Mixed Finite Element Scheme Research Articles

The Reduced-Dimension Method for Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors of the Extended Fisher–Kolmogorov Equation

A reduced-dimension (RD) method based on the proper orthogonal decomposition (POD) technology and the linearized Crank–Nicolson mixed finite element (CNMFE) scheme for solving the 2D nonlinear extended Fisher–Kolmogorov (EFK) equation is proposed. The method reduces CPU runtime and error accumulation by reducing the dimension of the unknown CNMFE solution coefficient vectors. For this purpose, the CNMFE scheme of the above EFK equation is established, and the uniqueness, stability and convergence of the CNMFE solutions are discussed. Subsequently, the matrix-based RDCNMFE scheme is derived by applying the POD method. Furthermore, the uniqueness, stability and error estimates of the linearized RDCNMFE solution are proved. Finally, numerical experiments are carried out to validate the theoretical findings. In addition, we contrast the RDCNMFE method with the CNMFE method, highlighting the advantages of the dimensionality reduction method.

A mixed finite element spatial discretization scheme and a higher‐order accurate temporal discretization scheme for a strongly discontinuous electromagnetic interface condition in ideal sliding electrical contact problems

AbstractSliding electrical contact involves multiple conductors sliding in contact at different speeds, with current flowing through the contact surfaces. The Lagrangian method is commonly used to describe the electromagnetic field in order to overcome the trouble of convective dominance, especially in high‐speed sliding electrical contact problems. However, to maintain correct field continuity, magnetic vector potential and scalar potential taken as variables cannot be continuous simultaneously at the ideal sliding electrical contact interface. This involves a strongly discontinuous condition for variables. Further, commonly used spatial–temporal discretization algorithms are invalid, for example, the classic finite element (CFEM) framework does not allow discontinuous variables, and the backward Euler method in time domain introduces a significant interface error source associated with the velocity of relative motion. To accurately handle strongly discontinuous conditions in numerical calculations, a mixed nodal finite element scheme and a higher‐order accurate temporal discretization scheme are introduced. In this scheme, classical finite element method is performed in each subdomain, and the derivative terms at the boundary are added as new variables. The effectiveness and accuracy of the above methods are verified by comparing them with a standard solution in a two‐dimensional railgun model and analyzing the current density distribution in a three‐dimensional railgun model.

Pressure-robust finite element scheme for the time-dependent fully coupled Stokes–Darcy-transport problem

We present a pressure-robust mixed finite element scheme for the time-dependent fully coupled Stokes–Darcy-transport problem. The P1c⊕RT0 discretization and Raviart–Thomas mixed elements are used to discretize the velocity in the free flow region and the porous medium region, respectively. We approximate the pressure with piecewise constant finite elements and the concentration with piecewise linear finite elements. We perform a divergence-free reconstruction of the velocity in the Stokes domain. Our scheme preserves local mass conservation in the sense of projection and has pressure-robust property. The stability and error analysis of the method are obtained. The errors of velocity and concentration are independent of pressure, and our method works well even with small viscosity. The theoretical results are validated with numerical examples.

A fully implicit edge/face centered discontinuous Galerkin / mixed finite element scheme for the advection-dispersion equation

Mixed Finite Element (MFE) method is a robust numerical technique for solving elliptic and parabolic partial differential equations (PDEs). However, MFE can generate solutions with strong unphysical oscillations and/or large numerical diffusion for hyperbolic type PDEs. For its part, Discontinuous Galerkin (DG) finite element method is well adapted to solve hyperbolic systems and can accurately reproduce solutions involving sharp fronts. Therefore, the combination of DG and MFE is a good strategy for solving hyperbolic/parabolic problems such as advection – diffusion/dispersion equations. The classical formulation of the two methods is based on operator and time splitting allowing for separate solutions to advection with an explicit scheme and to dispersion with an implicit scheme. However, this kind of approach has the following drawbacks: (i) it lacks efficiency, as two systems with different unknowns are solved at each time step, (ii) it induces errors generated by the splitting, (iii) it can be CPU wise-expensive because of the CFL constraint, and (iv) it cannot be employed for steady-state transport simulations.To overcome these difficulties, we develop in this work a fully implicit edge/face centered DG-MFE formulation where the two methods share the same unknowns. In this formulation, the DG method is developed on lumping regions associated with the mesh edges/faces instead of mesh elements. Thus, the traces of concentration at mesh edges/faces, which are the Degrees Of Freedom (DOF) of the hybrid-MFE, are also part of the DOFs of the DG. The temporal discretization is based on the Crank–Nicolson method for both advection and dispersion. Numerical tests are performed to validate the new scheme by comparison against an analytical solution and to show its ability to handle steady-state transport simulations.The procedure is developed for 2D triangular meshes but can easily be extended to other 2D and 3D shape elements.

A Mixed Finite Element Method for the Multi-Term Time-Fractional Reaction–Diffusion Equations

In this work, a fully discrete mixed finite element (MFE) scheme is designed to solve the multi-term time-fractional reaction–diffusion equations with variable coefficients by using the well-known L1 formula and the Raviart–Thomas MFE space. The existence and uniqueness of the discrete solution is proved by using the matrix theory, and the unconditional stability is also discussed in detail. By introducing the mixed elliptic projection, the error estimates for the unknown variable u in the discrete L∞(L2(Ω)) norm and for the auxiliary variable λ in the discrete L∞((L2(Ω))2) and L∞(H(div,Ω)) norms are obtained. Finally, three numerical examples are given to demonstrate the theoretical results.

Optimal time two-mesh mixed finite element method for a nonlinear fractional hyperbolic wave model

<abstract><p>In this article, a second-order time discrete algorithm with a shifted parameter $ \theta $ combined with a fast time two-mesh (TT-M) mixed finite element (MFE) scheme was considered to look for the numerical solution of the nonlinear fractional hyperbolic wave model. The second-order backward difference formula including a shifted parameter $ \theta $ (BDF2-$ \theta $) with the weighted and shifted Grünwald difference (WSGD) approximation for fractional derivative was used to discretize time direction at time $ t_{n-\theta} $, the $ H^1 $-Galerkin MFE method was applied to approximate the spatial direction, and the fast TT-M method was used to save computing time of the developed MFE system. A priori error estimates for the fully discrete TT-M MFE system were analyzed and proved in detail, where the second-order space-time convergence rate in both $ L^2 $-norm and $ H^1 $-norm were derived. Detailed numerical algorithms with smooth and weakly regular solutions were provided. Finally, some numerical examples were provided to illustrate the feasibility and effectiveness for our scheme.</p></abstract>

An H1-Galerkin Space-Time Mixed Finite Element Method for Semilinear Convection–Diffusion–Reaction Equations

In this paper, the semilinear convection–diffusion–reaction equation is split into a lower-order system by introducing the auxiliary variable q=a(x)ux. An H1-Galerkin space-time mixed finite element method for the lower-order system is then constructed. The proposed method applies the finite element method to discretize the time and space directions simultaneously and does not require checking the Ladyzhenskaya–Babusˇka–Brezzi (LBB) compatibility constraints, which differs from the traditional mixed finite element method. The uniqueness of the approximate solutions u and q are proven. The L2(L2) norm optimal order error estimates of the approximate solution u and q are derived by introducing the space-time projection operator. The numerical experiment is presented to verify the theoretical results. Furthermore, by comparing with the classical H1-Galerkin mixed finite element scheme, the proposed scheme can easily improve computational accuracy and time convergence order by changing the basis function.

Two-grid algorithm for two-dimensional linear Schrödinger equation by the mixed finite element method

In this paper, we propose a new two-grid algorithm for solving two-dimensional Schrödinger equation in semi-discrete and fully discrete mixed finite element scheme, respectively. With this algorithm, the solution of the Schrödinger equation on a fine grid can be reduced to the original problem on a much coarser grid together with two Poisson equations on the fine grid. In addition, we obtain error results of the two-grid solution. At last, a numerical experiment is provided to demonstrate the efficiency of the two-grid algorithm.

Variational schemes and mixed finite elements for large strain isotropic elasticity in principal stretches: Closed‐form tangent eigensystems, convexity conditions, and stabilised elasticity

AbstractA new computational framework for large strain elasticity in principal stretches is presented. Distinct from existing literature, the proposed formulation makes direct use of principal stretches rather than their squares that is, eigenvalues of Cauchy‐Green strain tensor. The proposed framework has three key features. First, the eigen‐decomposition of the tangent elasticity and initial (geometric) stiffness operators is obtained in closed‐form from principal information alone. Crucially, these newly found eigenvalues describe the general convexity conditions of isotropic hyperelastic energies. In other words, convexity is postulated concisely through tangent eigenvalues supplementing the original work of Ball (Arch Ration Mech Anal. 1976; 63(4): 337–403). Consequently, this novel finding opens the door for designing efficient automated Newton‐style algorithms with stabilised tangents via closed‐form semipositive definite projection or spectral shifting that converge irrespective of mesh resolution, quality, loading scenario and without relying on path‐following techniques. A critical study of closed‐form tangent stabilisation in the context of isotropic hyperelasticity is therefore undertaken in this work. Second, in addition to high order displacement‐based formulation, mixed Hu‐Washizu variational principles are formulated in terms of principal stretches by introducing stretch work conjugate Lagrange multipliers that enforce principal stretch‐stress compatibility. This is similar to enhanced strain methods. However, the resulting mixed finite element scheme is cost‐efficient, specially compared to approximating the entire strain tensors since the formulation is in the scalar space of singular values. Third, the proposed framework facilitates simulating rigid and stiff systems and those that are nearly‐inextensible in principal directions, a constituent of elasticity that cannot be easily studied using standard formulations.

Two‐grid mixed finite element method for two‐dimensional time‐dependent Schrödinger equation

In this paper, we study the semidiscrete mixed finite element scheme and construct a two‐grid algorithm for the two‐dimensional time‐dependent Schrödinger equation. We analyze error results of the mixed finite element solution in ‐norm by some projection operators. Then, we propose a two‐grid method of the semidiscrete mixed finite element. With this method, the solution of the Schrödinger equation on a fine grid is reduced to the solution of original problem on a much coarser grid together with the solution of two elliptic equations on the fine grid. We also obtain the error estimate of two‐grid solution with exact solution in ‐norm. Finally, a numerical experiment indicates that our two‐grid algorithm is more efficient than the standard mixed finite element method.

Novel analysis approach for the convergence of a second order time accurate mixed finite element scheme for parabolic equations

Novel analysis approach for the convergence of a second order time accurate mixed finite element scheme for parabolic equations

Analysis of fully discrete mixed finite element scheme for stochastic Navier–Stokes equations with multiplicative noise

This paper is concerned with stochastic incompressible Navier–Stokes equations with multiplicative noise in two dimensions with respect to periodic boundary conditions. Based on the Helmholtz decomposition of the multiplicative noise, semi-discrete and fully discrete time-stepping algorithms are proposed. The convergence rates for mixed finite element methods based time-space approximation with respect to convergence in probability for the velocity and the pressure are obtained. Furthermore, with establishing some stability and using the negative norm technique, the partial expectations of the $$H^1$$ and $$L^2$$ norms of the velocity error are proved to converge optimally.

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>

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>

A robust upwind mixed hybrid finite element method for transport in variably saturated porous media

Abstract. The mixed finite element (MFE) method is well adapted for the simulation of fluid flow in heterogeneous porous media. However, when employed for the transport equation, it can generate solutions with strong unphysical oscillations because of the hyperbolic nature of advection. In this work, a robust upwind MFE scheme is proposed to avoid such unphysical oscillations. The new scheme is a combination of the upwind edge/face centered finite volume method with the hybrid formulation of the MFE method. The scheme ensures continuity of both advective and dispersive fluxes between adjacent elements and allows to maintain the time derivative continuous, which permits employment of high-order time integration methods via the method of lines (MOL). Numerical simulations are performed in both saturated and unsaturated porous media to investigate the robustness of the new upwind MFE scheme. Results show that, contrarily to the standard scheme, the upwind MFE method generates stable solutions without under and overshoots. The simulation of contaminant transport into a variably saturated porous medium highlights the robustness of the proposed upwind scheme when combined with the MOL for solving nonlinear problems.

A posteriori error estimates for Biot system using a mixed discretization for flow

We first derive convergence and a priori stability, and next reliability and efficiency of a posteriori error indicators for a Biot poroelastic model coupled with an elastic model in R3, solved by a continuous Galerkin scheme (CG) for the displacement and a mixed finite element scheme for the flow. The coupled system is decoupled by a fixed stress splitting algorithm. The numerical implementation of the residual based error indicators is simple, even for the mixed discretization, but at the expense of two suboptimal bounds. The scheme is tested on two benchmark problems via several numerical experiments.

A posteriori error analysis of Banach spaces-based fully-mixed finite element methods for Boussinesq-type models

Abstract In this paper we consider Banach spaces-based fully-mixed variational formulations recently proposed for the Boussinesq and the Oberbeck–Boussinesq models, and develop reliable and efficient residual-based a posteriori error estimators for the 2D and 3D versions of the associated mixed finite element schemes. For the reliability analysis, we employ the global inf-sup condition for each sub-model, namely Navier–Stokes and heat equations in the case of Boussinesq, along with suitable Helmholtz decomposition in nonstandard Banach spaces, the approximation properties of the Raviart–Thomas and Clément interpolants, further regularity on the continuous solutions, and small data assumptions. In turn, the efficiency estimates follow from inverse inequalities and the localization technique through bubble functions in adequately defined local Lp spaces. Finally, several numerical results including natural convection in 3D differentially heated enclosures, are reported with the aim of confirming the theoretical properties of the estimators and illustrating the performance of the associated adaptive algorithm.

How to prove optimal convergence rates for adaptive least-squares finite element methods∗

Abstract The convergence analysis with rates for adaptive least-squares finite element methods (ALSFEMs) combines arguments from the a posteriori analysis of conforming and mixed finite element schemes. This paper provides an overview of the key arguments for the verification of the axioms of adaptivity for an ALSFEM for the solution of a linear model problem. The formulation at hand allows for the simultaneous analysis of first-order systems of the Poisson model problem, the Stokes equations, and the linear elasticity equations. Following [Carstensen and Park, SIAM J. Numer. Anal. 53 (1), 2015], the adaptive algorithm is driven by an alternative residual-based error estimator with exact solve and includes a separate marking strategy for quasi-optimal data resolution of the right-hand side. This presentation covers conforming discretisations for an arbitrary polynomial degree and mixed homogeneous boundary conditions.

A posteriori error analysis of a momentum conservative Banach spaces based mixed-FEM for the Navier–Stokes problem

In this paper we develop an a posteriori error analysis of a new momentum conservative mixed finite element method recently introduced for the steady-state Navier–Stokes problem in two and three dimensions. More precisely, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces, such as local estimates, and suitable Helmholtz decompositions, we derive a reliable and efficient residual-based a posteriori error estimator for the corresponding mixed finite element scheme on arbitrary (convex or non-convex) polygonal and polyhedral regions. On the other hand, inverse inequalities, the localization technique based on bubble functions, among other tools, are employed to prove the efficiency of the proposed a posteriori error indicator. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.

A Reduced-Order Extrapolation (ROE) Method for Solution Coefficient Vectors in the Mixed Finite Element (MFE) Method for the Two-Dimensional (2D) Fourth-Order Hyperbolic Equation

This study focuses on a reduced-order extrapolation method for the coefficient vectors of the mixed finite element solution for two-dimensional fourth-order hyperbolic equation. We first establish the mixed finite element scheme for the equation and give the matrix model of the mixed finite element scheme and the existence, stability and error estimates of its solutions. Then, we derive a reduced-order extrapolation mixed finite element matrix model with a small number of unknowns, where the proper orthogonal decomposition method is used to save central processing unit time, and prove the existence, stability and error estimates of the reduced-order extrapolation mixed finite element solutions with the help of matrix knowledge. More importantly, the reduced-order extrapolation mixed finite element matrix model have the same basis functions and error accuracy as the mixed finite element matrix model. Finally, some numerical experiments confirm the effectiveness of the reduced-order extrapolation mixed finite element matrix model, where the central processing unit time is greatly reduced and the accuracy is maintained.