Difference Schemes For Elliptic Equations Research Articles

Applications of Multigrid Algorithms to Finite Difference Schemes for Elliptic Equations with Variable Coefficients

This paper is devoted to a study of multigrid algorithms applied to finite difference schemes. If the elliptic equation has variable coefficients, the analysis of multigrid algorithms in the existent literature only gave a convergence rate depending on the number of levels. In this paper, for multigrid algorithms applied to finite difference schemes for elliptic equations with variable coefficients, we establish a convergence rate independent of the number of levels. Our convergence analysis does not require any additional regularity of the solution and is valid for commonly used smoothing operators including the standard Gauss--Seidel method. Under guidance of the general theory, we give details of implementation of the inherited multigrid V(1,1) algorithm. Furthermore, we will provide numerical examples to illustrate the general theory and demonstrate that the inherited multigrid algorithm is efficient for numerical solutions of elliptic equations with variable coefficients. In particular, we will consider elliptic equations on an L-shaped domain whose solutions do not have full regularity and show that the multigrid V(1,1) algorithm performs well in such situations.

A Note on Bitsadze-Samarskii Type Nonlocal Boundary Value Problems: Well-Posedness

The Bitsadze-Samarskii type nonlocal boundary value problem for the differential equation in a Hilbert space H with the self-adjoint positive definite operator A is considered. The well posedness of this problem in Hölder spaces without a weight is established. The coercivity inequalities for the solution of the Neumann-Bitsadze-Samarskii type nonlocal boundary value problem for elliptic equations are obtained. The first order of accuracy difference scheme for the approximate solution of this problem is presented. The well posedness of this difference scheme in difference analogue of Hölder spaces is established. In applications, the stability, the almost coercivity and the coercivity estimates for solutions of difference schemes for elliptic equations are obtained.

On Finite Difference Schemes for Elliptic Equations with Discontinuous Coefficients

Abstract. We study the convergence of finite difference schemes for approximating elliptic equations of second order with discontinuous coefficients. Two of these finite difference schemes arise from the discretization by the finite element method using bilinear shape functions. We prove an convergence for the gradient, if the solution is locally in H3. Thus, in contrast to the first order convergence for the gradient obtained by the finite element theory we show that the gradient is superclose. From the Bramble–Hilbert Lemma we derive a higher order compact (HOC) difference scheme that gives an approximation error of order four for the gradient. A numerical example is given.

On Bitsadze–Samarskii type nonlocal boundary value problems for elliptic differential and difference equations: Well-posedness

The well-posedness of the Bitsadze–Samarskii type nonlocal boundary value problem in Hölder spaces with a weight is established. The coercivity inequalities for the solution of the nonlocal boundary value problem for elliptic equations are obtained. The stable second order of accuracy difference scheme for the approximate solution of the problem is presented. The well-posedness of this difference scheme in difference analogue of Hölder spaces is established. For applications, the almost coercivity and the coercivity estimates for solutions of difference schemes for elliptic equations are obtained.

On a difference scheme of fourth order of accuracy for the Bitsadze–Samarskii type nonlocal boundary value problem

The Bitsadze–Samarskii type nonlocal boundary value problem math image for the differential equation in a Hilbert space H with the self‐adjoint positive definite operator A with a closed domain D(A) ⊂ H is considered. Here, f(t) be a given abstract continuous function defined on [0,1] with values in H, φ and ψ be the elements of D(A), and λj are the numbers from the set [0,1]. The well‐posedness of the problem in Hölder spaces with a weight is established. The coercivity inequalities for the solution of the nonlocal boundary value problem for elliptic equations are obtained. The fourth order of accuracy difference scheme for approximate solution of the problem is presented. The well‐posedness of this difference scheme in difference analogue of Hölder spaces is established. For applications, the stability, the almost coercivity, and the coercivity estimates for the solutions of difference schemes for elliptic equations are obtained. Mathematical Methods in the Applied Sciences. Copyright © 2012 John Wiley & Sons, Ltd.

Well-posedness of the difference schemes for elliptic equations in [formula omitted] spaces

The second order of accuracy difference scheme for the approximate solutions of the nonlocal boundary-value problem − v ″ ( t ) + A v ( t ) = f ( t ) ( 0 ≤ t ≤ 1 ) , v ( 0 ) = v ( 1 ) , v ′ ( 0 ) = v ′ ( 1 ) for differential equations in an arbitrary Banach space E with a strongly positive operator A is considered. The well-posedness of this difference scheme in C τ β , γ ( E ) spaces is established. In applications, a series of coercivity inequalities in difference analogues of various Hölder norms for the solutions of difference schemes of the second order of accuracy over one variable for the approximate solutions of the nonlocal boundary value problem for elliptic equations are obtained.

High-order finite difference schemes for elliptic problems with intersecting interfaces

In this work we derive high-order (second and fourth of convergence, but with first and third-order local truncation error, respectively) compact finite difference schemes for elliptic equations with intersecting interfaces. The approach uses the differential equation and the jump (interface) relations as additional identities which can be differentiated to eliminate higher order local truncation errors. Numerical experiments are carried out to demonstrate the high-order accuracy and to show that our method is effective to sharp contrast in the diffusion coefficients of the problems.

Higher-Order, Cartesian Grid Based Finite Difference Schemes for Elliptic Equations on Irregular Domains

Second and fourth order Cartesian grid based finite difference methods are proposed for elliptic and parabolic partial differential equations, and associated eigenvalue problems on irregular domains with general boundary conditions. Our methods are based on the continuation of a solution idea using multivariable Taylor's expansion of the solution about selected boundary points, and the core ideas of the immersed interface method. The methods offer systematic treatment of the general boundary conditions in two- and three-dimensional domains and are directly applied to semi-discretize heat equations on irregular domains. Convergence analysis and numerical examples are presented. The validity and effectiveness of the proposed methods are demonstrated through our numerical results including computations of the eigenvalues of the associated eigenvalue problem.

Difference Schemes for Elliptic Equations with Mixed Derivatives

AbstractIn this paper, the a priori estimates of stability in the energy and the uniform norms are proved for the monotone and conservative difference schemes approximating elliptic equations with mixed derivatives. The estimates are obtained without any assumption about the symmetry of the coe±cient matrix of the initial differential equation.

Theory of Difference Schemes

3R3. Theory of Difference Schemes. Pure and Applied Mathematics Series, Vol 240. - AA Samarskii (Fac of Comput Math and Cybernetics, Moscow MV Lomonosov State Univ, Moscow, Russia). Marcel Dekker, New York. 2001. 761 pp. ISBN 0-8247-0468-1. $225.00. Reviewed by VD Radulescu (Dept of Math, Univ of Craiova, 13, St AI Cuza, Craiova, 1100, Romania).The study of difference schemes is one of the central subjects in Numerical Analysis. Because of the variety and importance of their applications, in particular to Applied Mathematics, difference schemes caused developments in various areas of mathematics.This monograph is intended as an introduction to difference schemes at the advanced undergraduate and beginning graduate level. The author aimed at breaching the gap that too often exists between engineering and example-oriented textbooks on the one hand, and needlessly abstract mathematical formulations on the other. The book is divided into ten chapters, followed by a list of symbols and some concluding remarks. After several necessary prerequisites exposed in the first two chapters, the author develops in Chapters 3–5 concrete difference schemes for equations of elliptic, parabolic, and hyperbolic types. Chapter 3 focuses on homogeneous difference schemes for ordinary differential equations, by means of which the author solves the canonical problem of the theory of difference schemes in which a primary family of difference schemes is specified and schemes of a desired quality should be selected within the primary family. Chapter 4 is devoted to various difference approximations of second-order elliptic equations. The approximation technique for the Laplace operator and formulations of difference boundary conditions are described for regions of arbitrary shape. Chapter 5 provides the general theory of difference schemes in which it seems reasonable to eliminate constraints on the structure and implicit form of difference operators. Such a theory treats difference schemes as operator equations and operator-difference equations, which are analogs of difference schemes for time-dependent equations of mathematical physics.Chapter 6 includes a priori estimates expressing stability of two-layer and three-layer schemes in terms of the initial data and the right-hand side of the corresponding equations. This chapter also includes many good examples illustrating the practical use of general stability theory with regard to particular schemes to assist the users in subsequent implementations. The author’s strategy is that the stability is the most pressing problem in any algorithm, since it is a necessary rather than a sufficient condition for accuracy. The main purpose of the next three chapters is to show how the results of the general theory of difference schemes are aimed at starting principles for constructing difference schemes of a prescribed quality. The difference schemes for elliptic equations are viewed as operator equations of the first kind, while the difference schemes relating to analogs of nonstationary equations of mathematical physics are treated as difference equations with operator coefficients in an abstract space of any dimension. In the last chapter, several economical schemes and iterative methods for multidimensional problems in mathematical physics are developed. The book is well written and is strongly influenced by the well-known Russian school of Numerical Analysis. The point of view taken here, rigorous presentation without excessive formalism, is however nonstandard. In conclusion, Samarskii’s book Theory of Difference Schemes is original, interesting, and transmits a message in a clear and efficient way. On the whole, this monograph is an excellent contribution to the literature of numerical methods and computer algorithms for solving mathematical-physics problems. The book qualifies to be a reference work that certainly would be a valuable addition to libraries of universities and research laboratories pursuing research in Applied Mathematics.

Compact difference schemes for elliptic equations with mixed derivatives on arbitrary grids

- We study difference schemes with indefinite coefficients and indefinite right-hand sides on stencils that consist of any number of arbitrarily located points. In the two-dimensional case we obtain linear systems of equations for the calculation of coefficients in ordinary (noncompact) and compact schemes and conditions on the right-hand side under which the approximation order of differential equations would be equal to a predetrmined natural number. Analogously we study the behaviour of ordinary and compact approximations of universal boundary conditions, including the conditions of the first, second, and third kinds. In the three-dimensional case we establish the number of conditions which specify coefficients and the right-hand sides of schemes. We consider particular examples of schemes in the plane and in the space.

Optimal finite difference grids and rational approximations of the square root I. Elliptic problems

The main objective of this paper is optimization of second-order finite difference schemes for elliptic equations, in particular, for equations with singular solutions and exterior problems. A model problem corresponding to the Laplace equation on a semi-infinite strip is considered. The boundary impedance (Neumann-to-Dirichlet map) is computed as the square root of an operator using the standard three-point finite difference scheme with optimally chosen variable steps. The finite difference approximation of the boundary impedance for data of given smoothness is the problem of rational approximation of the square root on the operator's spectrum. We have implemented Zolotarev's optimal rational approx-imant obtained in terms of elliptic functions. We have also found that a geometrical progression of the grid steps with optimally chosen parameters is almost as good as the optimal approximant. For bounded operators it increases from second to exponential the convergence order of the finite difference impedance with the convergence rate proportional to the inverse of the logarithm of the condition number. For the case of unbounded operators in Sobolev spaces associated with elliptic equations, the error decays as the exponential of the square root of the mesh dimension. As an example, we numerically compute the Green function on the boundary for the Laplace equation. Some features of the optimal grid obtained for the Laplace equation remain valid for more general elliptic problems with variable coefficients. © 2000 John Wiley & Sons, Inc.

Optimal finite difference grids and rational approximations of the square root I. Elliptic problems

The main objective of this paper is optimization of second-order finite difference schemes for elliptic equations, in particular, for equations with singular solutions and exterior problems. A model problem corresponding to the Laplace equation on a semi-infinite strip is considered. The boundary impedance (Neumann-to-Dirichlet map) is computed as the square root of an operator using the standard three-point finite difference scheme with optimally chosen variable steps. The finite difference approximation of the boundary impedance for data of given smoothness is the problem of rational approximation of the square root on the operator's spectrum. We have implemented Zolotarev's optimal rational approx-imant obtained in terms of elliptic functions. We have also found that a geometrical progression of the grid steps with optimally chosen parameters is almost as good as the optimal approximant. For bounded operators it increases from second to exponential the convergence order of the finite difference impedance with the convergence rate proportional to the inverse of the logarithm of the condition number. For the case of unbounded operators in Sobolev spaces associated with elliptic equations, the error decays as the exponential of the square root of the mesh dimension. As an example, we numerically compute the Green function on the boundary for the Laplace equation. Some features of the optimal grid obtained for the Laplace equation remain valid for more general elliptic problems with variable coefficients. © 2000 John Wiley & Sons, Inc.

Rate of convergence of the method of group over-relaxation for solving variational difference schemes for elliptic equations of order 2m in a two-dimensional region

WE show that, to obtain the solution of the system to an accuracy O( h 2 m ) we need to perform O( ln h −1 h m ) iterations. The influence of errors in performing the arithmetic operations on the computational results is investigated. To preserve an accuracy O( h 2 m ) we show that the elementary arithmetic operation should have a relative accuracy of order h 7 m − 1.5 . We described in [1] a modified type of variational difference schemes for solving elliptic equations of order 2 m, m ⩾ 1, in an arbitrary region. Estimates of the accuracy and condition number were established (the condition number is of order h 2 m , where h is the mesh step). In the present paper we consider the solution of the schemes described in [1], based on Hermitian interpolations [2], for equations in a two-dimensional region, using the method of group over-relaxation. We estimate the number of iterations sufficient to achieve an accuracy O( h 2 m ) in the norm of space L 2, and we prove that this number is of order O( h − m ln h −1) for certain values of the relaxation parameter. To realize one iterative step, O( h −2) arithmetic operations are required. We also examine the numerical stability of the method, and show that, to obtain a solution with accuracy O( h 2 m ) in the norm of L 2, it is sufficient for the relative accuracy of the elementary arithmetic operations to be of order h 7 m − 1.5 . Aspects of the convergence of the method of over-relaxation were discussed in [3]; some of the devices used in [3] will be employed in the present paper.

Difference schemes for elliptic equations with special boundary conditions

On the boundary S of the rectangle a linear combination of the derivative of the unknown function along the inward normal to S, the second derivative along the tangent to the boundary and the values of the unknown function on S are specified. Problems of this type occur, for example, in magnetohydrodynamics [ll in the flow of a conducting fluid with velocity u (x1, x,1 in a magnetic field B in a channel of rectangular section. It is assumed that the walls of the channel are conducting only on segments of finite length. Then we obtain for the potential u in the rectangle G a boundary value problem with boundary conditions of the type indicated above on two opposite sides of the rectangle and u F 0 on the other two sides. In the rectangle itself Au = -f, f = f(v, B). In section 1 we consider the boundary value problem for an elliptic selfconjugate equation which includes the problem of the potential. In section 2 we consider the problem for Poisson’s equation in the rectangle G. On all the sides of the rectangle we specify the boundary conditions