Discrete H1 Research Articles

Novel Numerical Analysis of Multi-Term Time Fractional Viscoelastic Non-Newtonian Fluid Models for Simulating Unsteady Mhd Couette Flow of a Generalized Oldroyd-B Fluid

Abstract In this paper, we consider the application of the finite difference method for a class of novel multi-term time fractional viscoelastic non-Newtonian fluid models. An important contribution of the work is that the new model not only has a multi-term time derivative, of which the fractional order indices range from 0 to 2, but also possesses a special time fractional operator on the spatial derivative that is challenging to approximate. There appears to be no literature reported on the numerical solution of this type of equation. We derive two new different finite difference schemes to approximate the model. Then we establish the stability and convergence analysis of these schemes based on the discrete H 1 norm and prove that their accuracy is of O(τ + h 2) and O(τ min{3–γs ,2–αq ,2–β}+h 2), respectively. Finally, we verify our methods using two numerical examples and apply the schemes to simulate an unsteady magnetohydrodynamic (MHD) Couette flow of a generalized Oldroyd-B fluid model. Our methods are effective and can be extended to solve other non-Newtonian fluid models such as the generalized Maxwell fluid model, the generalized second grade fluid model and the generalized Burgers fluid model.

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H1 and L2 norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.

A node-centered finite volume method for a fracture model on triangulations

In this paper, a node-centered finite volume method based on triangulations for a fracture model is presented, in which we restrict the pressure to the linear finite element space while the velocity can be approximated by constant vectors element by element. The numerical scheme is established just associated with the pressure to avoid the saddle-point problem. Error estimates of O(h) accuracy for the discrete H1 semi-norm and the discrete L2 norm of pressure p and the (L2)2 norm of velocity u are developed on general triangulations. Under an additional assumption about essentially symmetric control volumes, the error estimates for the pressure p can be improved to O(h3/2). Finally, numerical experiments are carried out to verify the accuracy and convergence rates for the proposed finite volume scheme.

English

Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete H1 norm best approximation error estimates for H2 functions hold for arbitrary triangulations. However, the constants in similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on an example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily low speed. The results complement analogous findings for conforming P1 finite elements.

Space–time Legendre–Gauss–Lobatto collocation method for two-dimensional generalized sine-Gordon equation

In this paper, we present a space–time Legendre–Gauss–Lobatto (LGL) collocation method for solving the generalized two-dimensional sine-Gordon equation with nonhomogeneous Dirichlet boundary conditions. The proposed method is based on the LGL collocation method to discretize in space, then use the LGL collocation method or block LGL collocation method to discretize in time. Our formulation has high-order accuracy in both space and time. We introduce a new H1 projection for the error analysis of collocation method for nonhomogeneous Dirichlet boundary conditions. This projection differs from the classical H1 projection. The new H1 projection is identically equal to Lagrange interpolation on boundary. The approximation properties of the new H1 projection are obtain. We derive error bounds in both discrete L2 and H1 norms for the (spatially) semidiscrete formulation, the analysis is based on new H1 projection results.

A weak Galerkin finite element scheme with boundary continuity for second-order elliptic problems

A new weak Galerkin (WG) finite element method for solving the second-order elliptic problems on polygonal meshes by using polynomials of boundary continuity is introduced and analyzed. The WG method is utilizing weak functions and their weak derivatives which can be approximated by polynomials in different combination of polynomial spaces. Different combination gives rise to different weak Galerkin finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of certain combination of polynomial spaces that minimize the degree of freedom in the numerical scheme, yet without losing the accuracy of the numerical approximation. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete H1 norm and the standard L2 norm. In addition, the paper also presents some numerical experiments to demonstrate the power of the WG method. The numerical results show that the WG method achieves the optimal convergence order with a relatively low computational cost, which reveals a great promise of the flexibility and accuracy of the WG method.

Global energy‐tracking identities and global energy‐tracking splitting FDTD schemes for the Drude Models of Maxwell's equations in three‐dimensional metamaterials

In this article, we study the Drude models of Maxwell's equations in three‐dimensional metamaterials. We derive new global energy‐tracking identities for the three dimensional electromagnetic problems in the Drude metamaterials, which describe the invariance of global electromagnetic energy in variation forms. We propose the time second‐order global energy‐tracking splitting FDTD schemes for the Drude model in three dimensions. The significant feature is that the developed schemes are global energy‐preserving, unconditionally stable, second‐order accurate both in time and space, and computationally efficient. We rigorously prove that the new schemes satisfy these energy‐tracking identities in the discrete form and the discrete variation form and are unconditionally stable. We prove that the schemes in metamaterials are second order both in time and space. The superconvergence of the schemes in the discrete H1 norm is further obtained to be second order both in time and space. Their approximations of divergence‐free are also analyzed to have second‐order accuracy both in time and space. Numerical experiments confirm our theoretical analysis results. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 763–785, 2017

A mortar finite volume method for a fractured model in porous media

In this paper, we consider a mortar finite volume method for a fractured model of flow in porous media. In this model, the permeability coefficients are variable between the fracture and the surrounding porous media. A finite volume method based on Raviart–Thomas elements combined with the mortar technique of domain decomposition is presented, in which sub-domains are triangulated independently and the meshes do not match at interfaces. The great advantage of the method is avoiding solving the saddle-point problem, since the numerical scheme is just related to the pressure p, and the velocity u can be expressed by p. We also prove error estimates of order h on the discrete H1 norm between the exact solution p and the mortar finite volume solution P and the (L2)2 norm between u and U. Finally, numerical experiments have been performed to show the consistency of the convergence rates with the theoretical analysis.

Stability of fully discrete schemes with interpolation-type fractional formulas for distributed-order subdiffusion equations

Two fully discrete methods are investigated for simulating the distributed-order sub-diffusion equation in Caputo’s form. The fractional Caputo derivative is approximated by the Caputo’s BDF1 (called L1 early) and BDF2 (or L1-2 when it was first introduced) approximations, which are constructed by piecewise linear and quadratic interpolating polynomials, respectively. It is shown that the first scheme, using the BDF1 formula, possesses the discrete minimum-maximum principle and nonnegativity preservation property such that it is stable and convergent in the maximum norm. The method using the BDF2 formula is shown to be stable and convergent in the discrete H1 norm by using the discrete energy method. For problems of distributed order within a certain region, the method is also proven to preserve the discrete maximum principle and nonnegativity property. Extensive numerical experiments are provided to show the effectiveness of numerical schemes, and to examine the initial singularity of the solution. The applicability of our numerical algorithms to a problem with solution which lacks the smoothness near the initial time is examined by employing a class of power-type nonuniform meshes.

Weak Galerkin method with [formula omitted]-order finite elements for second order parabolic equations

In this paper, we propose a weak Galerkin method with a stabilization term for a model problem of second order parabolic differential equations. We establish both the continuous time and the discrete time weak Galerkin finite element schemes, which allow the use of totally discontinuous piecewise polynomial basis and the finite element partitions on shape regular polygons. In addition, we adapt the combination of polynomial spaces {Pr(T0),Pr−1(e¯),[Pr−1(T)]2} that reduces the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation. We show as well that the continuous time weak Galerkin finite element method preserves the energy conservation law. The optimal convergence order estimates in both discrete H1 and L2 norms are obtained. We also present numerical experiments to illustrate the theoretical results.

A class of stable and conservative finite difference schemes for the Cahn-Hilliard equation

In this paper, we propose a class of stable finite difference schemes for the initial-boundary value problem of the Cahn-Hilliard equation. These schemes are proved to inherit the total mass conservation and energy dissipation in the discrete level. The dissipation of the total energy implies boundness of the numerical solutions in the discrete H 1 norm. This in turn implies boundedness of the numerical solutions in the maximum norm and hence the stability of the difference schemes. Unique existence of the numerical solutions is proved by the fixed-point theorem. Convergence rate of the class of finite difference schemes is proved to be O(h 2 + τ2) with time step τ and mesh size h. An efficient iterative algorithm for solving these nonlinear schemes is proposed and discussed in detail.

Convergence analysis of compact difference schemes for diffusion equation with nonlocal boundary conditions

Compact difference schemes for solving the diffusion equation with nonlocal boundary conditions are considered in this paper. Fourth-order compact difference is used to approximate the second order spatial derivative, and the integrals in the boundary conditions are approximated by the composite Simpson quadrature formula. The backward Euler and Crank–Nicolson schemes are presented as the fully discrete schemes. Error estimates in the discrete h1 and l∞ norms are given by the energy method, showing both schemes are fourth-order accurate in space, and they have first-order and second-order accuracy in time, respectively. Numerical results are provided to confirm the theoretical results.

A weak Galerkin finite element method with polynomial reduction

The weak Galerkin (WG) is a novel numerical method based on variational principles for weak functions and their weak partial derivatives defined as distributions. In the implementation, the weak partial derivatives and the weak functions are approximated by polynomials with various degrees of freedom. The accuracy and the computational complexity of the corresponding WG scheme is significantly impacted by the selection of such polynomials. This paper presents an optimal combination for the polynomial spaces that minimize the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation. For illustrative purpose, the authors use the second order elliptic equation to demonstrate the basic ideas of polynomial reduction. Consequently, a new weak Galerkin finite element method is proposed and analyzed. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete H1 norm and the standard L2 norm. In addition, the paper presents some numerical results to demonstrate the power of the WG method in dealing with finite element partitions with arbitrary polygons in 2D or polyhedra in 3D. The numerical examples include various finite element partitions such as triangular mesh, quadrilateral mesh, honeycomb mesh in 2D and mesh with deformed cubes in 3D.

A compact finite difference method for a class of time fractional convection-diffusion-wave equations with variable coefficients

This paper is concerned with numerical methods for a class of time fractional convection-diffusion-wave equations. The convection coefficient in the equation may be spatially variable and the time fractional derivative is in the Caputo sense with the order ? (1 <? < 2). The class of the equations includes time fractional convection-diffusion-wave/diffusion-wave equations with or without damping as its special cases. In order to overcome the difficulty caused by variable coefficient problems, we first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference method for the spatial derivative and by the L1 approximation coupled with the Crank-Nicolson technique for the time derivative. The local truncation error and the solvability of the method are discussed in detail. A rigorous theoretical analysis of the stability and convergence is carried out using a discrete energy analysis method. The optimal error estimates in the discrete H1, L2 and L? norms are obtained under the mild condition that the time step is smaller than a positive constant, which depends solely upon physical parameters involved (this condition is no longer required for the special case of constant coefficients). Applications using three model problems give numerical results that demonstrate the effectiveness and the accuracy of the proposed method.

Energy Identities of ADI-FDTD Method with Periodic Structure

In this paper, a new kind of energy identities for the Maxwell equations with periodic boundary conditions is proposed and then proved rigorously by the energy methods. By these identities, several modified energy identities of the ADI-FDTD scheme for the two dimensional (2D) Maxwell equations with the periodic boundary conditions are derived. Also by these identities it is proved that 2D-ADI-FDTD is approximately energy conserved and unconditionally stable in the discrete L2 and H1 norms. Experiments are provided and the numerical results confirm the theoretical analysis on stability and energy conservation.

Convergence of Finite Volume Scheme for a Three–Dimensional Poisson Equation

We construct and analyze a finite volume scheme for numerical solution of a three-dimensional Poisson equation. We derive optimal convergence rates in the discrete H1 norm and sub-optimal convergence in the maximum norm, where we use the maximal available regularity of the exact solution and minimal smoothness requirement on the source term. The theoretical results are justified through implementing some canonical examples in 3D. Bibliography: 26 titles. Illustrations: 4 figures.

A higher-order compact LOD method and its extrapolations for nonhomogeneous parabolic differential equations

A higher-order compact locally one-dimensional (LOD) finite difference method for two-dimensional nonhomogeneous parabolic differential equations is proposed. The resulting scheme consists of two one-dimensional tridiagonal systems, and all computations are implemented completely in one spatial direction as for one-dimensional problems. The solvability and the stability of the scheme are proved almost unconditionally. The error estimates are obtained in the discrete H1,L2 and L∞ norms, and show that the proposed compact LOD method has the accuracy of the second-order in time and the fourth-order in space. Two Richardson extrapolation algorithms are presented to increase the accuracy to the fourth-order and the sixth-order in both time and space when the time step is proportional to the spatial mesh size. Numerical results demonstrate the accuracy of the compact LOD method and the high efficiency of its extrapolation algorithms.

A modified weak Galerkin finite element method for a class of parabolic problems

In this paper, we consider the solution of parabolic equation using the modified weak Galerkin finite element procedure, which is named as MWG-FEM, based on the conception of the modified weak derivative over discontinuous functions with heterogeneous properties, in which the classical gradient operator is replaced by a modified weak gradient operator. Optimal order error estimates in a discrete L2 norm and H1 norm are established for the corresponding modified weak Galerkin finite element solutions. Finally, we numerically verify the convergence theory for the MWG-FEM through some examples.

Stability and Super Convergence Analysis of ADI-FDTD for the 2D Maxwell Equations in a Lossy Medium

Several new energy identities of the two dimensional(2D) Maxwell equations in a lossy medium in the case of the perfectly electric conducting boundary conditions are proposed and proved. These identities show a new kind of energy conservation in the Maxwell system and provide a new energy method to analyze the alternating direction implicit finite difference time domain method for the 2D Maxwell equations (2D-ADI-FDTD). It is proved that 2D-ADI-FDTD is approximately energy conserved, unconditionally stable and second order convergent in the discrete L2 and H1 norms, which implies that 2D-ADI-FDTD is super convergent. By this super convergence, it is simply proved that the error of the divergence of the solution of 2D-ADI-FDTD is second order accurate. It is also proved that the difference scheme of 2D-ADI-FDTD with respect to time t is second order convergent in the discrete H1 norm. Experimental results to confirm the theoretical analysis on stability, convergence and energy conservation are presented.

A weak Galerkin finite element method for second-order elliptic problems

This paper introduces a finite element method by using a weakly defined gradient operator over generalized functions. The use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations. This article intends to provide a general framework for managing differential operators on generalized functions. As a demonstrative example, the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem, in which the classical gradient operator is replaced by the discrete weak gradient. The resulting numerical scheme is called a weak Galerkin (WG) finite element method. It can be seen that the weak Galerkin method allows the use of totally discontinuous functions in the finite element procedure. For the second order elliptic problem, an optimal order error estimate in both a discrete H1 and L2 norms are established for the corresponding weak Galerkin finite element solutions. A superconvergence is also observed for the weak Galerkin approximation.