Element Approximation In Space Research Articles

Iterative solution methods for high-order/hp–DGFEM approximation of the linear Boltzmann transport equation

In this article we consider the iterative solution of the linear system of equations arising from the discretisation of the poly-energetic linear Boltzmann transport equation using a high-order/hp–version discontinuous Galerkin finite element approximation in space, angle, and energy. In particular, we develop preconditioned Richardson iterations which may be understood as generalisations of source iteration in the mono-energetic setting, and derive computable a posteriori bounds for the solver error incurred due to inexact linear algebra, measured in a relevant problem-specific norm. We prove that the convergence of the resulting schemes and a posteriori solver error estimates are independent of the mesh size h and polynomial degree p. We also discuss how the poly-energetic Richardson iteration may be employed as a preconditioner for the generalised minimal residual (GMRES) method. Furthermore, we show that standard implementations of GMRES based on minimising the Euclidean norm of the residual vector can be utilized to yield computable a posteriori solver error estimates at each iteration, through judicious selections of left- and right-preconditioners for the original linear system. The effectiveness of poly-energetic source iteration and preconditioned GMRES, as well as their respective a posteriori solver error estimates, is demonstrated through numerical examples arising in the modelling of photon transport. While the convergence of poly-energetic source iteration is independent of h and p, we observe that the number of iterations required to attain convergence when employing GMRES only depends mildly on h and p. Moreover, this latter approach is highly effective in the low energy regime.

The Gauge–Uzawa methods for incompressible fluids with microstructure

In this paper, the Gauge–Uzawa method for the time‐dependent incompressible micropolar fluid problem is proposed. Firstly, the first‐order and second‐order Euler semi‐implicit Gauge–Uzawa methods for the micropolar fluid are constructed, and the unconditionally stability analysis and error estimate of the corresponding first‐order scheme are deduced. Secondly, the fully discrete scheme with finite element approximation in space is presented, and the stability result of the corresponding first‐order scheme is derived. Finally, some numerical results are given, which test the theoretical analysis results.

An Explicit–Implicit Spectral Element Scheme for the Nonlinear Space Fractional Schrödinger Equation

In this paper, we solve the space fractional nonlinear Schrödinger equation (SFNSE) by developing an explicit–implicit spectral element scheme, which is formulated based on the Legendre spectral element approximation in space and the Crank–Nicolson leap frog (CNLF) difference discretization in time. Both mass and energy conservative properties are discussed for the spectral element scheme. Numerical stability and convergence of the scheme are proved. Numerical experiments are performed to confirm the high accuracy and efficiency of the proposed numerical scheme.

On the two-step BDF finite element methods for the incompressible Navier–Stokes problem under boundary conditions of friction type

This article concerns the numerical approximation of the two-dimensional nonstationary Navier–Stokes equations with slip boundary conditions of friction type. It studies the well-posedness and error analysis properties of the numerical scheme in L2-norm for all positive time using the two-step backward differentiation formula (BDF) in time and the finite element approximation in space. Error estimates are proved under feasible assumptions on the regularity of the solution and with the aid of different versions of discrete Grownwall lemmas. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

Unconditional superconvergence analysis of two modified finite element fully discrete schemes for nonlinear Burgers' equation

In this paper, two modified nonconforming finite element fully discrete schemes are considered for solving the nonlinear Burgers' equation. The schemes are constructed by using the nonconforming Wilson element approximation in space combining with the backward Euler(BE) and 2-step backward differentiation formula (BDF2) respectively in time. We present the superconvergence results of the two proposed schemes in modified energy norm of order O(h2+τ) and O(h2+τ2), where h and τ are the space subdivision parameter and time step respectively. The proof involves the time-space error splitting technique, the discrete embedding inequality, the technique of taking difference between two time, the discrete derivative transfer technique and a new modified projection operator. Our analysis not only gets rid of the restriction between temporal and spatial step sizes, but also provides one order higher error estimate results than that of corresponding traditional finite element fully discrete schemes. Finally, numerical experiments are carried out to demonstrate the effective of theoretical analysis.

An energy-conserving finite element method for nonlinear fourth-order wave equations

In this paper, an energy-conserving finite element method is developed and intensively analyzed for a class of nonlinear fourth-order wave equations in a general sense for the first time, where the two-level, Crank-Nicolson type of temporal discretization scheme is designed to cooperate with the Lagrange finite element approximation in space in order to achieve the conservation of discrete energy at each time step. The energy conservation is crucial in engineering field to preserve the total energy as constant for dynamic vibration problems of beams and thin plates that can be modeled by the presented wave equations. In addition, the optimal spatial convergence properties in both L2- and H1-norm and the second-order temporal approximation rate are obtained for finite element solutions of u (deflection), Δu (bending moment) and/or ut (deflection speed) at the same time. Numerical experiments are carried out to validate all attained theoretical results.

Strong Convergence Order for the Scheme of Fractional Diffusion Equation Driven by Fractional Gaussian Noise

Fractional Gaussian noise models the time series with long-range dependence; when the Hurst index $H\in(1/2,1)$, it has positive correlation reflecting a persistent autocorrelation structure. This paper studies the numerical method for solving the stochastic fractional diffusion equation driven by fractional Gaussian noise. Using the operator theoretical approach, we present the regularity estimate of the mild solution and the fully discrete scheme with finite element approximation in space and backward Euler convolution quadrature in time. The $\mathcal{O}(\tau^{H-\rho\alpha})$ convergence rate in time and $\mathcal{O}(h^{\min(2,2-2\rho,\frac{2H}{\alpha}-2\rho-\epsilon)})$ in space are obtained, showing the relationship between the regularity of noise and convergence rates, where $\rho$ is a parameter to measure the regularity of noise and $\alpha\in(0,1)$. Finally, numerical experiments are performed to support the theoretical results.

Finite element approximation and analysis of a viscoelastic scalar wave equation with internal variable formulations

We consider linear scalar wave equations with a hereditary integral term of the kind used to model viscoelastic solids. The kernel in this Volterra integral is a sum of decaying exponentials (The so-called Maxwell, or Zener model) and this allows the introduction of one of two types of families of internal variables, each of which evolve according to an ordinary differential equation (ODE). There is one such ODE for each decaying exponential, and the introduction of these ODEs means that the Volterra integral can be removed from the governing equation. The two types of internal variable are distinguished by whether the unknown appears in the Volterra integral, or whether its time derivative appears; we call the resulting problems the displacement and velocity forms. We define fully discrete formulations for each of these forms by using continuous Galerkin finite element approximations in space and an implicit ‘Crank–Nicolson’ type of finite difference method in time. We prove stability and a priori bounds, and using the FEniCS environment, https://fenicsproject.org/ (The FEniCS project version 1.5, Archive of Numerical Software, 3 (100), 9–23, 2015.) give some numerical results. These bounds do not require Grönwall’s inequality and so can be regarded to be of high quality, allowing confidence in long time integration without an a priori exponential build up of error. As far as we are aware this is the first time that these two formulations have been described together with accompanying proofs of such high quality stability and error bounds. The extension of the results to vector-valued viscoelasticity problems is straightforward and summarised at the end. The numerical results are reproducible by acquiring the python sources from https://github.com/Yongseok7717, or by running a custom built docker container (instructions are given).

Stability Analysis of the Crank-Nicolson Finite Element Method for the Navier-Stokes Equations Driven by Slip Boundary Conditions

This paper is devoted to the study of numerical approximation for a class of two-dimensional Navier-Stokes equations with slip boundary conditions of friction type. The objective is to establish the well-posedness and stability of the numerical scheme in L 2 -norm and H 1 -norm for all positive time using the Crank-Nicholson scheme in time and the finite element approximation in space. The resulting variational structure dealing with is in the form of inequality, and obtaining H 1 -estimate is more involved because of the presence of the nondifferentiable term appearing at the boundary where slip occurs. We prove that the numerical scheme is stable in L 2 and H 1 -norms with the aid of different versions of discrete Grownwall lemmas, under a CFL-type condition. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

On Parameter Identification for Reaction-Dominated Pore-Scale Reactive Transport Using Modified Bee Colony Algorithm

Parameter identification is an important research topic with a variety of applications in industrial and environmental problems. Usually, a functional has to be minimized in conjunction with parameter identification; thus, there is a certain similarity between the parameter identification and optimization. A number of rigorous and efficient algorithms for optimization problems were developed in recent decades for the case of a convex functional. In the case of a non-convex functional, the metaheuristic algorithms dominate. This paper discusses an optimization method called modified bee colony algorithm (MBC), which is a modification of the standard bees algorithm (SBA). The SBA is inspired by a particular intelligent behavior of honeybee swarms. The algorithm is adapted for the parameter identification of reaction-dominated pore-scale transport when a non-convex functional has to be minimized. The algorithm is first checked by solving a few benchmark problems, namely finding the minima for Shekel, Rosenbrock, Himmelblau and Rastrigin functions. A statistical analysis was carried out to compare the performance of MBC with the SBA and the artificial bee colony (ABC) algorithm. Next, MBC is applied to identify the three parameters in the Langmuir isotherm, which is used to describe the considered reaction. Here, 2D periodic porous media were considered. The simulation results show that the MBC algorithm can be successfully used for identifying admissible sets for the reaction parameters in reaction-dominated transport characterized by low Pecklet and high Damkholer numbers. Finite element approximation in space and implicit time discretization are exploited to solve the direct problem.

Error Analysis of Euler Semi-implicit Scheme for the Nonstationary Magneto-hydrodynamics Problem with Temperature Dependent Parameters

In this article we consider a fully discrete Euler semi-implicit scheme for the nonstationary electromagnetically and thermally driven flow, which is describing the motion of a nonisothermal incompressible magneto-hydrodyna-mics fluid subject to generalized Boussinesq problem with temperature dependent parameters. A prototypical time-stepping scheme, which is comprised of the Euler semi-implicit discretization in time and conforming mixed finite element approximation in space is studied in detail. We obtain that the proposed scheme is unconditionally stable and derive some optimal error estimates for the fluid velocity, the fluid magnetic and the fluid temperature. Moreover, a suboptimal error estimate for the fluid pressure is proved. Numerical results are provided to verify the theoretical rates of the scheme.

On the Rayleigh-Bénard-Marangoni problem: Theoretical and numerical analysis

<p style='text-indent:20px;'>This paper is devoted to the theoretical and numerical analysis of the non-stationary Rayleigh-Bénard-Marangoni (RBM) system. We analyze the existence of global weak solutions for the non-stationary RBM system in polyhedral domains of <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^3 $\end{document}</tex-math></inline-formula> and the convergence, in the norm of <inline-formula><tex-math id="M2">\begin{document}$ L^{2}(\Omega), $\end{document}</tex-math></inline-formula> to the corresponding stationary solution. Additionally, we develop a numerical scheme for approximating the weak solutions of the non-stationary RBM system, based on a mixed approximation: finite element approximation in space and finite differences in time. After proving the unconditional well-posedness of the numerical scheme, we analyze some error estimates and establish a convergence analysis. Finally, we present some numerical simulations to validate the behavior of our scheme.

Computational identification of adsorption and desorption parameters for pore scale transport in periodic porous media

Computational identification of unknown adsorption and desorption rates is discussed in conjunction with reactive flow considered at pore scale. The reactive transport is governed by incompressible Stokes equations, coupled with convection–diffusion equation for species transport. The surface reactions, namely adsorption and desorption, are accounted via Robin boundary conditions. Henry and Langmuir isotherms are considered. Measured concentration of the specie at the outlet of the domain has to be provided to carry out the identification procedure. Deterministic and stochastic parameter identification approaches are considered. The influence of the noise in the measurements on the accuracy of the identified parameters is discussed. Multistage identification procedure is suggested for the considered class of problems. The proposed identification approach is applicable for different geometries (random and periodic) and for a range of process parameters. In this paper the potential of the approach is demonstrated in identifying parameters of Langmuir isotherm for low Peclet and low Damkoler numbers reactive flow in a 2D periodic porous media with circular inclusions. Simulation results for random porous media and other regime parameters are subject of follow up papers. Finite element approximation in space and implicit time discretization are exploited.

A Second Order Energy Stable Linear Scheme for a Thin Film Model Without Slope Selection

In this paper we present a second order accurate, energy stable numerical scheme for the epitaxial thin film model without slope selection, with a mixed finite element approximation in space. In particular, an explicit treatment of the nonlinear term, $$\frac{\nabla u}{1+|\nabla u|^2}$$ , greatly simplifies the computational effort; only one linear equation with constant coefficients needs to be solved at each time step. Meanwhile, a second order Douglas–Dupont regularization term, $$A\tau \varDelta ^2 ( u^{n+1} - u^n)$$ , is added in the numerical scheme, so that an unconditional long time energy stability is assured. In turn, we perform an $$\ell ^\infty (0,T; L^2)$$ convergence analysis for the proposed scheme, with an $$O (\tau ^2 + h^q)$$ error estimate derived. In addition, an optimal convergence analysis is provided for the nonlinear term using $$Q_q$$ finite elements, which shows that the spatial convergence order can be improved to $$q+1$$ on regular rectangular mesh. A few numerical experiments are presented, which confirms the efficiency and accuracy of the proposed second order numerical scheme.

Numerical Solution of Time-Dependent Problems with a Fractional-Power Elliptic Operator

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves a fractional power of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, standard two-level schemes are used. The approximate solution at a new time-level is obtained as a solution of a discrete problem with the fractional power of the elliptic operator. A Pade-type approximation is constructed on the basis of special quadrature formulas for an integral representation of the fractional power elliptic operator using explicit schemes. A similar approach is applied in the numerical implementation of implicit schemes. The results of numerical experiments are presented for a test two-dimensional problem.

Two-level schemes for the advection equation

The advection equation is the basis for mathematical models of continuum mechanics. In the approximate solution of nonstationary problems it is necessary to inherit main properties of the conservatism and monotonicity of the solution. In this paper, the advection equation is written in the symmetric form, where the advection operator is the half-sum of advection operators in conservative (divergent) and non-conservative (characteristic) forms. The advection operator is skew-symmetric. Standard finite element approximations in space are used. The standard explicit two-level scheme for the advection equation is absolutely unstable. New conditionally stable regularized schemes are constructed, on the basis of the general theory of stability (well-posedness) of operator-difference schemes, the stability conditions of the explicit Lax–Wendroff scheme are established. Unconditionally stable and conservative schemes are implicit schemes of the second (Crank–Nicolson scheme) and fourth order. The conditionally stable implicit Lax–Wendroff scheme is constructed. The accuracy of the investigated explicit and implicit two-level schemes for an approximate solution of the advection equation is illustrated by the numerical results of a model two-dimensional problem.

Error estimates of a semidiscrete finite element method for fractional stochastic diffusion‐wave equations

In this paper, we consider the Galerkin finite element method for solving the fractional stochastic diffusion‐wave equations driven by multiplicative noise, which can be used to describe the propagation of mechanical waves in viscoelastic media with random effects. The optimal strong convergence error estimates with respect to the semidiscrete finite element approximation in space are established. Finally, a numerical example is presented to verify the reliability of the theoretical study.

Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

Abstract The aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Korteweg–de Vries equation. The method is based on a combination of a discrete variational derivative method in time and finite volume element approximation in space. The resulting scheme is accurate, robust and energy-preserving. In addition, for comparison, we also develop a momentum-preserving finite volume element scheme and an implicit midpoint finite volume element scheme. Finally, a complete numerical study is developed to investigate the accuracy, conservation properties and long time behaviors of the energy-preserving scheme, in comparison with the momentum-preserving scheme and the implicit midpoint scheme, for the complex modified Korteweg–de Vries equation.

Splitting schemes with respect to physical processes for double-porosity poroelasticity problems

Abstract We consider unsteady poroelasticity problem in fractured porous medium within the classical Barenblatt double-porosity model. For numerical solution of double-porosity poroelasticity problems we construct splitting schemes with respect to physical processes, where transition to a new time level is associated with solving separate problem for the displacements and fluid pressures in pores and fractures. The stability of schemes is achieved by switching to three-level explicit-implicit difference scheme with some of the terms in the system of equations taken from the lower time level and by choosing a weight parameter used as a regularization parameter. The computational algorithm is based on the finite element approximation in space. The investigation of stability of splitting schemes is based on the general stability (well-posedness) theory of operator-difference schemes. A priori estimates for proposed splitting schemes and the standard two-level scheme are provided. The accuracy and stability of considered schemes are demonstrated by numerical experiments.

Modeling large-deforming fluid-saturated porous media using an Eulerian incremental formulation

The paper deals with modeling fluid saturated porous media subject to large deformation. An Eulerian incremental formulation is derived using the problem imposed in the spatial configuration in terms of the equilibrium equation and the mass conservation. Perturbation of the hyperelastic porous medium is described by the Biot model which involves poroelastic coefficients and the permeability governing the Darcy flow. Using the material derivative with respect to a convection velocity field we obtain the rate formulation which allows for linearization of the residuum function. For a given time discretization with backward finite difference approximation of the time derivatives, two incremental problems are obtained which constitute the predictor and corrector steps of the implicit time-integration scheme. Conforming mixed finite element approximation in space is used. Validation of the numerical model implemented in the SfePy code is reported for an isotropic medium with a hyperelastic solid phase. The proposed linearization scheme is motivated by the two-scale homogenization which will provide the local material poroelastic coefficients involved in the incremental formulation.