Finite Element Approximation In Space Research Articles

A space-time mixed finite element method for reduced fracture flow models on nonmatching grids

This paper is concerned with the numerical solution of the flow problem in a fractured porous medium where the fracture is treated as a lower dimensional object embedded in the rock matrix. We consider a space-time mixed variational formulation of such a reduced fracture model with mixed finite element approximations in space and discontinuous Galerkin discretization in time. Different spatial and temporal grids are used in the subdomains and in the fracture to adapt to the heterogeneity of the problem. Analysis of the numerical scheme, including well-posedness of the discrete problem, stability and a priori error estimates, is presented. Using substructuring techniques, the coupled subdomain and fracture system is reduced to a space-time interface problem which is solved iteratively by GMRES. Each GMRES iteration involves solution of time-dependent problems in the subdomains using the method of lines with local spatial and temporal discretizations. The convergence of GMRES is proved by using the field-of-values analysis and the properties of the discrete space-time interface operator. Numerical experiments are carried out to illustrate the performance of the proposed iterative algorithm and the accuracy of the numerical solution.

A second-order in time, BGN-based parametric finite element method for geometric flows of curves

Over the last two decades, the field of geometric curve evolutions has attracted significant attention from scientific computing. One of the most popular numerical methods for solving geometric flows is the so-called BGN scheme, which was proposed by Barrett et al. (2007) [8], due to its favorable properties (e.g., its computational efficiency and the good mesh property). However, the BGN scheme is limited to first-order accuracy in time, and how to develop a higher-order numerical scheme is challenging. In this paper, we propose a fully discrete, temporal second-order parametric finite element method, which integrates with two different mesh regularization techniques, for solving geometric flows of curves. The scheme is constructed based on the BGN formulation and a semi-implicit Crank-Nicolson leap-frog time stepping discretization as well as a linear finite element approximation in space. More importantly, we point out that the shape metrics, such as manifold distance and Hausdorff distance, instead of function norms, should be employed to measure numerical errors. Extensive numerical experiments demonstrate that the proposed BGN-based scheme is second-order accurate in time in terms of shape metrics. Moreover, by employing the classical BGN scheme as mesh regularization techniques, our proposed second-order schemes exhibit good properties with respect to the mesh distribution. In addition, an unconditional interlaced energy stability property is obtained for one of the mesh regularization techniques.

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.

An extended finite element method for the Nernst-Planck-Poisson equations

The Nernst-Planck-Poisson (NPP) system of equations is used to study the ion diffusion mechanism in materials, commonly used in storage batteries and solid oxide fuel cells. In particular, NPP equations are used to predict the concentration of charged defects and electric potential. Depending on the size of the computational domain, the distributions exhibit steep gradients near the boundaries. The traditional finite element method, when employed, requires extremely refined mesh to capture the steep gradient as they employ simple polynomials. To alleviate the mesh dependence, in this work, we propose to augment the traditional finite element approximation space with a suitable ansatz to capture the steep gradient within the framework of the extended finite element method. The robustness and the accuracy of the proposed framework is demonstrated by comparing it with an overkill finite element solution.

Constraints for eliminating the Gibbs phenomenon in finite element approximation spaces

One of the major challenges in finite element methods is the mitigation of spurious oscillations near sharp layers and discontinuities known as the Gibbs phenomenon. In this paper, we propose a set of functionals to identify spurious oscillations in best approximation problems in finite element spaces. Subsequently, we adopt these functionals in the formulation of constraints in an effort to eliminate the Gibbs phenomenon. By enforcing these constraints in best approximation problems, we can entirely eliminate over- and undershoot in one-dimensional continuous approximations, and significantly suppress them in one- and higher-dimensional discontinuous approximations.

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.

Extraction and application of super-smooth cubic B-splines over triangulations

The space of C1 cubic Clough–Tocher splines is a classical finite element approximation space over triangulations for solving partial differential equations. However, for such a space there is no B-spline basis available, which is a preferred choice in computer aided geometric design and isogeometric analysis. A B-spline basis is a locally supported basis that forms a convex partition of unity. In this paper, we explore several alternative C1 cubic spline spaces over triangulations equipped with a B-spline basis. They are defined over a Powell–Sabin refined triangulation and present different types of C2 super-smoothness. The super-smooth B-splines are obtained through an extraction process, i.e., they are expressed in terms of less smooth basis functions. These alternative spline spaces maintain the same optimal approximation power as Clough–Tocher splines. This is illustrated with a selection of numerical examples in the context of least squares approximation and finite element approximation for second and fourth order boundary value problems.

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.

Error Estimation and Adaptivity for Stochastic Collocation Finite Elements Part I: Single-Level Approximation

A general adaptive refinement strategy for solving linear elliptic partial differential equations with random data is proposed and analysed herein. The adaptive strategy extends the a posteriori error estimation framework introduced by Guignard and Nobile [SIAM J. Numer. Anal., 56 (2018), pp. 3121--3143] to cover problems with a nonaffine parametric coefficient dependence. A suboptimal, but nonetheless reliable and convenient implementation of the strategy involves approximation of the decoupled PDE problems with a common finite element approximation space. Computational results obtained using such a single-level strategy are presented in this paper (part I). Results obtained using a potentially more efficient multilevel approximation strategy, where meshes are individually tailored, will be discussed in part II of this work. The codes used to generate the numerical results are available on GitHub.

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.

Corrected finite element methods on unfitted meshes for Stokes moving interface problem

We propose an explicit in time corrected finite element method for a problem with a moving elastic interface in Stokes flow, using unfitted meshes and standard finite element approximation spaces. Piecewise polynomial correction functions are computed to accurately capture the discontinuities of the solution resulting from the elastic force applied to the fluid and are used to correct the load vector. The method is proven to be optimal for the steady state problem. Additionally, we propose a trigonometric interpolant to approximate and evolve the interface and explicit Euler scheme to advance in time. Numerical examples demonstrate the third and second order accuracy of the method for the velocity and pressure, respectively, and the stability of the numerical scheme.

A condensed generalized finite element method (CGFEM) for interface problems

Extensive developments on various generalizations of the Finite Element Method (FEM) for the interface problems, with unfitted mesh, have been made in the last few decades. Typical generalizations and techniques include the immersed FEM, the penalized FEM, the generalized or the extended FEM (GFEM/XFEM). The GFEM/XFEM achieves good approximation by augmenting the finite element approximation space with enrichment functions, which require additional degrees of freedom (DOF). However these additional DOFs, in general, deteriorate the conditioning of the GFEM/XFEM. In this study, we propose a condensed GFEM (CGFEM) for the interface problem based on the partition of unity method and a local least square scheme. Compared to the conventional GFEM/XFEM, the CGFEM possesses the same number of DOFs as in the FEM, yields good approximation, and is well-conditioned. In comparison with the immersed FEM, the construction of shape functions of CGFEM is independent of the equation types and the material coefficients of the interface problem. As a result, the CGFEM could be applied to more general interface problems, e.g., problems with anisotropic materials and elasticity systems, in a unified approach. Moreover, the shape functions of CGFEM are continuous, and thus do not need any penalty terms and parameters. The optimal order of convergence of the CGFEM has been analyzed and established theoretically in this paper. Numerical experiments for isotropic and anisotropic interface problems and comparisons with the conventional GFEM/XFEM have also been presented to illuminate the theory and effectiveness of CGFEM.

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.

The treatment of constraints due to standard boundary conditions in the context of the mixed Web-spline finite element method

PurposeThis paper aims to propose a new boundary condition and a web-spline basis of finite element space approximation to remedy the problems of constraints due to homogeneous and non-homogeneous; Dirichlet boundary conditions. This paper considered the two-dimensional linear elasticity equation of Navier–Lamé with the condition CAB. The latter allows to have a total insertion of the essential boundary condition in the linear system obtained; without using a numerical method as Lagrange multiplier. This study have developed mixed finite element; method using the B-splines Web-spline space. These provide an exact implementation of the homogeneous; Dirichlet boundary conditions, which removes the constraints caused by the standard; conditions. This paper showed the existence and the uniqueness of the weak solution, as well as the convergence of the numerical solution for the quadratic case are proved. The weighted extended B-spline; approach have become a much more workmanlike solution.Design/methodology/approachIn this paper, this study used the implementation of weighted finite element methods to solve the Navier–Lamé system with a new boundary condition CA, B (Koubaiti et al., 2020), that generalises the well-known basis, especially the Dirichlet and the Neumann conditions. The novel proposed boundary condition permits to use a single Matlab code, which summarises all kind of boundary conditions encountered in the system. By using this model is possible to save time and programming recourses while reap several programs in a single directory.FindingsThe results have shown that the Web-spline-based quadratic-linear finite elements satisfy the inf–sup condition, which is necessary for existence and uniqueness of the solution. It was demonstrated by the existence of the discrete solution. A full convergence was established using the numerical solution for the quadratic case. Due to limited regularity of the Navier–Lamé problem, it will not change by increasing the degree of the Web-spline. The computed relative errors and their rates indicate that they are of order 1/H. Thus, it was provided their theoretical validity for the numerical solution stability. The advantage of this problem that uses the CA, B boundary condition is associated to reduce Matlab programming complexity.Originality/valueThe mixed finite element method is a robust technique to solve difficult challenges from engineering and physical sciences using the partial differential equations. Some of the important applications include structural mechanics, fluid flow, thermodynamics and electromagnetic fields (Zienkiewicz and Taylor, 2000) that are mainly based on the approximation of Lagrange. However, this type of approximation has experienced a great restriction in the level of domain modelling, especially in the case of complicated boundaries such as that in the form of curvilinear graphs. Recently, the research community tried to develop a new way of approximation based on the so-called B-spline that seems to have superior results in solving the engineering problems.

Extended group finite element method

Interpolation methods for nonlinear finite element discretizations are commonly used to eliminate the computational costs associated with the repeated assembly of the nonlinear systems. While the group finite element formulation interpolates nonlinear terms onto the finite element approximation space, we propose the use of a separate approximation space that is tailored to the nonlinearity. In many cases, this allows for the exact reformulation of the discrete nonlinear problem into a quadratic problem with algebraic constraints. Furthermore, the substitution of the nonlinear terms often shifts general nonlinear forms into trilinear forms, which can easily be described by third-order tensors. The numerical benefits as well as the advantages in comparison to the original group finite element method are studied using a wide variety of academic benchmark problems.

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.

Development of Partition of Unity Finite Element Method for the Modelling of Acoustic Waves Propagating through Porous Domain

The numerical simulation of propagating sound fields in two dimensions using Partition of Unity Finite Element Method(PUFEM) technique is implemented in this work. PUFEM is one of the most practical methods developed to alleviate high computational cost of conventional Finite Element Method when we deal with the medium and high frequency problem. The main idea of PUFEM is to expand the finite element approximation space by incorporating a group of wave functions which are solutions of the governing equation into the target problem. With the intention of further accelerating the computational process, an advanced 2D Exact Integration Technique(EIT) for accelerating the computation process of highly oscillating integrals is proposed and the related PUFEM triangular element is created. The performance and practical interest of this new PUFEM element combined with EIT are evaluated for the wave propagation problem where a porous absorber is included.