Local Discontinuous Galerkin Finite Element Research Articles

Local discontinuous Galerkin method for the Riesz space distributed-order Sobolev equation

We present and analyze a local discontinuous Galerkin finite element method to solve the Riesz space distributed-order Sobolev equation. To approximate the distributed-order Riesz space derivative, the Gauss quadrature as the high computational accuracy method is proposed. A multi-term fractional equation is then constructed from the considered equation by approximating the Riesz space derivative. Moreover, stability analysis is provided for a semi-discrete scheme. We provide numerical results to justify the theoretical analysis by solving the ordinary differential equation. We obtain this after implementing the local discontinuous Galerkin scheme on the Riesz space distributed-order Sobolev equation with the Crank–Nicolson scheme as a time marching method and the Laplace transform technique.

Development of a LDG method on polytopal mesh with optimal order of convergence

A Local discontinuous Galerkin (LDG) finite element method on triangular mesh was introduced and analyzed in Castillo et al. (2000) for elliptic problem with suboptimal order of convergence for the flux variable. The purpose of this work is to develop a LDG finite element method on polytopal mesh that achieves optimal convergence rate for both unknowns, potential u and its gradient q. In our new LDG method, u and q are approximated by polynomials of degree k and k−1 respectively. Optimal order of convergence are obtained for both unknowns. Numerical results in 2d and 3d are presented to confirm the theory.

The local discontinuous Galerkin finite element method for a multiterm time-fractional initial-boundary value problem

The local discontinuous Galerkin finite element method for a multiterm time-fractional initial-boundary value problem

Numerical analysis of variable-order fractional KdV-Burgers-Kuramoto equation

<abstract><p>In this paper, a fully discrete local discontinuous Galerkin finite element method is proposed to solve the KdV-Burgers-Kuramoto equation with variable-order Riemann-Liouville time fractional derivative. The method proposed in this paper is based on the finite difference method in time and local discontinuous Galerkin method in space. For all $ \epsilon(t)\in (0, 1) $ with variable order, we prove the scheme is unconditional stable and convergent. Finally, numerical examples are provided to verify the theoretical analysis and the order of convergence for the proposed method.</p></abstract>

Numerical Methods for the Time Fractional Convection-Diffusion-Reaction Equation

In this article, efficient methods are derived for seeking numerical solution to the time fractional convection-diffusion-reaction equation whose solution very likely exhibits a weak regularity at the starting time. Here, the time fractional derivative in the Caputo sense with order in (0, 1) is discretized by the finite difference methods with uniform and non-uniform meshes and the spatial derivative by the local discontinuous Galerkin finite element method. The derived numerical schemes are stable and convergent. Finally, some numerical experiments are presented to verify the theoretical analysis.

A discontinuous finite element approximation to singular Lane-Emden type equations

In this article, we develop the local discontinuous Galerkin finite element method for the numerical approximations of a class of singular second-order ordinary differential equations known as the Lane-Emden type equations equipped with initial or boundary conditions. These equations have been considered via different models that naturally appear for example in several phenomena in astrophysical science. By converting the governing equations into a first-order systems of differential equations, the approximate solution is sought in a piecewise discontinuous polynomial space while the natural upwind fluxes are used at element interfaces. The existence-uniqueness of the weak formulation is provided and the numerical stability of the method in the L∞ norm is established. Five illustrative test problems are given to demonstrate the applicability and validity of the scheme. Comparisons between the numerical results of the proposed method with existing results are carried out in order to show that the new approximation algorithm provides accurate solutions even near the singular point.

A combined hybrid mixed element method for incompressible miscible displacement problem with local discontinuous Galerkin procedure

AbstractIn this article, we propose a combined hybrid discontinuous mixed finite element method for miscible displacement problem with local discontinuous Galerkin method. Here, to obtain more accurate approximation and deal with the discontinuous case, we use the hybrid mixed element method to approximate the pressure and velocity, and use the local discontinuous Galerkin finite element method for the concentration. Compared with other combined methods, this method can improve the efficiency of computation, deal with the discontinuous problem well and keep local mass balance. We study the convergence of this method and give the corresponding optimal error estimates in L∞(L2) for velocity and concentration and the super convergence in L∞(H1) for pressure. Finally, we also present some numerical examples to confirm our theoretical analysis.

The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: Mathematical analysis

In Li and Wang (2019) [11], three kinds of typical Caputo-type partial differential equations are numerically studied via the local discontinuous Galerkin (LDG) finite element methods, including Caputo-type reaction-diffusion equation, Caputo-type reaction-diffusion-wave equation, and Caputo-type cable equation without regularity analysis. In this article, we study the existence, uniqueness, and regularity of the solutions to these equations. Besides, the error estimate for Caputo-type cable equation has been greatly improved.

LDG schemes with second order implicit time discretization for a fractional sub-diffusion equation

In this paper, we propose new local discontinuous Galerkin (LDG) schemes for solving a time fractional sub-diffusion equation. The new LDG schemes is constructed rely on the splitting of time fractional derivative and space derivative. The key idea of semi-discrete LDG scheme is to split the time fractional derivative into a weak singular integral and a classic first order derivative. By using second time discretization to approximate the weak singular integral and the classic first order derivative, two fully discrete LDG schemes with second order implicit time discretization for the time fractional diffusion equation are presented. The LDG finite element approximations is used in space variable, and Crank–Nicholson, second order backward differentiation formulas are used for the temporal approximation. The error estimates of the presented semi-discrete and fully-discrete LDG schemes are both analyzed. The analysis results show that our numerical schemes are unconditional stable with the second order accuracy in temporal discretization. Numerical experiments are presented to confirm the theoretical results.

3-D Numerical Modeling for the Magnetization of Superconductors Using a Local Discontinuous Galerkin Finite Element Method

A numerical methodology is proposed to discretize a nonlinear low-frequency approximation to Maxwell’s equations using a local discontinuous Galerkin (DG) finite element method, with an upwind-like numerical flux, for modeling superconductors. In this paper, we focus on high-temperature superconductors (HTS) and the electrical resistivity is modeled using a power law. Nodal elements and the Whitney element are used. Numerical studies have been performed to verify the proposed methodology: a problem with a manufactured solution, the nonlinear magnetic front problem, and the magnetization of HTS wires. Based on the final time that can be reached for a given time-step size, the proposed strategy is compared with the $\mathbf {H}$ formulation discretized using the Galerkin finite element method with the Whitney element for the magnetic front problem. The proposed local DG strategy allows the use of a larger time-step size over a longer time interval, particularly, when we use the Whitney element. The proposed methodology can also capture sharp gradients of the current density with limited spurious oscillations. The numerical results are in agreement with Bean’s model for large values of power-law’s exponent. The proposed local DG strategy could be generalized to more complex electrical resistivity models, including multiphysics models.

The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: Numerical analysis

In this article, three kinds of typical Caputo-type partial differential equations are numerically studied via the finite difference methods/the local discontinuous Galerkin finite element methods, including Caputo-type reaction–diffusion equation, Caputo-type reaction–diffusion-wave equation, and Caputo-type cable equation. The derived numerical schemes are unconditionally stable and convergent. The numerical experiments are also displayed which support the theoretical analysis.

Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation

Abstract In this paper we develop and analyze the local discontinuous Galerkin (LDG) finite element method for solving the general Lax equation. The local discontinuous Galerkin method has the flexibility for arbitrary h and p adaptivity, and allows for hanging nodes. By choosing the numerical fluxes carefully we prove stability and give an error estimate. Finally some numerical examples are computed to show the convergence order and excellent numerical performance of proposed method.

Local discontinuous Galerkin method for distributed-order time and space-fractional convection–diffusion and Schrödinger-type equations

We develop a local discontinuous Galerkin finite element method for the distributed-order time and Riesz space-fractional convection–diffusion and Schrodinger-type equations. The stability of the presented schemes is proved and optimal order of convergence $$\mathcal {O}(h^{N+1}+(\Delta t)^{1+\frac{\theta }{2}}+\theta ^{2})$$ for the Riesz space-fractional diffusion and Schrodinger-type equations with distributed order in time, an order of convergence of $$\mathcal {O}(h^{N+\frac{1}{2}}+(\Delta t)^{1+\frac{\theta }{2}}$$ $$+\theta ^{2})$$ is provided for the Riesz space-fractional convection–diffusion equations with distributed order in time where h, $$\theta $$ and $$\Delta t$$ are space step size, the distributed-order variables and the step sizes in time, respectively. Finally, the performed numerical examples confirm the optimal convergence order and illustrate the effectiveness of the method.

High order well-balanced central local discontinuous Galerkin-finite element methods for solving the Green–Naghdi model

In this paper, a hybrid numerical method, combining the central local discontinuous Galerkin method with continuous finite element method, is proposed to solve the fully nonlinear weakly dispersive Green–Naghdi model describing a large spectrum of shallow water waves. In our numerical approach, the Green–Naghdi model is first rewritten as balance laws coupled with an elliptic equation in terms of new variables adapted for numerical studies. Then we discretize the balance laws with well-balanced central local discontinuous Galerkin methods and the elliptic part with continuous finite element methods. Numerical tests are presented to illustrate the performance of the proposed schemes.

Fully Discrete Local Discontinuous Galerkin Approximation for Time-Space Fractional Subdiffusion/Superdiffusion Equations

We focus on developing the finite difference (i.e., backward Euler difference or second-order central difference)/local discontinuous Galerkin finite element mixed method to construct and analyze a kind of efficient, accurate, flexible, numerical schemes for approximately solving time-space fractional subdiffusion/superdiffusion equations. Discretizing the time Caputo fractional derivative by using the backward Euler difference for the derivative parameter (0<α<1) or second-order central difference method for (1<α<2), combined with local discontinuous Galerkin method to approximate the spatial derivative which is defined by a fractional Laplacian operator, two high-accuracy fully discrete local discontinuous Galerkin (LDG) schemes of the time-space fractional subdiffusion/superdiffusion equations are proposed, respectively. Through the mathematical induction method, we show the concrete analysis for the stability and the convergence under theL2norm of the LDG schemes. Several numerical experiments are presented to validate the proposed model and demonstrate the convergence rate of numerical schemes. The numerical experiment results show that the fully discrete local discontinuous Galerkin (LDG) methods are efficient and powerful for solving fractional partial differential equations.

Variational multiscale element free Galerkin (VMEFG) and local discontinuous Galerkin (LDG) methods for solving two-dimensional Brusselator reaction–diffusion system with and without cross-diffusion

The finite element method (FEM) is one of the basic methods for solving deterministic and stochastic partial differential equations. This method is proposed in the 19 decade and after years several modifications for this well-known technique such as mortal FEM, discontinuous Galerkin FEM, extended FEM, least squares FEM, spectral FEM, mixed FEM, immersed FEM, adaptive FEM, etc. have been proposed. Also, for improving the accuracy and for considering complex domain, shape functions of the moving least squares approximation have been replaced with the traditional shape function of the finite element technique. The name of this improved method is element free Galerkin (EFG) method which is classified in the category of meshless methods. The EFG method has been applied for solving many problems in engineering. In the current paper, we select two numerical procedures that are extracted from FEM. The discontinuous Galerkin approach is a useful technique for solving problems that their exact solutions have discontinuity. It should be noted that many enriched ideas have been explained for improving accuracy of EFG method. In the current paper, the local discontinuous Galerkin and variational multiscale EFG methods are applied for obtaining numerical solution of two-dimensional Brusselator system with and without cross-diffusion. The Brusselator system is a theoretical model for a type of autocatalytic reaction. Up to best of authors’ knowledge, the accuracy of the obtained numerical solutions using local discontinuous Galerkin method is very related to the used time-discrete scheme. Therefore, fourth-order exponential time differencing method has been employed for discretizing the time variable. Also, to achieve a full discretization scheme, the local discontinuous Galerkin finite element method is used. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes.

A global–local discontinuous Galerkin finite element for finite-deformation analysis of multilayered shells

We present a global–local finite element formulation for the analysis of doubly-curved laminated composite shells. The proposed formulation is applicable to a wide range of problems, including those involving finite strains/rotations, nonlinear constitutive behavior, and static or dynamic structural response. Moreover, in dynamic analysis applications, it can be combined with either explicit or implicit time integration schemes. The global–local framework is based on the superposition of a global displacement field, spanning the thickness of the entire laminate, and local (i.e. layerwise) displacement fields associated with each layer in the laminate. This approach affords highly-resolved representations in regions of critical interest, and allows a smooth transition from higher to lower resolution zones. Discontinuous Galerkin fluxes are used to enforce interlaminar continuity conditions in perfectly-bonded laminates, and the cohesive-zone methodology is used to represent interfacial delamination. Stresses and surface normals are referred to the initial configuration in the proposed total Lagrangian formulation. Assumed natural strain and enhanced assumed strain techniques are employed to alleviate shear locking and other pathologies that are known to afflict bilinear shell elements. The performance of the proposed finite element is examined with the aid of several numerical examples involving thick as well as thin shells.

Local discontinuous Galerkin approximation of non-Fickian diffusion model in viscoelastic polymers

In this paper, we investigate a fully discrete local discontinuous Galerkin approximation of a non-linear non-Fickian diffusion model in viscoelastic polymers. For the spatial discretization, we adopt local discontinuous Galerkin finite element method and for the time discretization we use backward Euler method. We derive the stability estimate and a priori error estimate for the discrete scheme. Numerical examples are given to verify the theoretical findings.

A fully discrete local discontinuous Galerkin method for one-dimensional time-fractional Fisher's equation

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional time-fractional Fisher's equation, which is obtained from the standard one-dimensional Fisher's equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0<α<1). The proposed LDG is based on the LDG finite element method for space and finite difference method for time. We prove that the method is stable, and the numerical solution converges to the exact one with order O(hk+1+τ2−α), where h, τ and k are the space step size, time step size, polynomial degree, respectively. The numerical experiments reveal that the LDG is very effective.

Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation

This paper aims to develop a fully discrete local discontinuous Galerkin finite element method for numerical simulation of the time-fractional telegraph equation, where the fractional derivative is in the sense of Caputo. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. The stability and convergence of this discontinuous approach are discussed and theoretically proven. Finally numerical examples are performed to illustrate the effectiveness and the accuracy of the method.