High‐order compact schemes for fractional differential equations with mixed derivatives

In this article, we consider two‐dimensional fractional subdiffusion equations with mixed derivatives. A high‐order compact scheme is proposed to solve the problem. We establish a sufficient condition and show that the scheme converges with fourth order in space and second order in time under this condition.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2141–2158, 2017     

Some nonconforming mixed box schemes for elliptic problems

In this article, we introduce three schemes for the Poisson problem in 2D on triangular meshes, generalizing the FVbox scheme introduced by Courbet and Croisille [1]. In this kind of scheme, the approximation is performed on the mixed form of the problem, but contrary to the standard mixed method, with a pair of trial spaces different from the pair of test spaces. The latter is made of Galerkin‐discontinuous spaces on a unique mesh. The first scheme uses as trial spaces the P¹ nonconforming space of Crouzeix‐Raviart both for u and for the flux p = ?u. In the two others, the quadratic nonconforming space of Fortin and Soulie is used. An important feature of all these schemes is that they are equivalent to a first scheme in u only and an explicit representation formula for the flux p = ?u. The numerical analysis of the schemes is performed using this property. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 355–373, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10003     

Optimal convergence analysis of mixed finite element methods for fourth‐order elliptic and parabolic problems

By using a special interpolation operator developed by Girault and Raviart (finite element methods for Navier‐Stokes Equations, Springer‐Verlag, Berlin, 1986), we prove that optimal error bounds can be obtained for a fourth‐order elliptic problem and a fourth‐order parabolic problem solved by mixed finite element methods on quasi‐uniform rectangular meshes. Optimal convergence is proved for all continuous tensor product elements of order k ≥ 1. A numerical example is provided for solving the fourth‐order elliptic problem using the bilinear element. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

High‐order compact difference schemes for the modified anomalous subdiffusion equation

In this article, two kinds of high‐order compact finite difference schemes for second‐order derivative are developed. Then a second‐order numerical scheme for a Riemann–Liouvile derivative is established based on a fractional centered difference operator. We apply these methods to a fractional anomalous subdiffusion equation to construct two kinds of novel numerical schemes. The solvability, stability, and convergence analysis of these difference schemes are studied by using Fourier method. The convergence orders of these numerical schemes are and , respectively. Finally, numerical experiments are displayed which are in line with the theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 213–242, 2016     

On the subdomain‐Galerkin/least squares method for 2‐ and 3‐D mixed elliptic problems with reaction terms

In this article we apply the subdomain‐Galerkin/least squares method, which is first proposed by Chang and Gunzburger for first‐order elliptic systems without reaction terms in the plane, to solve second‐order non‐selfadjoint elliptic problems in two‐ and three‐dimensional bounded domains with triangular or tetrahedral regular triangulations. This method can be viewed as a combination of a direct cell vertex finite volume discretization step and an algebraic least‐squares minimization step in which the pressure is approximated by piecewise linear elements and the flux by the lowest order Raviart‐Thomas space. This combined approach has the advantages of both finite volume and least‐squares methods. Among other things, the combined method is not subject to the Ladyzhenskaya‐Babus?ka‐Brezzi condition, and the resulting linear system is symmetric and positive definite. An optimal error estimate in the H¹(Ω) × H(div; Ω) norm is derived. An equivalent residual‐type a posteriori error estimator is also given. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 738–751, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10030.     

On the refined maximum principle for degenerate elliptic and parabolic problems

We extend the refined maximum principle in [H. Berestycki, L. Nirenberg, S.R.S. Varadhan, The principal eigenvalue and the maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994) 47–92] to degenerate elliptic and parabolic equations with unbounded coefficients. Then we discuss the well-posedness of the corresponding Dirichlet boundary value problems.     

Parallel difference schemes with interface extrapolation terms for quasi-linear parabolic systems

In this paper some new parallel difference schemes with interface extrapolation terms for a quasi-linear parabolic system of equations are constructed. Two types of time extrapolations are proposed to give the interface values on the interface of sub-domains or the values adjacent to the interface points, so that the unconditional stable parallel schemes with the second accuracy are formed. Without assuming heuristically that the original boundary value problem has the unique smooth vector solution, the existence and uniqueness of the discrete vector solutions of the parallel difference schemes constructed are proved. Moreover the unconditional stability of the parallel difference schemes is justified in the sense of the continuous dependence of the discrete vector solution of the schemes on the discrete known data of the original problems in the discrete W2(2,1) (Q△) norms. Finally the convergence of the discrete vector solutions of the parallel difference schemes with interface extrapolation terms to the unique generalized solution of the original quasi-linear parabolic problem is proved. Numerical results are presented to show the good performance of the parallel schemes, including the unconditional stability, the second accuracy and the high parallelism.     

Higher‐order finite volume element methods based on Barlow points for one‐dimensional elliptic and parabolic problems

The article is devoted to a kind of higher‐order finite volume element methods, where the dual partitions are constructed by Barlow points, for elliptic and parabolic problems in one space dimension. Techniques to derive the stability and to control the nonsymmetry are presented. Superconvergence and the optimal order errors in the H¹‐ and L²‐norms are obtained. Numerical results illustrate the theoretical findings. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 977–994, 2015     

SOME BOUNDARY VALUE PROBLEMS FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS OF SECOND ORDER

The present article deals with some boundary value problems for nonlinear elliptic equations with degenerate rank 0 including the oblique derivative problem.Firstly the formulation and estimates of solutions of the oblique derivative problem are given, and then by the above estimates and the method of parameter extension,the existence of solutions of the above problem is proved.In this article,the complex analytic method is used,namely the corresponding problem for degenerate elliptic complex equations of first order is firstly discussed,afterwards the above problem for the degenerate elliptic equations of second order is solved.     

Iterative methods and high-order difference schemes for 2D elliptic problems with mixed derivative

We derive a fourth-order compact finite difference scheme for a two-dimensional elliptic problem with a mixed derivative and constant coefficients. We conduct experimental study on numerical solution of the problem discretized by the present compact scheme and the traditional second-order central difference scheme. We study the computed accuracy achieved by each scheme and the performance of the Gauss-Seidel iterative method, the preconditioned GMRES iterative method, and the multigrid method, for solving linear systems arising from the difference schemes.     

A weak Galerkin finite element method for the second order elliptic problems with mixed boundary conditions

In this paper, a weak Galerkin finite element method is proposed and analyzed for the second-order elliptic equation with mixed boundary conditions. Optimal order error estimates are established in both discrete $H^1$ norm and the standard $L^2$ norm for the corresponding WG approximations. The numerical experiments are presented to verify the efficiency of the method.     

四维抛物型方程的高精度分支稳定的显式差分格式

对四维抛物型方程构造了一个高精度显格式,格式的稳定性条件为r=Δt/Δx2=△t/Δy2=△t/△z2=Δt/Δw2<1/2,截断误差阶达到O(Δt2 Δx4),通过数值实验,表明本文理论分析的正确性和文中格式较同类格式的优越性.     

Compact difference schemes for heat equation with Neumann boundary conditions

In this article, two recent proposed compact schemes for the heat conduction problem with Neumann boundary conditions are analyzed. The first difference scheme was proposed by Zhao, Dai, and Niu (Numer Methods Partial Differential Eq 23, (2007), 949–959). The unconditional stability and convergence are proved by the energy methods. The convergence order is O(τ² + h^2.5) in a discrete maximum norm. Numerical examples demonstrate that the convergence order of the scheme can not exceeds O(τ² + h³). An improved compact scheme is presented, by which the approximate values at the boundary points can be obtained directly. The second scheme was given by Liao, Zhu, and Khaliq (Methods Partial Differential Eq 22, (2006), 600–616). The unconditional stability and convergence are also shown. By the way, it is reported how to avoid computing the values at the fictitious points. Some numerical examples are presented to show the theoretical results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

High‐order approximation of 2D convection‐diffusion equation on hexagonal grids

We derive a fourth‐order finite difference scheme for the two‐dimensional convection‐diffusion equation on an hexagonal grid. The difference scheme is defined on a single regular hexagon of size h over a seven‐point stencil. Numerical experiments are conducted to verify the high accuracy of the derived scheme, and to compare it with the standard second‐order central difference scheme. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

An unfitted finite-element method for elliptic and parabolic interface problems

Bhupen Deka Department of Mathematics, Assam University, Silchar-788011, India ^{A finite-element discretization, independent of the locationof the interface, is proposed and analysed for linear ellipticand parabolic interface problems. We establish error estimatesof optimal order in the H1-norm and almost optimal order inthe L2-norm for elliptic interface problems. An extension toparabolic interface problems is also discussed and an optimalerror estimate in the L2(0, T;H1())-norm and an almost optimalorder estimate in the L2(0, T;L2())-norm are derived for thespatially discrete scheme. A fully discrete scheme based onthe backward Euler method is analysed and an optimal order errorestimate in the L2(0, T;H1())-norm is derived. The interfacesare assumed to be of arbitrary shape and smooth for our purpose.}     

Unconditional stability of alternating difference schemes with intrinsic parallelism for two‐dimensional parabolic systems

The difference method with intrinsic parallelism for two dimensional parabolic system is studied. The general alternating difference schemes, in particular those with variable time steplengthes, are constructed and proved to be unconditionally stable. The two dimensional alternating group explicit scheme, alternating block explicit‐implicit scheme, alternating block Crank‐Nicolson scheme and block ADI scheme are the special cases of the general schemes constructed here. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 625–636, 1999     

Peripheral and ADI hopscotch methods for two dimensional parabolic and elliptic equations with a mixed term

The peripheral and ADI hopscotch methods are extended to solve problems in two space dimensions with a mixed derivative term. The method is compared numerically with existing hopscotch methods.     

Fourth‐order compact difference schemes for 1D nonlinear Kuramoto–Tsuzuki equation

In this article, first, we establish some compact finite difference schemes of fourth‐order for 1D nonlinear Kuramoto–Tsuzuki equation with Neumann boundary conditions in two boundary points. Then, we provide numerical analysis for one nonlinear compact scheme by transforming the nonlinear compact scheme into matrix form. And using some novel techniques on the specific matrix emerged in this kind of boundary conditions, we obtain the priori estimates and prove the convergence in norm. Next, we analyze the convergence and stability for one of the linearized compact schemes. To obtain the maximum estimate of the numerical solutions of the linearized compact scheme, we use the mathematical induction method. The treatment is that the convergence in norm is obtained as well as the maximum estimate, further the convergence in norm. Finally, numerical experiments demonstrate the theoretical results and show that one of the linearized compact schemes is more accurate, efficient and robust than the others and the previous. It is worthwhile that the compact difference methods presented here can be extended to 2D case. As an example, we present one nonlinear compact scheme for 2D Ginzburg–Landau equation and numerical tests show that the method is accurate and effective. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2080–2109, 2015     

Analysis of expanded mixed methods for fourth-order elliptic problems

The recently proposed expanded mixed formulation for numerical solution of second-order elliptic problems is here extended to fourth-order elliptic problems. This expanded formulation for the differential problems under consideration differs from the classical formulation in that three variables are treated, i.e., the displacement, the stress, and the moment tensors. It works for the case where the coefficient of the differential equations is small and does not need to be inverted, or for the case in which the stress tensor of the equations does not need to be symmetric. Based on this new formulation, various mixed finite elements for fourth-order problems are considered; error estimates of quasi-optimal or optimal order depending upon the mixed elements are derived. Implementation techniques for solving the linear system arising from these expanded mixed methods are discussed, and numerical results are presented. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 483–503, 1997     

A nonconforming mixed finite element for second-order elliptic problems

In a recent work, Hiptmair [Mathematisches Institut, M9404, 1994] has constructed and analyzed a family of nonconforming mixed finite elements for second-order elliptic problems. However, his analysis does not work on the lowest order elements. In this article, we show that it is possible to construct a nonconforming mixed finite element for the lowest order case. We prove the convergence and give estimates of optimal order for this finite element. Our proof is based on the use of the properties of the so-called nonconforming bubble function to control the consistency terms introduced by the nonconforming approximation. We further establish an equivalence between this mixed finite element and the nonconforming piecewise quadratic finite element of Fortin and Soulie [J. Numer. Methods Eng., 19, 505–520, 1983]. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 445–457, 1997