meshes using a new family of finite element differential forms on cubical meshes. The key theoretical contribution is a generalization of the linear degree, in the construction of the serendipity family of differential forms, and this generalization is essential in the unisolvency proof of the new family of finite element differential forms.

Residual-based a posteriori error estimators for mixed finite element methods for fourth order elliptic singularly perturbed problems

We consider mixed finite element approximation of a singularly perturbed fourth-order elliptic problem with two different boundary conditions, and present a new measure of the error, whose components are balanced with respect to the perturbation parameter. With different boundary conditions, the simply supported plate model and the clamped plate model are considered. In particular, a balanced energy norm has been defined. Based on the new norm, residual-based a posteriori estimators are developed for both problems, which are uniform with respect to both the perturbation parameter and the mesh function. A novel analysis approach is introduced for the clamped plate model to address certain difficulty of the problem. Numerical examples are provided to confirm theoretical findings.

A mixed elasticity formulation for fluid–poroelastic structure interaction

We develop a mixed finite element method for the coupled problem arising in the interaction between a free fluid governed by the Stokes equations and flow in deformable porous medium modeled by the Biot system of poroelasticity. Mass conservation, balance of stress, and the Beavers–Joseph–Saffman condition are imposed on the interface. We consider a fully mixed Biot formulation based on a weakly symmetric stress-displacement-rotation elasticity system and Darcy velocity-pressure flow formulation. A velocity-pressure formulation is used for the Stokes equations. The interface conditions are incorporated through the introduction of the traces of the structure velocity and the Darcy pressure as Lagrange multipliers. Existence and uniqueness of a solution are established for the continuous weak formulation. Stability and error estimates are derived for the semi-discrete continuous-in-time mixed finite element approximation. Numerical experiments are presented to verify the theoretical results and illustrate the robustness of the method with respect to the physical parameters.

The macroelement analysis for axisymmetric Stokes equations

We consider the mixed finite element approximation of the axisymmetric Stokes problem (ASP) on a bounded polygonal domain in the rz-plane. Standard stability results on mixed methods do not apply due to the singular coefficients in the differential operator and due to the singular or vanishing weights in the associated function spaces. We develop new finite element analysis in these weighted spaces, and propose macroelement conditions that are sufficient to ensure the well-posedness of the mixed methods for the ASP. These conditions are local, relatively easy to verify, and therefore will be useful for validating the stability of a variety of mixed finite element methods. These new conditions can not only re-verify existing stable mixed methods for the ASP, but also lead to the discovery of new stable conservative mixed methods. We report numerical test results that confirm the theory.

Finite element approximation of steady flows of colloidal solutions

We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions–Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method.

Superconvergence analysis of a mixed finite element approximation for the nonlinear fourth-order Rosenau-RLW equation

The focus of this paper is on a implicit Backward-Euler (BE) scheme with the mixed finite element method (FEM) for the two-dimentional general Rosenau-RLW equation. In which the bilinear element is used to approximate the exact solution v and the variable p=−Δv, and the zero-order nédéléc's finite element to the variable q→=ϕ∇v, respectively. An important point is that with the proposed scheme we are able to bound the numerical solution in H1-norm, so that we can efficiently solve the nonlinear term. Moreover, the stability, existence and uniqueness of approximate solution are demonstrated. Based on the combination of the interpolation and projection technique, the superconvergence estimates of order O(h2+τ) for v in H1-norm and the introducing variable q→ in L2-norm, and optimal error estimate of order O(h+τ) for introducing variable p in H1-norm is proved. Finally, numerical example is done to certify our theoretical results.

A mixed finite element method for solving coupled wave equation of Kirchhoff type with nonlinear boundary damping and memory term

This paper is concerned with the numerical approximation of the solution of the coupled wave equation of Kirchhoff type with nonlinear boundary damping and memory term using a mixed finite element method. The Raviart‐Thomas mixed finite element method is one of the most prominent techniques to discretize the second‐order wave equations; therefore, we apply this scheme for space discretization. Furthermore, an L2‐in‐space error estimate is presented for this mixed finite element approximation. Finally, the efficiency of the method is verified by a numerical example.

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.

Mixed Finite Element Method for Modified Poisson–Nernst–Planck/Navier–Stokes Equations

In this paper, a complete mixed finite element method is developed for a modified Poisson–Nernst–Planck/Navier–Stokes (PNP/NS) coupling system, where the original Poisson equation in PNP system is replaced by a fourth-order elliptic equation to more precisely account for electrostatic correlations in a simplified form of the Landau–Ginzburg-type continuum model. A stabilized mixed weak form is defined for each equation of the modified PNP/NS model in terms of primary variables and their corresponding vector-valued gradient variables, based on which a stable Stokes-pair mixed finite element is thus able to be utilized to discretize all solutions to the entire modified PNP/NS model in the framework of Stokes-type mixed finite element approximation. Semi- and fully discrete mixed finite element schemes are developed and are analyzed for the presented modified PNP/NS equations, and optimal convergence rates in energy norms are obtained for both schemes. Numerical experiments are carried out to validate all attained theoretical results.

Analysis of pore collapse and shear-enhanced compaction in hydrocarbon reservoirs using coupled poro-elastoplasticity and permeability

The withdrawal of fluid from a reservoir results in a decline of the fluid pressure followed by a consequent change in stress state in porous rocks. Stress change may cause irreversible deformation and compaction. Such compaction is generally the result of pore collapse and shear-enhanced compaction caused by changes at a microscopic level in the porous rocks. Pore collapse and shear-enhanced compaction are considered as potential problems during reservoir production and drilling operations. The purpose of this paper is to analyze the pore collapse and shear-enhanced compaction in hydrocarbon reservoirs using coupled poro-elastoplasticity and permeability. In this coupling, the poro-elastoplasticity analysis includes the linear component based on Biot’s theory and the nonlinear component based on a cap plasticity model. The fluid flow formulation is defined by Darcy’s law, including a nonlinear permeability model. The numerical approximation is implemented using continuous finite element approximations for rock deformation and mixed finite element approximation for pore pressure and flux. Several numerical simulations are performed to indicate the onset of pore collapse and shear-enhanced compaction and evaluate their effects on reservoir performance.