Mixed Finite Element Research Articles

On non-uniqueness of pressures in problems of fluid filtration in fractured-porous media

The paper proposes a new approach to the numerical solution of the singular Neumann problem for a system of elliptic equations, in which, for example, the problem of filtration of a two-phase incompressible fluid in a fractured-porous medium is reduced within the framework of the dual porosity model. The one-dimensionality of the kernel of the corresponding operator is established, after which an extended generalized mixed formulation is considered in which the singularity is eliminated by introducing some additional bilinear form. The extended formulation is approximated by the mixed finite element method, and the properties of the resulting system of linear algebraic equations with a saddle point matrix are established. Moreover, some numerical examples illustrate the theoretical results.

Flux-mortar mixed finite element methods with multipoint flux approximation

The flux-mortar mixed finite element method was recently developed in Boon et al. (2022) for a general class of domain decomposition saddle point problems on non-matching grids. In this work we develop the method for Darcy flow using the multipoint flux approximation as the subdomain discretization. The subdomain problems involve solving positive definite cell-centered pressure systems. The normal flux on the subdomain interfaces is the mortar coupling variable, which plays the role of a Lagrange multiplier to impose weakly continuity of pressure. We present well-posedness and error analysis based on reformulating the method as a mixed finite element method with a quadrature rule. We develop a non-overlapping domain decomposition algorithm for the solution of the resulting algebraic system that reduces it to an interface problem for the flux-mortar, as well as an efficient interface preconditioner. A series of numerical experiments is presented illustrating the performance of the method on general grids, including applications to flow in complex porous media.

The deflection platform phenomenon of functionally graded flexoelectric simply supported nanobeam

Direct flexoelectricity appears in dielectrics as a kind of coupling effect of electric field and elastic strain gradient. Its strong size dependence makes it effective in nanostructures. In this paper, based on the flexoelectric theory and mixed finite element method (FEM), the mechanical properties of the two-dimensional functionally graded (FG) flexoelectric nanobeam are studied, and the phenomenon of the deflection platform is found. By applying various types of loads to FG nanobeams, the influence of geometric parameters, length scale, and material distribution on the phenomenon of the deflection platform is discussed. According to the simulation results, it is found that the aspect ratio and the length scale of the beam will change the shape of the deflection platform, making it change regularly among various shapes. The geometric size will greatly increase the height of the platform, but it has little effect on the dimensionless deflection. Reasonable material distribution can enhance the internal coupling effect, increase the height of the platform and reduce the required load. Our results suggest that when studying nano-scale flexoelectric models, the deflection platforms or similar phenomena should be considered.

Superconvergence analysis of a conservative mixed finite element method for the nonlinear Klein–Gordon–Schrödinger equations

AbstractIn this paper, a linearized mass and energy conservative mixed finite element method (MFEM) is proposed for solving the nonlinear Klein–Gordon–Schrödinger equations. Optimal error estimates without grid‐ratio condition are derived by some rigorous analysis and an error splitting technique, that is, one is the temporal error which is only τ‐dependent and the other is the spatial error which is only h‐dependent. Furthermore, the superclose results are obtained by using the idea of combination of interpolation and projection. Besides, a so‐called “lifting” approach also play an important role to obtain the superclose results. With the above achievements, the global superconvergent properties are deduced through the interpolated post processing operators. Finally, three numerical examples are given to validate the convergence order, unconditional stability, mass, and energy conservation.

Numerical simulations of a mixed finite element method for damped plate vibration problems

<abstract><p>The mixed finite element method can reduce the requirement for the smoothness of the finite element space and simplify the interpolation space for finite elements, and hence is especially effective in solving high order differential equations. In this work, we establish a mixed finite element scheme for the initial boundary conditions of damped plate vibrations and prove the existence and uniqueness of the solution of the semi-discrete and backward Euler fully discrete schemes. We use linear element approximation for the introduced intermediate variables, conduct the error analysis, and obtain the optimal order error estimate. We verify the efficiency and the accuracy of the mixed finite element scheme via numerical case studies and quantify the influence of the damping coefficient on the frequency and amplitude of the vibration.</p></abstract>

Error estimates of mixed finite elements combined with Crank-Nicolson scheme for parabolic control problems

<abstract><p>In this paper, a mixed finite element method combined with Crank-Nicolson scheme approximation of parabolic optimal control problems with control constraint is investigated. For the state and co-state, the order $ m = 1 $ Raviart-Thomas mixed finite element spaces and Crank-Nicolson scheme are used for space and time discretization, respectively. The variational discretization technique is used for the control variable. We derive optimal priori error estimates for the control, state and co-state. Some numerical examples are presented to demonstrate the theoretical results.</p></abstract>

Two-grid $ H^1 $-Galerkin mixed finite elements combined with $ L1 $ scheme for nonlinear time fractional parabolic equations

<abstract><p>In this paper, we propose a two-grid algorithm for nonlinear time fractional parabolic equations by $ H^1 $-Galerkin mixed finite element discreitzation. First, we use linear finite elements and Raviart-Thomas mixed finite elements for spatial discretization, and $ L1 $ scheme on graded mesh for temporal discretization to construct a fully discrete approximation scheme. Second, we derive the stability and error estimates of the discrete scheme. Third, we present a two-grid method to linearize the nonlinear system and discuss its stability and convergence. Finally, we confirm our theoretical results by some numerical examples.</p></abstract>

Analysis of a linearized poromechanics model for incompressible and nearly incompressible materials

In this work, we thoroughly analyze the linearized version of a poromechanics model developed to simulate soft tissues perfusion. This is a fully unsteady model in which the fluid and solid equations are strongly coupled through the interstitial pressure. As such, it generalizes Darcy, Brinkman and Biot equations of poroelasticity. The mathematical and numerical analysis of this model was initially performed for a compressible porous material. Here, we focus on the nearly incompressible case with a semigroup approach, which also allows us to prove the existence of weak solutions. We show the existence and uniqueness of strong and weak solutions in the incompressible limit case, for which a divergence constraint on the mixture velocity appears. Due to the special form of the coupling, the underlying problem is not coercive. Nevertheless, by using the notion of $ T $-coercivity, we obtain stability estimates and well-posedness results. Our study also provides guidelines to propose stable and robust approximations of the problem with mixed finite elements. In particular, we recover an inf-sup condition that is independent of the porosity. Finally, we numerically investigate the elliptic regularity of the associated steady-state problem and illustrate the sensitivity of the solution with respect to the different model parameters.

A posteriori error estimates for mixed finite element discretizations of the Neutron Diffusion equations

We analyse a posteriori error estimates for the discretization of the neutron diffusion equations with mixed finite elements. We provide guaranteed and locally efficient estimators on a base block equation, the one-group neutron diffusion equation. We pay particular attention to AMR strategies on Cartesian meshes, since such structures are common for nuclear reactor core applications. We exhibit a robust marker strategy for this specific constraint, the direction marker strategy.

Penalty function method for geometrically nonlinear buckling analysis of imperfect truss with multi-freedom constraints based on mixed FEM

This paper is concerned with the approach to implementing the Penalty function method for imposing multi-freedom constraints in geometrically nonlinear analysis of imperfect trusses based on mixed finite element formulation. Using a finite element model based on displacement formulation, it is required to incorporate both the dependent boundary relations and initial length imperfection to the nonlinear master stiffness system of equations for solving the geometrically nonlinear problem of imperfect truss with multi-freedom constraints. For decreasing the mathematical complexion of the incorporating process, the author proposes a novel mixed finite truss element considering initial imperfection, used in building the model for solving the geometrically nonlinear problem of truss with multi-freedom constraints. The modified nonlinear stiffness equation is constructed by employing the penalty function method to convert a constrained problem into an unconstrained problem by extremizing the augmented energy function established based on the proposed mixed finite element formulation. For solving the nonlinear equilibrium equation of imperfect trusses with multi-freedom constraints, the incremental equilibrium equation is constructed, and the incremental-iterative algorithm for calculation is established utilizing the arc-length method, used for writing the calculation program for investigating geometrically nonlinear behavior of imperfect truss with multi-freedom constraints. The results of the numerical test show the influence of initial imperfection and choosing weight values on the equilibrium path of the truss.

Comparison of a hybrid model and experimental measurements for a dielectric-coated coaxial ECR thruster

Electrostatic probe and thrust balance measurements of a coaxial electron-cyclotron-resonance plasma thruster with magnetic nozzle are compared against numerical simulations of the device that solve self-consistently the plasma transport problem with a hybrid particle-in-cell/fluid approach and the microwave electromagnetic fields using mixed finite elements. A simple phenomenological anomalous transport model similar to those used in Hall thruster modeling is applied. Reasonable average relative errors are reported on the ion current density (8.7%) and plasma density (12.8%) profiles along the plume. Good agreement is found in terms of relative errors on thruster performance parameters as the 90%-current divergence angle (0%–3%), utilization efficiency (3%–10%), peak ion energy (9%–15%), and energy efficiency (2%–17%). The comparison suggests that enhanced cross-field diffusion is present in the plasma. Differences in the experimental and numerical behavior of electron temperature point to the areas of the model that could be improved. These include the electron heat flux closure relation, which must correctly account for the axial electron cooling observed.

A posteriori error estimates of mixed discontinuous Galerkin method for a class of Stokes eigenvalue problems

<abstract><p>For a class of Stokes eigenvalue problems including the classical Stokes eigenvalue problem and the magnetohydrodynamic Stokes eigenvalue problem a residual type a posteriori error estimate of the mixed discontinuous Galerkin finite element method using $ \mathbb{P}_{k}-\mathbb{P}_{k-1} $ element $ (k\geq 1) $ is studied in this paper. The a posteriori error estimators for approximate eigenpairs are given. The reliability and efficiency of the posteriori error estimator for the eigenfunction are proved and the reliability of the estimator for the eigenvalue is also analyzed. The numerical results are provided to confirm the theoretical predictions and indicate that the method considered in this paper can reach the optimal convergence order $ O(dof^{\frac{-2k}{d}}) $.</p></abstract>

A Posteriori Error Estimate of Two Grid Mixed Finite Element Methods for Semilinear Elliptic Equations

In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.

Unconditional stability and convergence analysis of fully discrete stabilized finite volume method for the time-dependent incompressible MHD flow

In this paper, we consider the stability and convergence of fully discrete stabilized finite volume method for the unsteady incompressible magnetohydrodynamics (MHD) equations on the quasi-uniform and regular triangulations. The spatial discretization is based on the lowest equal-order mixed finite element pair for the velocity, pressure and the magnetic fields, while the time discretization is based on the backward Euler semi-implicit scheme. In order to overcome the restriction of the discrete inf-sup condition, the local pressure projection stabilized method on the element is employed. This stabilized method has the advantages of parameter-free, no need to calculate the high-order derivatives or the edge-based data structures and it can be implemented at the element level. The unconditional stability of numerical scheme is established by using the Gronwall lemma, mathematical induction and choosing different test functions. Optimal error estimates of numerical approximations in various norms are also presented by constructing the corresponding dual problem. Finally, some numerical results are proposed to verify the established theoretical findings and show the performances of the considered numerical method.

Superclose analysis of $H^1$-Galerkin mixed finite element methods combined with two-grid scheme for semilinear parabolic equations

Superclose analysis of $H^1$-Galerkin mixed finite element methods combined with two-grid scheme for semilinear parabolic equations

A two-grid discretization method for nonlinear Schrödinger equation by mixed finite element methods

A two-grid discretization method for nonlinear Schrödinger equation by mixed finite element methods

Effect of the vertical component of ground motion on a rubble masonry wall model

In this paper the influence of the vertical component of ground motion on the performance of an unreinforced masonry wall is analysed using sets of one-component and two-component ground motions. The set of motions represents the actual seismicity of L'Aquila, while the investigated wall mimics an experimental specimen with two unconnected external leaves and a rubble core. The model falls within the mixed finite element method – discrete element method and accounts for crack formation, complete separation and new contact formation. The modelling strategy is capable to simulate the out-of-plane seismic response and the progressive loss of compactness of the wall up to collapse with the separation between the two external leaves. The vertical component increases the fragility of the wall and confirms the relevance of vertical ground motion for very vulnerable constructions. Nonetheless, to worsen the response, the vertical component needs to overcome specific, non-negligible intensity measure thresholds.

Two-dimensional model for composite beams and its state-space based mixed finite element solution by DQM

Most one-dimensional composite beam theories are based on simplified shear deformation assumptions, or even ignore shear deformation, which have limitations in analyzing multilayer beams with significant material differences. Therefore, this paper firstly proposed a two-dimensional analytical model for composite beams through the equivalent transformation of cross section. Based on the mixed variational principle, the state equations are then derived by finite element meshing and interpolation along the length of the beam, with nodal displacements and their energy-conjugated stresses as state variables. Subsequently, the differential quadrature method (DQM) is introduced to solve the equations and a two-dimensional analysis method for composite beams is established. Due to the use of displacements and stresses as fundamental variables in the state equations, various transfer characteristics of displacements and stresses at the interlayer interfaces of composite beams can be conveniently handled. This method is finally verified by numerical examples of steel-concrete composite beams and concrete beams with corrugated steel webs. Since no assumptions of displacement and stress distributions along the thickness of beams are required, the present method can provide benchmarks for various one-dimensional beam theories.

Well-posedness of generalized Stokes-Brinkman equations modeling moving solid phases

<abstract><p>Fluid flow through a free-fluid region and the adjacent porous medium has been studied in various problems, such as water flow in rice fields. For the problem with self-propelled solid phases, we provide a generalized Stokes equation for the free-fluid domain and the Brinkman equation in a macroscopic scale due to the movement of self-propelled solid phases rather than a single solid in the porous medium. The model is derived with the assumption that the porosity is not a constant. The porosity in the mathematical model varies depending on the propagation of the solid phases. These two models can be matched at the free-fluid/porous-medium interface and are developed for real world problems. We show the proof of the well-posedness of the discretized form of the weak formulation obtained from applying a mixed finite element scheme to the generalized Stokes-Brinkman equations. The proofs of the continuity and coercive property of the linear and bilinear functionals in the discretized equation are illustrated. We present the existence and uniqueness of the generalized Stokes-Brinkman equations for the numerical problem in two dimensions. The system of equations can be applied to fluid flow propelled by moving solid phases, such as mucus flow in the trachea.</p></abstract>

Variational discretization combined with fully discrete splitting positive definite mixed finite elements for parabolic optimal control problems

Variational discretization combined with fully discrete splitting positive definite mixed finite elements for parabolic optimal control problems