Mixed Finite Element Discretization Research Articles

An equilibrated estimator for mixed finite element discretizations of the curl-curl problem

Abstract We propose a new a posteriori error estimator for mixed finite element discretizations of the curl-curl problem. This estimator relies on a Prager–Synge inequality, and therefore leads to fully guaranteed constant-free upper bounds on the error. The estimator is also locally efficient and polynomial-degree-robust. The construction is based on patch-wise divergence-constrained minimization problems, leading to a cheap embarrassingly parallel algorithm. Crucially, the estimator operates without any assumption on the topology of the domain, and unconventional arguments are required to establish the reliability estimate. Numerical examples illustrate the key theoretical results, and suggest that the estimator is suited for mesh adaptivity purposes.

A MEEVC discretization for two-dimensional incompressible Navier-Stokes equations with general boundary conditions

In this work, we present a mass, energy, enstrophy and vorticity conserving (MEEVC) mixed finite element discretization for two-dimensional incompressible Navier-Stokes equations as an alternative to the original MEEVC scheme proposed in A. Palha and M. Gerritsma (2017) [5]. The present method can incorporate general boundary conditions. Conservation properties are proven. Supportive numerical experiments with both exact and inexact quadrature are provided.

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.

Saddle point least squares discretization for convection-diffusion

We consider a model convection-diffusion problem and present our recent analysis and numerical results regarding mixed finite element formulation and discretization in the singular perturbed case when the convection term dominates the problem. Using the concepts of optimal norm and saddle point reformulation, we found new error estimates for the case of uniform meshes. We compare the standard linear Galerkin discretization to a saddle point least square discretization that uses quadratic test functions, and explain the non-physical oscillations of the discrete solutions. We also relate a known upwinding Petrov–Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error norms. The results can be extended to the multidimensional case in order to find efficient approximations for more general singular perturbed problems including convection dominated models.

An augmented Lagrangian-based preconditioning technique for a class of block three-by-three linear systems

We propose an augmented Lagrangian-based preconditioner to accelerate the convergence of Krylov subspace methods applied to linear systems of equations with a block three-by-three structure such as those arising from mixed finite element discretizations of the coupled Stokes–Darcy flow problem. We analyze the spectrum of the preconditioned matrix and we show how the new preconditioner can be efficiently applied. Numerical experiments are reported to illustrate the effectiveness of the preconditioner in conjunction with flexible GMRES for solving linear systems of equations arising from a 3D test problem.

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.

Two-Grid Method for a Fully Discrete Mixed Finite Element Solution of the Time-Dependent Schrödinger Equation

We study the backward Euler fully discrete mixed finite element method for the time-dependent Schrödinger equation; the error result of the mixed finite element solution is obtained in the L2-norm with order O(τ+hk+1). Then, a two-grid method is presented with a backward Euler fully discrete scheme. Using this method, we solve the original problem on a much coarser grid and solve elliptic equations on a fine grid. In addition, the error of the two-grid solution is also obtained in the L2-norm with order O(τ+hk+1+Hk+2). The numerical experiment is provided to demonstrate the efficiency of the algorithm.

Unconditional Superconvergence Error Estimates of Semi-Implicit Low-Order Conforming Mixed Finite Element Method for Time-Dependent Navier–Stokes Equations

In this paper, the unconditional superconvergence error analysis of the semi-implicit Euler scheme with low-order conforming mixed finite element discretization is investigated for time-dependent Navier–Stokes equations. In terms of the high-accuracy error estimates of the low-order finite element pair on the rectangular mesh and the unconditional boundedness of the numerical solution in L∞-norm, the superclose error estimates for velocity in H1-norm and pressure in L2-norm are derived firstly by dealing with the trilinear term carefully and skillfully. Then, the global superconvergence results are obtained with the aid of the interpolation post-processing technique. Finally, some numerical experiments are carried out to support the theoretical findings.

Eigenvalue Optimization with respect to Shape‐Variations in Electromagnetic Cavities

AbstractThe article considers the optimization of eigenvalues in electromagnetic cavities by means of shape variations. The field distribution and its frequency in a radio‐frequency cavity are governed by Maxwell's eigenvalue problem. To this end, we utilize a mixed formulation by Kikuchi (1987) and a mixed finite element discretization by means of Nédélec and Lagrange elements. The shape optimization is based on the method of mappings, where a Piola transformation is utilized to assert conformity of the mapped spaces. We derive the derivatives by the use of adjoint calculus for the constraining Maxwell eigenvalue problem. In two numerical examples, we demonstrate the functionality of this method.

High-Order Mixed Finite Element Variable Eddington Factor Methods

We apply high-order mixed finite element discretization techniques and their associated preconditioned iterative solvers to the Variable Eddington Factor (VEF) equations in two spatial dimensions. The mixed finite element VEF discretizations are coupled to a high-order Discontinuous Galerkin (DG) discretization of the discrete ordinates transport equation to form effective linear transport algorithms that are compatible with high-order (curved) meshes. This combination of VEF and transport discretizations is motivated by the use of high-order mixed finite element methods in hydrodynamics calculations at the Lawrence Livermore National Laboratory (LLNL). Due to the mathematical structure of the VEF equations, the standard Raviart Thomas (RT) mixed finite elements cannot be used to approximate the vector variable in the VEF equations. Instead, we investigate three alternatives based on the use of continuous finite elements for each vector component, a non-conforming RT approach where DG-like techniques are used, and a hybridized RT method. We present numerical results that demonstrate high-order accuracy, compatibility with curved meshes, and robust and efficient convergence in iteratively solving the coupled transport-VEF system and in the preconditioned linear solvers used to invert the discretized VEF 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.

Error analysis of a conservative finite element scheme for time-dependent inductionless MHD problem

In this paper, we develop and analyze a fully discrete mixed finite element method for unsteady inductionless magnetohydrodynamics problem. An Euler semi-implicit scheme is proposed with a mixed variational formulation based on the variables (u,p,j,ϕ), in which the Navier–Stokes equations are approximated by stable finite elements and the current density is discretized by the divergence-conforming finite element. This scheme has the feature that the discrete current density keeps charge conservation property. It is shown that both the continuous problem and its fully discrete Euler semi-implicit scheme are well-posed. We prove unconditionally convergence and error estimates for velocity, pressure and current density. Finally, numerical experiments have been performed to validate the theoretical analysis and the law of charge conservation.

Semi-Discrete and Fully Discrete Mixed Finite Element Methods for Maxwell Viscoelastic Model of Wave Propagation

Semi-Discrete and Fully Discrete Mixed Finite Element Methods for Maxwell Viscoelastic Model of Wave Propagation

Superconvergence of the MINI mixed finite element discretization of the Stokes problem: An experimental study in 3D

Stokes flows are a type of fluid flow where convective forces are small in comparison with viscous forces, and momentum transport is entirely due to viscous diffusion. Besides being routinely used as benchmark test cases in numerical fluid dynamics, Stokes flows are relevant in several applications in science and engineering including porous media flow, biological flows, microfluidics, microrobotics, and hydrodynamic lubrication. The present study concerns the discretization of the equations of motion of Stokes flows in three dimensions utilizing the MINI mixed finite element, focusing on the superconvergence of the method which was investigated with numerical experiments using five purpose-made benchmark test cases with analytical solution. Despite the fact that the MINI element is only linearly convergent according to standard mixed finite element theory, a recent theoretical development proves that, for structured meshes in two dimensions, the pressure superconverges with order 1.5, as well as the linear part of the computed velocity with respect to the piecewise-linear nodal interpolation of the exact velocity. The numerical experiments documented herein suggest a more general validity of the superconvergence in pressure, possibly to unstructured tetrahedral meshes and even up to quadratic convergence which was observed with one test problem, thereby indicating that there is scope to further extend the available theoretical results on convergence.

A Mixed Finite Element Discretization of Dynamical Optimal Transport

In this paper we introduce a new class of finite element discretizations of the quadratic optimal transport problem based on its dynamical formulation. These generalize to the finite element setting the finite difference scheme proposed by Papadakis et al. (SIAM J Imaging Sci, 7(1):212–238, 2014). We solve the discrete problem using a proximal splitting approach and we show how to modify this in the presence of regularization terms which are relevant for physical data interpolation.

Domain decomposition and partitioning methods for mixed finite element discretizations of the Biot system of poroelasticity

We develop non-overlapping domain decomposition methods for the Biot system of poroelasticity in a mixed form. The solid deformation is modeled with a mixed three-field formulation with weak stress symmetry. The fluid flow is modeled with a mixed Darcy formulation. We introduce displacement and pressure Lagrange multipliers on the subdomain interfaces to impose weakly continuity of normal stress and normal velocity, respectively. The global problem is reduced to an interface problem for the Lagrange multipliers, which is solved by a Krylov space iterative method. We study both monolithic and split methods. In the monolithic method, a coupled displacement-pressure interface problem is solved, with each iteration requiring the solution of local Biot problems. We show that the resulting interface operator is positive definite and analyze the convergence of the iteration. We further study drained split and fixed stress Biot splittings, in which case we solve separate interface problems requiring elasticity and Darcy solves. We analyze the stability of the split formulations. Numerical experiments are presented to illustrate the convergence of the domain decomposition methods and compare their accuracy and efficiency.

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.

A non-incremental numerical method for dynamic elastoplastic problems by the symplectic Brezis–Ekeland–Nayroles principle

The goal of the present work is the demonstration of the ability of the Symplectic Brezis–Ekeland–Nayroles (SBEN) principle for the numerical simulation of dynamic elastoplastic problems in small strains. This non-incremental approach is an alternative method to conventional step-by-step techniques, as it allows the computation of the whole evolution curve over the entire loading history. We show that the SBEN variational formulation yields to a constrained time–space minimization problem. The cost function depends on stress, displacements, and plastic multiplier fields, which, naturally, leads to a mixed finite element discretization. The details of the solution algorithm are illustrated through the numerical study of the elastoplastic response of the thin and thick pressurized tubes, including inertia effects. Moreover, the balance of momentum is handled using two approaches. Firstly, the Schaefer superposition technique is considered, where the balance of momentum is exactly satisfied. Secondly, the balance of momentum is considered as an optimization constraint and satisfied only at the Gauss integration point level of the finite element problem. The accuracy and efficiency of the SBEN principle are assessed by comparing the numerical results with the analytical solutions (for the thin tube) and the predictions derived by the conventional incremental finite element procedure.

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.

A Port-Hamiltonian formulation of linear thermoelasticity and its mixed finite element discretization

A port-Hamiltonian formulation for general linear coupled thermoelasticity and for the thermoelastic bending of thin structures is presented. The construction exploits the intrinsic modularity of port-Hamiltonian systems to obtain a formulation of linear thermoelasticity as an interconnection of the elastodynamics and heat equations. The derived model can be readily discretized by using mixed finite elements. The discretization is structure-preserving, since the main features of the system are retained at a discrete level. The proposed model and discretization strategy are validated against a benchmark problem of thermoelasticity, the Danilovskaya problem.