Finite Element Scheme Research Articles

An analysis of the bilinear finite volume method for the singularly-perturbed convection-diffusion problems on Shishkin mesh

In this paper, we study the bilinear finite volume element method for solving the singularly perturbed convection-diffusion problem on the Shishkin mesh. We first prove that the finite volume element scheme is ϵ-uniformly stable. Then, based on new expression of the finite volume bilinear form and some detailed integral calculations, an ϵ-uniform error estimation is derived in the ϵ-weighted gradient norm, including the L2-norm. This error estimate is better than the known result. Moreover, we also give the L∞-error estimate near the boundary layer regions. At last, numerical experiments show the effectiveness of our method.

Two novel linearized energy-conserving finite element schemes for nonlinear regularized long wave equation

In this paper, two linearized second-order energy-conserving schemes for the nonlinear regularized long wave (RLW) equation are introduced, the unconditional superclose and superconvergence results are presented by using the conforming finite element method (FEM). Initially, through a skillful decomposition of the nonlinear term, two linearized second-order fully discrete schemes are developed. Compared to the previous nonlinear approaches, these schemes significantly reduce the number of iterations and improve computational efficiency; moreover, they conserve energy, and ensure the boundedness of the numerical solution in the H1-norm directly, which represents an advancement over the L∞-norm boundedness reported in prior studies. Secondly, based on the boundedness of the FE solution, the Ritz projection operator and high-precision results of the linear triangular element, the error estimates for superclose and superconvergence are derived without any restrictions on the ratio between time step size Δt and spatial mesh size h. Finally, four numerical examples are provided to confirm the accuracy of the theoretical analysis and the effectiveness of the method.

A split-step finite element method for the space-fractional Schrödinger equation in two dimensions

In this article, we propose a split-step finite element method (FEM) for the two-dimensional nonlinear Schrödinger equation (NLS) with Riesz fractional derivatives in space. The space-fractional NLS is first spatially discretized by finite element scheme and the semi-discrete variational scheme is obtained. We prove that it maintains the mass and energy conservation laws. Then, we establish a fully discrete split-step finite element scheme for the considered problem, which avoids the iteration at each time layer, thereby significantly reducing computational cost. The discrete mass conservation property and error estimate of this split-step finite element scheme is derived. Finally, illustrative tests and the numerical simulation of dynamic of wave solutions are included to confirm its effectiveness and capability.

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.

Collocation Finite Element Method for the Fractional Fokker–Planck Equation

ABSTRACTIn this study, the approximate results of the fractional Fokker–Planck equations have been investigated. First, finite element schemes have been obtained using collocation finite element method based on the trigonometric quintic B‐spline basis functions. Then, the present method is tested on two fundamental problems having appropriate initial conditions. The newly obtained numerical results contained the error norms and for various temporal and spatial steps are compared with the exact ones and other solutions. More accurate results have been obtained for large numbers of spatial and temporal elements.

Optimal [formula omitted] error estimates of mass- and energy- conserved FE schemes for a nonlinear Schrödinger–type system

In this paper, we present an implicit Crank–Nicolson finite element (FE) scheme for solving a nonlinear Schrödinger–type system, which includes Schrödinger–Helmholz system and Schrödinger–Poisson system. In our numerical scheme, we employ an implicit Crank–Nicolson method for time discretization and a conforming FE method for spatial discretization. The proposed method is proved to be well-posedness and ensures mass and energy conservation at the discrete level. Furthermore, we prove optimal L2 error estimates for the fully discrete solutions. Finally, some numerical examples are provided to verify the convergence rate and conservation properties.

Uniform error analysis of an exponential IMEX-SAV method for the incompressible flows with large Reynolds number based on grad-div stabilization

Based on the grad-div stabilization and scalar auxiliary variable (SAV) methods, a first-order Euler implicit/explicit finite element scheme is studied for the Navier–Stokes equations with large Reynolds number. In the designing of numerical scheme, the nonlinear term is explicitly treated such that one only needs to solve a constant coefficient algebraic system at each time step. Meanwhile, the proposed scheme is unconditionally stable without any condition of the time step τ and mesh size h. In finite element discretization, we use the stable Taylor-Hood element for the approximation of the velocity and pressure. By a rigorous analysis, we derive an uniform L2 error estimate O(τ+h2) of the velocity in which the constant is independent of the viscosity coefficient. Finally, numerical experiments are given to support theoretical results and the efficiency of the proposed scheme.

Computational insights into tumor invasion dynamics: A finite element approach

The finite element scheme is proposed and analyzed for the solution of an acid-mediated tumor invasion model. The reaction–diffusion equation shows the evolution in the tumor cell density, H+ ions concentration, and healthy tissue density over time. The coupled non-linear partial differential equations are discretized in time with the implicit Euler method and in space with standard Galerkin finite element. To solve the non-linear and coupled terms of the system a fixed point iteration scheme is presented. Moreover, a mass-lumped scheme is adopted to reduce the computation cost. The cut-off method is used to compute the bounded solutions of the PDEs. Finally, The effects of proliferation rate and healthy tissue degradation rate are investigated.

The weak Galerkin finite element method for Stokes interface problems with curved interface

In this paper, we develop a weak Galerkin (WG) finite element scheme for the Stokes interface problems with curved interface. The conventional numerical schemes rely on the use of straight segments to approximate the curved interface and the accuracy is limited by geometric errors. Hence in our method, we directly construct the weak Galerkin finite element space on the curved cells to avoid geometric errors. For the integral calculation on curved cells, we employ non-affine transformations to map curved cells onto the reference element. The optimal error estimates are obtained in both the energy norm and the L2 norm. A series of numerical experiments are provided to validate the efficiency of the proposed WG method.

Two linear energy stable lumped mass finite element schemes for the viscous Cahn–Hilliard equation on curved surfaces in 3D

The evolution of a dynamic system on complex curved 3D surfaces is essential for the understanding of natural phenomena, the development of new materials, and engineering design optimization. In this work, we study the viscous Cahn–Hilliard equation on curved surfaces and develop two linear energy stable finite element schemes based on the lumped mass method. Two stabilizing terms are added to ensure both the unique solvability and unconditional energy stability. We prove rigorously that two schemes are unconditionally energy stable . Numerical experiments are presented to verify theoretical results and to show the robustness and accuracy of the proposed method.

Convergence and superconvergence analysis for a mass conservative, energy stable and linearized BDF2 scheme of the Poisson–Nernst–Planck equations

In this paper, we consider a linearized BDF2 finite element scheme for the Poisson–Nernst–Planck (PNP) equations. By employing a novel approach, we rigorously derive unconditional optimal error estimates of the numerical solutions in the l∞(L2) and l∞(H1) norms, as well as superconvergent results. The key of the convergence and superconvergence analysis lies in deriving the stability of the finite element solutions in some stronger norms. The advantage of this approach is that there is no need to introduce a corresponding time discrete system, so it is more concise than the error split technique in previous literatures. Finally, we carry out two numerical examples to confirm the theoretical findings.

L1‐type finite element method for time‐fractional diffusion‐wave equations on nonuniform grids

AbstractAs is well known, one of the main challenges in dealing with the time‐fractional diffusion‐wave equation is that the solutions have weak regularity at the initial values. This necessitates the use of nonuniform grids for error analysis. However, the theoretical analysis of nonuniform grids is relatively complex and lacks efficient and stable numerical methods. The main research problem addressed in this article is the time‐fractional diffusion‐wave equation on nonuniform grids. We use the order reduction method and discrete complementary convolution kernel to discretize the Caputo derivative of the equation and obtain the L1‐type finite element method on nonuniform grids. The properties related to discrete complementary convolutional kernels are often used for algorithm convergence analysis, which simplifies the process of finite element theory analysis and is efficient and innovative. This article demonstrates the solvability, stability, and convergence of the L1‐type finite element schemes on nonuniform grids. Finally, the consistency between the proposed finite element scheme and the convergence order results obtained from theoretical analysis was verified through experiments.

Machine learning algorithms on predicting the turbulent mixed convection flow in a driven-cavity with two horizontal cylinders

Turbulent mixed convection flow and heat transfer properties in a driven cavity with two circular cylinders arranged one above another are analyzed numerically with the incorporation of a finite element scheme, based on the Galerkin method of weighted residuals. A comparison of streamlines, isotherms, and local, average Nusselt number is provided to illustrate how the Richardson number Ri, Reynolds number Re and the eddy viscosity ε affect the transport phenomena inside the cavity. The local and average Nusselt number have been evaluated and it is found that strong convection at the top wall causes the average Nusselt number to progressively rise with respect to Ri and ε, while steady or slightly varying profiles are seen with regard to Re. In order to predict the thermal distribution inside the cavity due to these hot cylinders, Artificial Neural Network (ANN) and Gaussian Process Regression (GPR) have been developed with these model parameters as input and average Nusselt Number as target values. Several plots have been depicted to come to a conclusion that both models have been trained very effectively with the minimal observed Mean Square Error (MSE) values of 0.0026018 and 0.0026428 for ANN and GPR, respectively. In support to these findings, the coefficient of determination R2 suggests that ANN (R2 = 0.89) outperforms GPR (R2 = 0.87) in testing accuracy, hence it is recommended for prediction task.

Nonlinear dynamics of a flexible rod partially sliding in a rigid sleeve under the action of gravity and configurational force

We investigate various methods of analyzing systems with moving boundaries, using as an example a flexible rod sliding in an ideal frictionless sleeve in the field of gravity. Special attention is paid to the configurational force acting on the rod at the sleeve opening and thus determining the rod’s dynamics. The non-material kinematic description used in simulations is based on the re-parametrization of the Lagrangian arc length coordinate. The variational formulation uses the energy expressions written for the entire rod, comprising the free segment and the one inside the sleeve. A novel finite element scheme is efficient for highly flexible rods, which may undergo complete ejection. A simplified two degrees of freedom model, which accelerates simulations, shows a good agreement as the bending stiffness increases. An analytical study using Hamiltonian mechanics exploits the separation of variables into fast oscillations and slow axial motion. The adiabatic invariant approach leads to approximate closed-form solutions for the slow dynamics and yields the maximum injection depth of the rod into the sleeve.

Discrete strong extremum principles for finite element solutions of advection‐diffusion problems with nonlinear corrections

AbstractA nonlinear correction technique for finite element methods of advection‐diffusion problems on general triangular meshes is introduced. The classic linear finite element method is modified, and the resulting scheme satisfies discrete strong extremum principle unconditionally, which means that it is unnecessary to impose the well‐known restrictions on diffusion coefficients and geometry of mesh‐cell (e.g., “acute angle” condition), and we need not to perform upwind treatment on the advection term separately. Moreover, numerical example shows that when a discrete scheme does not satisfy the strong extremum principle, even if it maintains the global physical bound, non‐physical numerical oscillations may still occur within local regions where no numerical result is beyond the physical bound. Thus, it is worth to point out that our new nonlinear finite element scheme can avoid non‐physical oscillations around sharp layers in advection‐dominate regions, due to maintaining discrete strong extremum principle. Convergence rates are verified by numerical tests for both diffusion‐dominate and advection‐dominate problems.

A variational multiscale stabilized finite element method for double diffusive incompressible flow with application to mixed convection in a heated staggered cavity under magnetic field

In this work, a novel stabilized finite element scheme in a subgrid multiscale variational framework is proposed to efficiently solve double-diffusive incompressible Newtonian fluid flow in a complex geometry occurring in several real-life applications. Here, the fine scale effects are transferred to the course scale problem following algebraic subgrid analysis, and then, the course scale equations are computed and the key stabilization parameters are derived following the Fourier analysis. The influence of flow, heat, and mass transfer parameters such as Reynold’s Number, Hartmann Number, Richardson Number, and Lewis Number are analyzed under a magnetic field, through the streamlines, isotherms, and the local heat fluxes to understand the double-diffusive mixed convective process in a complex staggered cavity, which is relevant to engineering applications such as ventilation and contamination assessment systems. The study finds that maximum species contamination together with the maximum Sherwood Number (12.54) occurs when the Richardson number is 10, and the minimum contamination together with the minimum Sherwood Number (0.71) occurs when the Lewis number is 0.1 when other parameters are varied within the operational range. The results are validated with previously existing literature for simple cases.

A novel family of Q1-finite volume element schemes on quadrilateral meshes

A novel family of isoparametric bilinear finite volume element schemes are constructed and analyzed to solve the anisotropic diffusion problems on general convex quadrilateral meshes. These new schemes are obtained by employing a special quadrature rule to approximate the line integrals in classical Q1-finite volume element method. The new quadrature rule is a linear combination of trapezoidal and midpoint rules, and the weights depend on a parameter ωK. The novelty of this work is that, for any fully anisotropic diffusion tensor, we provide some specific ωK to ensure the coercivity result of the proposed schemes on arbitrary parallelogram, quasi-parallelogram, trapezoidal and some general convex quadrilateral meshes. More interesting is that, the parameter ωK can only involves the anisotropic diffusion tensor and the geometry of quadrilateral cell. An optimal H1 error estimate is also proved on quasi-parallelogram meshes. Finally, the theoretical findings are validated by several numerical examples.

Implementation of finite element scheme to study thermal and mass transportation in water-based nanofluid model under quadratic thermal radiation in a disk

Implementation of finite element scheme to study thermal and mass transportation in water-based nanofluid model under quadratic thermal radiation in a disk

Structure preserving finite element schemes for the Navier-Stokes-Cahn-Hilliard system with degenerate mobility

In this work we present two new numerical schemes to approximate the Navier-Stokes-Cahn-Hilliard system with degenerate mobility using finite differences in time and finite elements in space. The proposed schemes are conservative, energy-stable and preserve the maximum principle approximately (the amount of the phase variable being outside of the interval [0,1] goes to zero in terms of a truncation parameter). Additionally, we present several numerical results to illustrate the accuracy and the well behavior of the proposed schemes, as well as a comparison with the behavior of the Navier-Stokes-Cahn-Hilliard model with constant mobility.

Mass, momentum and energy preserving FEEC and broken-FEEC schemes for the incompressible Navier–Stokes equations

Abstract In this article we propose two finite-element schemes for the Navier–Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the point-wise divergence free constraint of the velocity, its total momentum and its energy, in addition to being pressure robust. Following the broken-FEEC approach, our second scheme uses fully discontinuous spaces and local conforming projections to define the discrete differential operators. It preserves the same invariants up to a dissipation of energy to stabilize numerical discontinuities. For both schemes we use a middle point time discretization that preserve these invariants at the fully discrete level and we analyze its well-posedness in terms of a CFL condition. While our theoretical results hold for general finite elements preserving the de Rham structure, we consider one application to tensor-product spline spaces. Specifically, we conduct several numerical test cases to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.