Mixed Finite Element Method Research Articles

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.

Numerical solution for a class of evolution differential equations with [formula omitted]-Laplacian and memory

In this paper, we study a partial integral differential equation with p-Laplacian using a mixed finite element method. Two stable and convergent fixed point schemes are proposed to solve the nonlinear algebraic system. Using the implementation of the method in Matlab environment, we numerically analyse the convergence with an example. Some other examples, which illustrate several asymptotic behaviours and some localization effects of the solutions, are presented.

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.

A Reduced Basis Method for Darcy Flow Systems that Ensures Local Mass Conservation by Using Exact Discrete Complexes

A solution technique is proposed for flows in porous media that guarantees local conservation of mass. We first compute a flux field to balance the mass source and then exploit exact co-chain complexes to generate a solenoidal correction. A reduced basis method based on proper orthogonal decomposition is employed to construct the correction and we show that mass balance is ensured regardless of the quality of the reduced basis approximation. The method is directly applicable to mixed finite and virtual element methods, among other structure-preserving discretization techniques, and we present the extension to Darcy flow in fractured porous media.

Convergence of a regularized finite element discretization of the two-dimensional Monge–Ampère equation

This paper proposes a regularization of the Monge–Ampère equation in planar convex domains through uniformly elliptic Hamilton–Jacobi–Bellman equations. The regularized problem possesses a unique strong solution u ε u_\varepsilon and is accessible to the discretization with finite elements. This work establishes uniform convergence of u ε u_\varepsilon to the convex Alexandrov solution u u to the Monge–Ampère equation as the regularization parameter ε \varepsilon approaches 0 0 . A mixed finite element method for the approximation of u ε u_\varepsilon is proposed, and the regularized finite element scheme is shown to be uniformly convergent. The class of admissible right-hand sides are the functions that can be approximated from below by positive continuous functions in the L 1 L^1 norm. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions u u .

The Mixed Finite Element Reduced-Dimension Technique with Unchanged Basis Functions for Hydrodynamic Equation

The mixed finite element (MFE) method is one of the most valid numerical approaches to solve hydrodynamic equations because it can be very suited to solving problems with complex computing domains. Regrettably, the MFE method for the hydrodynamic equations would include lots of unknowns. Especially, when it is applied to settling the practical engineering problems, it could contain hundreds of thousands and even tens of millions of unknowns. Thus, it would bring about many difficulties for actual applications, such as consuming a long CPU running time and accumulating many round-off errors, so as to be very difficult to obtain the desired numerical solutions. Therefore, we herein take the two-dimensional (2D) unsteady Navier–Stokes equation in hydrodynamics as an example. Using the proper orthogonal decomposition to lower the dimension of unknown Crank–Nicolson MFE (CNMFE) solution coefficient vectors for the 2D unsteady Navier–Stokes equation about vorticity–stream functions, we construct a reduced-dimension recursive CNMFE (RDRCNMFE) method with unchanged basis functions. In the circumstances, the RDRCNMFE method can keep the basis functions unchanged in an MFE subspace and has the same precision as the classical CNMFE method. We employ the matrix method to analyse the existence and stability along with errors to the RDRCNMFE solutions, leading to a very simple theory analysis. We use the numerical simulations for the backwards-facing step flow to verify the effectiveness of the RDRCNMFE method. The RDRCNMFE method with unchanged basis functions only reduces the dimension of the solution coefficient vectors of the CNMFE, which is completely different from previous order reduction methods which greatly affects the accuracy by reducing the dimension of the MFE subspace.

Optimal pointwise-in-time error analysis of a mixed finite element method for a multi-term time-fractional fourth-order equation

In this paper, the regularity of the solution for a multi-term time-fractional fourth-order equation is analyzed, and the derived result shows that the solution behaves a weak singularity at initial time t=0. By introducing an intermediate variable p=−Δu, we obtain a system of two second-order differential equations, which is discretised by using the L1 scheme on graded meshes in time and the mixed finite element method (MFEM) in space. It is shown that the optimal pointwise-in-time error estimate is derived for the proposed scheme. Moreover, the fast L1-MFEM is constructed to improve the computational efficiency. Finally, numerical experiments are provided to further verify our theoretical convergent result and the computational efficiency of the proposed two schemes.

Mixed methods and lower eigenvalue bounds

It is shown how mixed finite element methods for symmetric positive definite eigenvalue problems related to partial differential operators can provide guaranteed lower eigenvalue bounds. The method is based on a classical compatibility condition (inclusion of kernels) of the mixed scheme and on local constants related to compact embeddings, which are often known explicitly. Applications include scalar second-order elliptic operators, linear elasticity, and the Steklov eigenvalue problem.

Second-Order Collocation-Based Mixed FEM for Flexoelectric Solids

Flexoelectricity is an electromechanical coupling between the electric field and the mechanical strain gradient, as well as between the mechanical strains and the electric field gradient, observed in all dielectric materials, including those with centrosymmetry. Flexoelectricity demands C1-continuity for straightforward numerical implementation as the governing equations in the gradient theory are fourth-order partial differential equations. In this work, an alternative collocation-based mixed finite element method for direct flexoelectricity is used, for which a newly developed quadratic element with a high capability of capturing gradients is introduced. In the collocation method, mechanical strains and electric field through independently assumed polynomials are collocated with the mechanical strains and electric field derived from the mechanical displacements and electric potential at collocation points inside a finite element. The mechanical strain gradient and electric field are obtained by taking the directional derivative of the independent mechanical strain and electric field gradients. However, an earlier proposed linear element is unable to capture all mechanical strain gradient components and, thus, simulate flexoelectricity correctly. This problem is solved in the present work by using quadratic shape functions for the mechanical displacements and electric potential with fewer degrees of freedom than the traditional mixed finite element method. A Fortran user-element code is developed by the authors: first, for the linear and, after that, for the quadratic element. After verifying the linear element with numerical results from the literature, both linear and quadratic elements’ behaviors are tested for different problems. It is shown that the proposed second-order collocation-based mixed FEM can capture the flexoelectric behavior better compared to the existing linear formulations.

Superconvergence error analysis of an efficient mixed finite element method for time-dependent natural convection problem

Superconvergence error analysis of an efficient mixed finite element method for time-dependent natural convection problem

Mixed finite element method for a second order Dirichlet boundary control problem

The main aim of this article is to analyze mixed finite element method for the second order Dirichlet boundary control problem. We develop both a priori and a posteriori error analysis using the energy space based approach. We obtain optimal order a priori error estimates in the energy norm and L2-norm with the help of auxiliary problems. The reliability and the efficiency of proposed a posteriori error estimator is discussed using the Helmholtz decomposition. Numerical experiments are presented to confirm the theoretical findings.

The Mixed Finite Element Method Based on the Crank-Nicolson Scheme for Darcy Flows in Porous Media

This article is devoted to the a priori error estimates of the fully discrete Crank-Nicolson approximation for the Darcy flows. Optimal order error estimates in two different norms are established for the lowest order. Numerical experiments confirm the theoretical analysis regarding convergence rates.

A mixed finite element method for large deformation of flexoelectric materials

Flexoelectricity is a universal and size-dependent electromechanical phenomenon coupling the electric field with strain gradient in dielectric materials. we establishe a mixed variational principle and develop a mixed finite element method (FEM). This allows us to use C0-continuous interpolation functions in FEM instead of C1 that is difficult to construct but has to be used in conventional FEM to investigate flexoelectricity. A good agreement between the analytical solutions and our numerical results of an infinite cylinder problem validates the present developed mixed FEM. We further investigate the effects of geometrically nonlinearity on flexoelectricity in soft dielectrics under large deformation. It is found that the flexoelectricity can be greatly enhanced due to the large deformation of soft dielectrics even with much smaller flexoelectric coefficients than popular ceramics. We also demonstrate the necessarity of adopting finite strain model or considering geometrical nonlinearity to precisely describing flexoelectric response in soft dielectrics. The findings in this work also suggest tremendous potential application of soft materials in the field of electromechanical sensors via flexoelectricity.

A multipoint flux mixed finite element method with mass-conservative characteristic finite element method for incompressible miscible displacement problem

A multipoint flux mixed finite element method with mass-conservative characteristic finite element method for incompressible miscible displacement problem

An efficient discontinuous Galerkin - mixed finite element model for variable density flow in fractured porous media

Modeling variable density flow (VDF) in fractured porous media is computationally challenging. The challenges are mainly related to: (i) the high permeability contrast between the matrix and the fractures and, (ii) the nonlinearity induced by density variations. Due to their local mass conservation property, cell-centered methods, such as finite volumes (FV), mixed finite elements (MFE) or Discontinuous Galerkin (DG) methods are well suited for modeling mass transfer in highly heterogeneous domains. When applied for fractured media, these methods require small grid cells next to fractures because of the cross-flow equilibrium assumption (i.e. the pressure and concentration in the fracture and in the adjacent matrix grid-cells are assumed the same). To avoid this constraint, an efficient model is developed in this work using advanced cell-centered numerical methods for VDF in fractured porous media with cross-flow equilibrium assumed only across the fractures. The new model uses the hybrid MFE method for the flow discretization in the matrix and in the fracture continua. Mass transport in the matrix is modeled using a monotonic upwind MFE scheme. Advection-dominated transport in fractures is discretized with the DG method, which is well adapted for hyperbolic equations. The new model ensures continuity of pressure, concentration, fluid mass flux, advective and dispersive contaminant fluxes at matrix-fracture interfaces as well as at the intersection of several fractures. The time discretization is performed using high-order adaptive time integration techniques via the method of lines (MOL). The developed model is first validated by comparison against a 2D-2D model for a test problem dealing with linear flow and transport in a 2D domain involving a “+”-shaped barrier/fracture network. Then, the MFE-DG model is used for the simulation of the Henry saltwater intrusion problem and the results are validated by comparison against the semi-analytical solution in the case of unfractured and fractured aquifers. Finally, the simulation of a transport problem with variable viscosity in a fractured heterogeneous domain points out the superiority of the new model in terms of accuracy and efficiency when compared to a standard finite element model.

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.

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.