Mixed Finite Element Method Research Articles

Analysis of Weak Galerkin Mixed Finite Element Method Based on the Velocity–Pseudostress Formulation for Navier–Stokes Equation on Polygonal Meshes

Analysis of Weak Galerkin Mixed Finite Element Method Based on the Velocity–Pseudostress Formulation for Navier–Stokes Equation on Polygonal Meshes

Two Finite Element Approaches for the Porous Medium Equation That Are Positivity Preserving and Energy Stable

In this work, we present the construction of two distinct finite element approaches to solve the porous medium equation (PME). In the first approach, we transform the PME to a log-density variable formulation and construct a continuous Galerkin method. In the second approach, we introduce additional potential and velocity variables to rewrite the PME into a system of equations, for which we construct a mixed finite element method. Both approaches are first-order accurate, mass conserving, and proved to be unconditionally energy stable for their respective energies. The mixed approach is shown to preserve positivity under a CFL condition, while a much stronger property of unconditional bound preservation is proved for the log-density approach. A novel feature of our schemes is that they can handle compactly supported initial data without the need for any perturbation techniques. Furthermore, the log-density method can handle unstructured grids in any number of dimensions, while the mixed method can handle unstructured grids in two dimensions. We present results from several numerical experiments to demonstrate these properties.

Modeling and mathematical analysis of root water uptake in the rhizosphere

ABSTRACT This research focuses on elucidating a system of partial differential equations designed to model the process of water uptake by a single root within an unsaturated soil. This model extends a unidimensional root scenario by incorporating radial and axial hydraulic conductance in the root structure. The flow dynamics in the rhizosphere are represented by the Richards equation, while a diffusion equation is employed to characterize water movement within the root. The resulting system forms a strongly coupled, nonlinear parabolic-elliptic system. To establish the existence and uniqueness of a solution, we utilize the technique of semi-discretization and the Schauder-Tychonoff fixed point theorem. Additionally, we approximate the system using a mixed finite element method, and the subsequent numerical tests presented align well with the model predictions. This comprehensive approach contributes to our understanding of the intricate dynamics involved in root water uptake and provides a robust framework for further exploration in this field.

Exploring different FEM strategies for hydro-mechanical coupled gas injection simulation in clay materials

Over the last few decades, the study of gas injection in porous media, particularly under multi-field coupled conditions, has emerged as a prominent focus within the field of geotechnical engineering. This article presents a comprehensive comparison of three numerical strategies, evaluating their impact on computational efficiency and result accuracy during Hydro-Mechanical (HM) coupled simulations of gas injection in clay-based geomaterials. This comprehensive comparison encompasses three numerical simulation methods for the mechanical sub-problem: The Standard Finite Element Method (SFEM), the Standard Finite Element Method with Selective Integration (SFEM+SI), and the Mixed Finite Element Method (MFEM). The Heat and Gas Fracking model (HGFRAC) is introduced to illustrate the computational characteristics of these methods. The results indicate that the effective application of SFEM is heavily dependent on a high-precision mesh. Convergence issues may arise when dealing with relatively coarse meshes. Nevertheless, these convergence issues can be effectively mitigated by incorporating either the Selective Integration method or the MFEM formulations. In terms of computational efficiency, it is evident that the SFEM+SI method demonstrates higher efficiency than SFEM and MFEM. However, it is noteworthy that the computed gas flow patterns of SFEM and SFEM+SI can be affected by the alignment of the mesh. With MFEM, displacements and strains are calculated as independent unknowns, enhancing result accuracy and achieving mesh independence.

Mixed finite elements for Kirchhoff–Love plate bending

We present a mixed finite element method with triangular and parallelogram meshes for the Kirchhoff–Love plate bending model. Critical ingredient is the construction of low-dimensional local spaces and appropriate degrees of freedom that provide conformity in terms of a sufficiently large tensor space and that allow for any kind of physically relevant Dirichlet and Neumann boundary conditions. For Dirichlet boundary conditions and polygonal plates, we prove quasi-optimal convergence of the mixed scheme. An a posteriori error estimator is derived for the special case of the biharmonic problem. Numerical results for regular and singular examples illustrate our findings. They confirm expected convergence rates and exemplify the performance of an adaptive algorithm steered by our error estimator.

Numerical solutions to Hyperbolic Maxwell quasi-variational inequalities in Bean–Kim model for type-II superconductivity

This paper is devoted to the finite element analysis for the Bean–Kim model governed by the full 3D Maxwell equations. Describing type-II superconductivity at the macroscopic level, this model leads to a challenging coupled system consisting of the Faraday equation and a hyperbolic quasi-variational inequality (QVI) of the second kind with L1-type nonlinearity, that arises explicitly from the magnetic field dependency in the critical current. With the involved Maxwell coupling in the 3D H(curl)-setting, the hyperbolic QVI character poses the primary challenge in the numerical investigation. Two mixed finite element methods based on implicit Euler and leapfrog time-stepping are proposed. On the one hand, the implicit Euler method results in a nonstandard system of curl-curl elliptic QVI with a first-order curl-type nonlinearity. Though the well-posedness of this system is guaranteed, its numerical realization is not straightforward and requires the use of a two-stage iteration process of high computational complexity. On the other hand, by approximating the electric and magnetic fields at two different time step levels, the leapfrog method turns out to be more suitable as it naturally eliminates the notorious QVI structure. More importantly, utilizing suited subdifferential and optimization techniques, we are able to prove an efficiently computable explicit formula for its exact solution in terms of the electric field, which makes its numerical computation substantially more favorable than the Euler method. As further advantages, the leapfrog method applies to broad scenarios involving low regular data of bounded variation (BV) in time for both the applied current source and the temperature distribution. Through nonstandard technical arguments tailored to the BV data, our analysis proves the conditional stability and, eventually, the uniform convergence of the proposed leapfrog method. This paper is closed by 3D numerical tests showcasing the reasonable and efficient performance of the proposed numerical solution.

$$ H^1$$-Galerkin mixed finite element method for the vibration equation of beam with structural damping

$$ H^1$$-Galerkin mixed finite element method for the vibration equation of beam with structural damping

Robust mixed finite element methods for a quad-curl singular perturbation problem

Robust mixed finite element methods are developed for a quad-curl singular perturbation problem. Lower order H(gradcurl)-nonconforming but H(curl)-conforming finite elements are constructed, which are extended to nonconforming finite element Stokes complexes and the associated commutative diagrams. Then H(gradcurl)-nonconforming finite elements are employed to discretize the quad-curl singular perturbation problem, which possess the sharp and uniform error estimates with respect to the perturbation parameter. The Nitsche’s technique is exploited to achieve the optimal convergence rate in the case of the boundary layers. Numerical results are provided to verify the theoretical convergence rates. In addition, the regularity of the quad-curl singular perturbation problem is established.

A locking-free numerical method for the quasi-static linear thermo-poroelasticity model

In this paper, we propose a locking-free numerical method to solve the quasi-static linear thermo-poroelasticity model, which exhibits two types of locking phenomena: Poisson locking and nonphysical oscillations. By analyzing the regularity of solution of the original model, we find that the Poisson locking is caused by divu≈0 as λ→+∞. Moreover, when discretizing the diffusion equations using backward Euler method for time, one can see that divu1≈0 for some special parameters. If we directly use the continuous Galerkin mixed finite element method (FEM) to solve the original model, the pressure and temperature fields exhibit numerical oscillations at early times. To overcome these two locking phenomena, we introduce a new variable ξ=αp+βT−λdivu to reformulate the original problem into a new one. This new problem includes a built-in mechanism to maintain stability for the continuous Galerkin mixed FEM. We further prove the existence and uniqueness of the weak solution by using the standard Galerkin method in conjunction with regularity estimates. We also design a fully discrete time-stepping scheme that employs mixed FEM with P2−P1−P1−P1 element pairs for the space variables and the backward Euler method for the time variable. Optimal convergence is demonstrated in both space and time. Finally, numerical examples are provided to demonstrate the optimal convergence rates of the variables and the robustness of the proposed method with respect to ν, and to verify the absence of the locking phenomenon.

A mixed finite element method for the forward problem of electrical impedance tomography with the shunt model

We propose a mixed finite element method for the forward problem of electrical impedance tomography with the shunt model. The method is based on a constrained minimization formulation in terms of the current density, where parts of the Lagrange multiplier are interpreted as the voltages on the electrodes. We justify the well-posedness of this continuous formulation, estimate the regularity of its solution, and use Raviart–Thomas elements for its discretization. The resulting mixed method is then proved to be stable, and converge especially with respect to the voltages on the electrodes. Simulation on a 2D model also gives consistent results.

Solution of the Elliptic Interface Problem by a Hybrid Mixed Finite Element Method

This paper addresses the elliptic interface problem involving jump conditions across the interface. We propose a hybrid mixed finite element method on the triangulation where the interfaces are aligned with the mesh. The second-order elliptic equation is initially decomposed into two equations by introducing a gradient term. Subsequently, weak formulations are applied to these equations. Scheme continuity is enforced using the Lagrange multiplier technique. Finally, we derive an explicit formula for the entries of the matrix equation representing Lagrange multiplier unknowns resulting from hybridization. The method yields approximations of all variables, including the solution and gradient, with optimal order. Furthermore, the matrix representing the final linear algebra systems is not only symmetric but also positive definite. Numerical examples convincingly demonstrate the effectiveness of the hybrid mixed finite element method in addressing elliptic interface problems.

An efficient flux‐variable approximation scheme for Darcy's flow

AbstractWe present an efficient numerical method to approximate the flux variable for the Darcy flow model. An important feature of our new method is that the approximate solution for the flux variable is obtained without approximating the pressure at all. To accomplish this, we introduce a user‐defined parameter delta, which is typically chosen to be small so that it minimizes the negative effect resulting from the absence of the pressure, such as inaccuracy in both the flux approximation and the mass conservation. The resulting algebraic system is of significantly smaller degrees of freedom, compared to the one from the mixed finite element methods or least‐squares methods. We also interpret the proposed method as a single step iterate of the augmented Lagrangian Uzawa applied to solve the mixed finite element in a special setting. Lastly, the pressure recovery from the flux variable is discussed and an optimal‐order error estimate for the method is obtained. Several examples are provided to verify the proposed theory and algorithm, some of which are from more realistic models such as SPE10.

Precise analysis of geometrical nonlinear mechanical behavior of shear deformable laminated curved composite beams in local coordinates via mixed type finite elements

In the present article, geometrically nonlinear analyses of curved laminated composite beams are carried out using Mixed Finite Element Method (MFEM). Timoshenko beam theory is used to consider shear influence which significantly effects the structural behavior of composite layers. Energy-based functional is generally needed to perform geometric nonlinear analyzes of layered composites. However, obtaining energy-based functional in geometric nonlinear theory is quite challenging due to the difficulty of solving mathematically nonlinear differential equations. In this study, this situation is overcome and a functional suitable for performing geometric nonlinear analyzes of layered composite materials is obtained. Incremental formulation procedure is utilized as a solution method for nonlinear finite element analyzes. This is an innovative application of the non-linear mixed finite element method and has the ability to perform geometric nonlinear analysis of curved laminated composite beams with perfect precision. By obtaining the functional in local coordinates, it is possible to solve special problems in various fields such as construction, machinery, aerospace and biomechanics.

Banach spaces-based mixed finite element methods for the coupled Navier–Stokes and Poisson–Nernst–Planck equations

Banach spaces-based mixed finite element methods for the coupled Navier–Stokes and Poisson–Nernst–Planck equations

Analysis of a Mixed Finite Element Method for Stochastic Cahn-Hilliard Equation with Multiplicative Noise

Analysis of a Mixed Finite Element Method for Stochastic Cahn-Hilliard Equation with Multiplicative Noise

A unified-field monolithic fictitious domain-finite element method for fluid-structure-contact interactions and applications to deterministic lateral displacement problems

Based upon two overlapped, body-unfitted meshes, a type of unified-field monolithic fictitious domain-finite element method (UFMFD-FEM) is developed in this paper for moving interface problems of dynamic fluid-structure interactions (FSI), which is accompanied with high-contrast physical coefficients across the interface and with contacting collisions between the structure and fluidic channel wall in the immersed FSI case. In particular, the proposed novel numerical method consists of a monolithic, stabilized mixed finite element method within the frame of fictitious domain/immersed boundary method (FD/IBM) for generic fluid-structure-contact interaction (FSCI) problems in the Eulerian–updated Lagrangian description, while involving the no-slip type of interface conditions on the fluid-structure interface, and the repulsive contact force/stress on the structural surface when the immersed structure contacts the fluidic channel wall. The developed UFMFD-FEM for FSI/FSCI problems can deal with the structural motion with large rotational and translational displacements and/or large deformation in an accurate and efficient fashion, which is first validated by two benchmark FSI problems and one FSCI model problem, then by experimental results of a realistic FSCI scenario – the microfluidic deterministic lateral displacement (DLD) problem that is applied to isolate circulating tumor cells (CTCs) from blood cells in the blood fluid through a cascaded filter DLD microchip in practice, where a particulate fluid with the pillar obstacles effect in the fluidic channel, i.e., the effects of fluid-structure interaction and structure collision, play significant roles to sort particles (cells) of different sizes with tilted pillar arrays. The developed unified-field, monolithic fictitious domain-based mixed finite element method can be seamlessly extended to more sophisticated, high dimensional FSCI problems with contacting collisions between the moving elastic structure and fluidic channel wall.

Optimally convergent mixed finite element methods for the time-dependent 2D/3D stochastic closed-loop geothermal system with multiplicative noise

Optimally convergent mixed finite element methods for the time-dependent 2D/3D stochastic closed-loop geothermal system with multiplicative noise

Modeling gas flow in low-permeability formations: An efficient combination of mixed finite elements and high order time integration schemes

Numerical simulation of gas flow in low permeability formations is a challenging task due to the high nonlinearity induced by (i) the compressibility of the gas, (ii) the Klinkenberg slippage effect and (iii) the Langmuir adsorption of the gas on the pore surface. Because of these nonlinearities, modeling gas flow in low permeability formations requires a great deal of computational effort. In this work, we develop an efficient numerical model using advanced spatial and temporal discretization methods for a simultaneous solution of the coupled equations of gas flow, cubic Peng-Robinson equation of state, slippage effect and Langmuir adsorption. The spatial discretization is performed with the lumped hybrid formulation of the mixed finite element method which is well adapted for fluid flow in heterogeneous porous media. The time integration is performed with high-order methods via the method of lines (MOL) which allows large time steps and efficient solution of the nonlinear system of equations. Numerical experiments, performed for gas extraction in a heterogeneous domain, point out the high efficiency of the new model since it can be until 10 times more efficient than the classical first-order time discretization method. Results of global sensitivity analysis show that the intrinsic gas-phase permeability and the Klinkenberg factor are the most influential parameters controlling the pumping rate in the case of gas extraction in a homogeneous domain.

A Nonlinear Mixed Finite Element Method for the Analysis of Flexoelectric Semiconductors

Abstract In this paper, we develop a nonlinear mixed finite element method for flexoelectric semiconductors and analyze the mechanically tuned redistributions of free carriers and electric currents through flexoelectric polarization in typical structures. We first present a macroscopic theory for flexoelectric semiconductors by combining flexoelectricity and nonlinear drift-diffusion theory. To use C0 continuous elements, we derive an incremental constrained weak form by introducing Langrage multipliers, in which the kinematic constraints between the displacement and its gradient are guaranteed. Based on the weak form, we established a mixed C0 continuous nine-node quadrilateral finite element as well as an iterative process for solving nonlinear boundary-value problems. The accuracy and convergence of the proposed element are validated by comparing linear finite element method results against analytical solutions for the bending of a beam. Finally, the nonlinear element method is applied to more complex problems, such as a circular ring, a plate with a hole, and an isosceles trapezoid. Results indicate that mechanical loads and doping levels have distinct influences on electric properties.

Error estimates for a mixed finite element method for the Maxwell’s transmission eigenvalue problem

In this paper, we analyze a numerical method combining the Ciarlet-Raviart mixed finite element formulation and an iterative algorithm for the Maxwell’s transmission eigenvalue problem. The eigenvalue problem is first written as a nonlinear quad-curl eigenvalue problem. Then the real transmission eigenvalues are proved to be the roots of a non-linear function. They are the generalized eigenvalues of a related linear self-adjoint quad-curl eigenvalue problem. These generalized eigenvalues are computed by a mixed finite element method. We derive the error estimates using the spectral approximation of compact operators, the theory of mixed finite element method for quad-curl problems, and the derivatives of eigenvalues.