Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids

In this paper, we study the convergence of a finite difference scheme on nonuniform grids for the solution of second-order elliptic equations with mixed derivatives and variable coefficients in polygonal domains subjected to Dirichlet boundary conditions. We show that the scheme is equivalent to a fully discrete linear finite element approximation with quadrature. It exhibits the phenomenon of supraconvergence, more precisely, for s ∈ [1,2] order O(h ^s )-convergence of the finite difference solution, and its gradient is shown if the exact solution is in the Sobolev space H ^1+s (Ω). In the case of an equation with mixed derivatives in a domain containing oblique boundary sections, the convergence order is reduced to O(h ^3/2?ε) with ε > 0 if u ∈ H ³(Ω). The second-order accuracy of the finite difference gradient is in the finite element context nothing else than the supercloseness of the gradient. For s ∈ {1,2}, the given error estimates are strictly local.     

The upwind finite difference fractional steps method for combinatorial system of dynamics of fluids in porous media and its application

For combinatorial system of multilayer dynamics of fluids in porous media, the second order and first order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward and two-dimensional and three-dimensional schemes are used to form a complete set. Some techniques, such as implicit-explicit difference scheme, calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates, are adopted. Optimal order estimates in L ² norm are derived to determine the error in the second order approximate solution. This method has already been applied to the numerical simulation of migration-accumulation of oil resources. Keywords: combinatorial system, multilayer dynamics of fluids in porous media, two-class upwind finite difference fractional steps method, convergence, numerical simulation of energy sources.     

A fourth-order finite difference scheme for two-dimensional nonlinear elliptic partial differential equations

A finite difference scheme for the two-dimensional, second-order, nonlinear elliptic equation is developed. The difference scheme is derived using the local solution of the differential equation. A 13-point stencil on a uniform mesh of size h is used to derive the finite difference scheme, which has a truncation error of order h⁴. Well-known iterative methods can be employed to solve the resulting system of equations. Numerical results are presented to demonstrate the fourth-order convergence of the scheme. © 1995 John Wiley & Sons, Inc.     

Superconvergent error estimates for a class of discretization methods for a coupled first‐order system with discontinuous coefficients

Lithological discontinuities in a reservoir generate discontinuous coefficients for the first‐order system of equations used in the simulation of fluid flow in porous media. Systems of conservation laws with discontinuous coefficients also arise in many other physical applications. In this article, we present a class of discretization schemes that include variants of mixed finite element methods, finite volume element methods, and cell‐centered finite difference equations as special cases. Error estimates of the order O(h²) in certain discrete L²‐norms are established for both the primary independent variable and its flux, even in the presence of discontinuous coefficients in the flux term. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 267–283, 1999     

An approach combining the Ritz-Galerkin and finite difference methods

The nonconforming combination of Ritz-Galerkin and finite difference methods is presented for solving elliptic boundary value problems with singularities. The Ritz-Galerkin method is used in the subdomains including singularities, the finite difference method is used in the rest of the solution domain. Moreover, on the common boundary of two regions where two different methods are used, the continuity conditions are constrained only on the nodes of difference grids. Theoretical analysis and numerical experiments have shown that average errors of numerical solutions and their generalized derivatives can reach the convergence rate O(h^2-δ), where h is the mesh spacing of uniform difference grids, and δ is an arbitrarily small, positive number. This convergence rate is better than O(h), obtained by the nonconforming combination of the Ritz-Galerkin and finite element methods.     

Solving the equation −uxx − ϵuyy = f(x,y, u) by an O(h4) finite difference method

The semi‐linear equation −u_xx − ϵu_yy = f(x, y, u) with Dirichlet boundary conditions is solved by an O(h⁴) finite difference method, which has local truncation error O(h²) at the mesh points neighboring the boundary and O(h⁴) at most interior mesh points. It is proved that the finite difference method is O(h⁴) uniformly convergent as h → 0. The method is considered in the form of a system of algebraic equations with a nine diagonal sparse matrix. The system of algebraic equations is solved by an implicit iterative method combined with Gauss elimination. A Mathematica module is designed for the purpose of testing and using the method. To illustrate the method, the equation of twisting a springy rod is solved. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 395–407, 2000     

Exponential spline solutions for a class of two point boundary value problems over a semi-infinite range

In this paper, we developed numerical methods of order O(h ²) and O(h ⁴) based on exponential spline function for the numerical solution of class of two point boundary value problems over a Semi-infinite range. The present approach gives better approximations over all the existing finite difference methods. Properties of the infinite linear system are established. Convergence analysis and a bound on the approximate solution are discussed. Test problem with various kinds of boundary conditions is included to illustrate the practical usefulness and superiority of our methods.     

Efficient algorithms for solving tensor product finite element equations

Summary There are currently several highly efficient methods for solving linear systems associated with finite difference approximations of Poisson's equation in rectangular regions. These techniques are employed to develop both direct and iterative methods for solving the linear systems arising from the use ofC ⁰ quadratic orC ¹ cubic tensor product finite elements.     

Convergence and superconvergence analysis for nonlinear delay reaction–diffusion system with nonconforming finite element

In this article, we propose and analyze several numerical methods for the nonlinear delay reaction–diffusion system with smooth and nonsmooth solutions, by using Quasi-Wilson nonconforming finite element methods in space and finite difference methods (including uniform and nonuniform L1 and L2-1_σ schemes) in time. The optimal convergence results in the senses of L²-norm and broken H¹-norm, and H¹-norm superclose results are derived by virtue of two types of fractional Grönwall inequalities. Then, the interpolation postprocessing technique is used to establish the superconvergence results. Moreover, to improve computational efficiency, fast algorithms by using sum-of-exponential technique are built for above proposed numerical schemes. Finally, we present some numerical experiments to confirm the theoretical correctness and show the effectiveness of the fast algorithms.     

Superconvergence of Solution Derivatives of the Shortley–Weller Difference Approximation to Elliptic Equations with Singularities Involving the Mixed Type of Boundary Conditions

This paper presents a superconvergence analysis for the Shortley–Weller finite difference approximation of second-order self-adjoint elliptic equations with unbounded derivatives on a polygonal domain with the mixed type of boundary conditions. In this analysis, we first formulate the method as a special finite element/volume method. We then analyze the convergence of the method in a finite element framework. An O(h ^1.5)-order superconvergence of the solution derivatives in a discrete H ¹ norm is obtained. Finally, numerical experiments are provided to support the theoretical convergence rate obtained.     

FINITE RINGS WITH COMMUTING NILPOTENT ELEMENTS

As a generalization of Wedderburn's theorem, Herstein [5] proved that a finite ring R is commutative, if all nilpotent elements are contained in the center of R. However a finite ring with commuting nilpotent elements is not necessarily commutative. Recently, in [9] and [10], Simons described the structure of finite rings R with J(R)² = 0 in a variety with definable principal congruences. In this paper, we will consider the difference between the finite commutative rings and the finite rings in which any two nilpotent elements commute with each other. As a consequence, we describe the structure of finite rings R with [J(R), J(R)] = 0 in a variety with definable principal congruences.     

Two energy-conserving and compact finite difference schemes for two-dimensional Schrödinger-Boussinesq equations

In this paper, we study two compact finite difference schemes for the Schrödinger-Boussinesq (SBq) equations in two dimensions. The proposed schemes are proved to preserve the total mass and energy in the discrete sense. In our numerical analysis, besides the standard energy method, a “cut-off” function technique and a “lifting” technique are introduced to establish the optimal H¹ error estimates without any restriction on the grid ratios. The convergence rate is proved to be of O(τ² + h⁴) with the time step τ and mesh size h. In addition, a fast finite difference solver is designed to speed up the numerical computation of the proposed schemes. The numerical results are reported to verify the error estimates and conservation laws.

     

On the behavior of attractors under finite difference approximation

We recall that the long-time behavior of the Kuramoto-Sivashinsky equation is the same as that of a certain finite system of ordinary differential equations. We show how a particular finite difference scheme approximating the Kuramoto-Sivashinsky may be viewed as a small C ¹ perturbation of this system for the grid spacing sufficiently small. As a consequence one may make deductions about how the global attractor and the flow on the attractor behaves under this approximation. For a sufficiently refined grid the long-time behavior of the solutions of the finite difference scheme is a function of the solutions at certain grid points, whose number and position remain fixed as the grid is refined. Though the results are worked out explicitly for the Kuramoto-Sivashinsky equation, the results extend to other infinite-dimensional dissipative systems.     

Technical note: A fourth‐order finite difference scheme for a system of a 2D nonlinear elliptic partial differential equations

In this article we present a fourth‐order finite difference scheme, for a system of two‐dimensional, second‐order, nonlinear elliptic partial differential equations with mixed spatial derivative terms, using 13‐point stencils with a uniform mesh size h on a square region R subject to Dirichlet boundary conditions. The scheme of order h⁴ is derived using the local solution of the system on a single stencil. The resulting system of algebraic equations can be solved by iterative methods. The difference scheme can be easily modified to obtain formulae for grid points near the boundary. Computational results are given to demonstrate the performance of the scheme on some problems including Navier‐Stokes equations. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 43–53, 2001     

Attractors for Fully Discrete Finite Difference Scheme of Dissipative Zakharov Equations

A fully discrete finite difference scheme for dissipative Zakharov equations is analyzed. On the basis of a series of the time-uniform priori estimates of the difference solutions, the stability of the difference scheme and the error bounds of optimal order of the difference solutions are obtained in L²×H¹×H² over a finite time interval (0, T]. Finally, the existence of a global attractor is proved for a discrete dynamical system associated with the fully discrete finite difference scheme.     

Difference of two sets and estimation of clarke generalized Jacobian via quasidifferential

The notion of difference for two convex compact sets inR ⁿ , proposed by Rubinovet al, is generalized toR ^m×n . A formula of the difference for the two sets, which are convex hulls of a finite number of points, is developed. In the light of this difference, the relation between Clarke generalized Jacobian and quasidifferential, in the sense of Demyanov and Rubinov, for a nonsnooth function, is established. Based on the relation, the method of estimating Clarke generalized Jacobian via quasidifferential for a certain class of functions, is presented.     

Convergence of finite element method for linear second‐order wave equations with discontinuous coefficients

A finite element method is proposed and analyzed for hyperbolic problems with discontinuous coefficients. The main emphasize is given on the convergence of such method. Due to low global regularity of the solutions, the error analysis of the standard finite element method is difficult to adopt for such problems. For a practical finite element discretization, optimal error estimates in L^∞(L²) and L^∞(H¹) norms are established for continuous time discretization. Further, a fully discrete scheme based on a symmetric difference approximation is considered, and optimal order convergence in L^∞(H¹) norm is established. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Algebraic and Discrete Velte Decompositions

Starting from an algebraic equivalent of the Velte decomposition of the function space (H₀ ¹) ^d (where d = 2, 3) into three orthogonal subspaces, we consider finite difference and finite element classes for the approximate solution of the first-kind Stokes problem in two and three dimensions which admit discrete Velte decompositions.This revised version was published online in October 2005 with corrections to the Cover Date.     

Some remarks on orders of projective planes,planar difference sets and multipliers

In this paper we investigate how finite group theory, number theory, together with the geometry of substructures can be used in the study of finite projective planes. Some remarks concerning the function v(x)= x ² + x + 1are presented, for example, how the geometry of a subplane affects the factorization of v(x). The rest of this paper studies abelian planar difference sets by multipliers.Partially supported by NSA grant MDA904-90-H-1013.     

Solution of bi-linear systems arising from high order discretizations of poisson-type equations

Bi-linear systems of the formAV+WA=G are obtained by approximating Poisson-type equations using higher-order finite difference formulae whereV,W andG are known matrices. Solution of the bi-linear system requiresO(n ³) operations for ann×n mesh. However, due to the increased accuracy obtained when using a high-order discretization formula,n can be made much smaller than in the conventional methods and indicates that faster Poisson-solvers which are numerically stable can be obtained by considering the bilinear system rather than the composite matrix form.