Author: Sina ReicheltPublisher:
ISBN:
Category :
Languages : en
Pages :
Book Description
Error Estimates for Elliptic Equations with Not Exactly Periodic Coefficients
Author: Sina ReicheltSome a Posteriori Error Estimates for Elliptic Partial Differential Equations
Author: M. R. PhillipsAn Optimal Error Estimate in Stochastic Homogenization of Discrete Elliptic Equations
Author: Antoine GloriaPublisher:
ISBN:
Category :
Languages : en
Pages :
Book Description
This paper is the second of a series of articles on quantitatives estimates in stochastic homogenization of discrete elliptic equations. We consider a discrete elliptic equation on the d-dimensional lattice Zd with random coefficients A of the simplest type: They are identically distributed and independent from edge to edge. On scales large w. r. t. the lattice spacing (i. e. unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients ...
Error Estimates for Finite Difference Solutions of Second-order Elliptic Equations in Discrete Sobolev Spaces
Author: Hyun Soo DohPeriodic Homogenization of Elliptic Systems
Author: Zhongwei ShenPublisher: Springer
ISBN: 3319912143
Category : Mathematics
Languages : en
Pages : 295
Book Description
This monograph surveys the theory of quantitative homogenization for second-order linear elliptic systems in divergence form with rapidly oscillating periodic coefficients in a bounded domain. It begins with a review of the classical qualitative homogenization theory, and addresses the problem of convergence rates of solutions. The main body of the monograph investigates various interior and boundary regularity estimates that are uniform in the small parameter e>0. Additional topics include convergence rates for Dirichlet eigenvalues and asymptotic expansions of fundamental solutions, Green functions, and Neumann functions. The monograph is intended for advanced graduate students and researchers in the general areas of analysis and partial differential equations. It provides the reader with a clear and concise exposition of an important and currently active area of quantitative homogenization.
Estimates for Solutions of Elliptic Partial Differential Equations with Explicit Constants and Aspects of the Finite Element Method for Second-Order Equations
Author: Andrew William CameronPublisher:
ISBN:
Category :
Languages : en
Pages : 167
Book Description
The classic Lp -based estimates for solutions of elliptic partial differential equations satisfying general boundary conditions were obtained by Agmon, Douglis, and Nirenberg in 1959. In Chapter 2, we rework these estimates to make their dependence on p explicit. It has long been believed that p enters these estimates as a single multiplicative factor of (p [-] 1)[-]1 for p close to 1 and p for p large. This is verified for second-order equations with boundary conditions of order at most one. Poorer results are obtained for more general problems. Local estimates for solutions of homogeneous equations satisfying homogeneous boundary conditions are also established. These are shown to be independent of p. Now consider the finite element approximation of a solution of a second-order elliptic partial differential equation. A typical finite element space that we consider is the Lagrange space of continuous functions which are piecewise polynomials on the elements of an unstructured but quasiuniform triangulation of the domain. As proved by Schatz in 1998, the finite element error is localised in the sense 1 that its L[INFINITY] and W[INFINITY] norms in a region depend most strongly on the behaviour of the true solution at points closest to that region. In Chapter 3, we show that the pattern in the positive norm error estimates continues into the L[INFINITY] -based negative norms. In particular, the error is localised in the negative norms in the same sense that it is in the positive norms. 1 A class of a posteriori W[INFINITY] estimators for the finite element error was inves- tigated by Hoffman, Schatz, Wahlbin, and Wittum in 2001 for the homogeneous Neumann problem. In Chapter 4, we obtain analogous results for an analogous class of L[INFINITY] estimators. Conditions are given under which these are asymptotically equivalent and asymptotically exact. One specific concrete example is provided. In the finite element approximation for the homogeneous Dirichlet problem, the computational domain does not typically match the domain on which the original problem is posed. In Chapter 5, we investigate this issue in conjunction with numerical integration. We find that superparametric elements preserve the 1998 1 weighted L[INFINITY] and W[INFINITY] error estimates of Schatz.
A Posteriori Error Estimators for Elliptic Equations with Discontinuous Diffusion Coefficients
Author: Martin PetzoldtOrdinary Differential Equations and Integral Equations
Author: C.T.H. BakerPublisher: Gulf Professional Publishing
ISBN: 9780444506009
Category : Juvenile Nonfiction
Languages : en
Pages : 562
Book Description
/homepage/sac/cam/na2000/index.html7-Volume Set now available at special set price ! This volume contains contributions in the area of differential equations and integral equations. Many numerical methods have arisen in response to the need to solve "real-life" problems in applied mathematics, in particular problems that do not have a closed-form solution. Contributions on both initial-value problems and boundary-value problems in ordinary differential equations appear in this volume. Numerical methods for initial-value problems in ordinary differential equations fall naturally into two classes: those which use one starting value at each step (one-step methods) and those which are based on several values of the solution (multistep methods). John Butcher has supplied an expert's perspective of the development of numerical methods for ordinary differential equations in the 20th century. Rob Corless and Lawrence Shampine talk about established technology, namely software for initial-value problems using Runge-Kutta and Rosenbrock methods, with interpolants to fill in the solution between mesh-points, but the 'slant' is new - based on the question, "How should such software integrate into the current generation of Problem Solving Environments?" Natalia Borovykh and Marc Spijker study the problem of establishing upper bounds for the norm of the nth power of square matrices. The dynamical system viewpoint has been of great benefit to ODE theory and numerical methods. Related is the study of chaotic behaviour. Willy Govaerts discusses the numerical methods for the computation and continuation of equilibria and bifurcation points of equilibria of dynamical systems. Arieh Iserles and Antonella Zanna survey the construction of Runge-Kutta methods which preserve algebraic invariant functions. Valeria Antohe and Ian Gladwell present numerical experiments on solving a Hamiltonian system of Hénon and Heiles with a symplectic and a nonsymplectic method with a variety of precisions and initial conditions. Stiff differential equations first became recognized as special during the 1950s. In 1963 two seminal publications laid to the foundations for later development: Dahlquist's paper on A-stable multistep methods and Butcher's first paper on implicit Runge-Kutta methods. Ernst Hairer and Gerhard Wanner deliver a survey which retraces the discovery of the order stars as well as the principal achievements obtained by that theory. Guido Vanden Berghe, Hans De Meyer, Marnix Van Daele and Tanja Van Hecke construct exponentially fitted Runge-Kutta methods with s stages. Differential-algebraic equations arise in control, in modelling of mechanical systems and in many other fields. Jeff Cash describes a fairly recent class of formulae for the numerical solution of initial-value problems for stiff and differential-algebraic systems. Shengtai Li and Linda Petzold describe methods and software for sensitivity analysis of solutions of DAE initial-value problems. Again in the area of differential-algebraic systems, Neil Biehn, John Betts, Stephen Campbell and William Huffman present current work on mesh adaptation for DAE two-point boundary-value problems. Contrasting approaches to the question of how good an approximation is as a solution of a given equation involve (i) attempting to estimate the actual error (i.e., the difference between the true and the approximate solutions) and (ii) attempting to estimate the defect - the amount by which the approximation fails to satisfy the given equation and any side-conditions. The paper by Wayne Enright on defect control relates to carefully analyzed techniques that have been proposed both for ordinary differential equations and for delay differential equations in which an attempt is made to control an estimate of the size of the defect. Many phenomena incorporate noise, and the numerical solution of stochastic differential equations has developed as a relatively new item of study in the area. Keven Burrage, Pamela Burrage and Taketomo Mitsui review the way numerical methods for solving stochastic differential equations (SDE's) are constructed. One of the more recent areas to attract scrutiny has been the area of differential equations with after-effect (retarded, delay, or neutral delay differential equations) and in this volume we include a number of papers on evolutionary problems in this area. The paper of Genna Bocharov and Fathalla Rihan conveys the importance in mathematical biology of models using retarded differential equations. The contribution by Christopher Baker is intended to convey much of the background necessary for the application of numerical methods and includes some original results on stability and on the solution of approximating equations. Alfredo Bellen, Nicola Guglielmi and Marino Zennaro contribute to the analysis of stability of numerical solutions of nonlinear neutral differential equations. Koen Engelborghs, Tatyana Luzyanina, Dirk Roose, Neville Ford and Volker Wulf consider the numerics of bifurcation in delay differential equations. Evelyn Buckwar contributes a paper indicating the construction and analysis of a numerical strategy for stochastic delay differential equations (SDDEs). This volume contains contributions on both Volterra and Fredholm-type integral equations. Christopher Baker responded to a late challenge to craft a review of the theory of the basic numerics of Volterra integral and integro-differential equations. Simon Shaw and John Whiteman discuss Galerkin methods for a type of Volterra integral equation that arises in modelling viscoelasticity. A subclass of boundary-value problems for ordinary differential equation comprises eigenvalue problems such as Sturm-Liouville problems (SLP) and Schrödinger equations. Liviu Ixaru describes the advances made over the last three decades in the field of piecewise perturbation methods for the numerical solution of Sturm-Liouville problems in general and systems of Schrödinger equations in particular. Alan Andrew surveys the asymptotic correction method for regular Sturm-Liouville problems. Leon Greenberg and Marco Marletta survey methods for higher-order Sturm-Liouville problems. R. Moore in the 1960s first showed the feasibility of validated solutions of differential equations, that is, of computing guaranteed enclosures of solutions. Boundary integral equations. Numerical solution of integral equations associated with boundary-value problems has experienced continuing interest. Peter Junghanns and Bernd Silbermann present a selection of modern results concerning the numerical analysis of one-dimensional Cauchy singular integral equations, in particular the stability of operator sequences associated with different projection methods. Johannes Elschner and Ivan Graham summarize the most important results achieved in the last years about the numerical solution of one-dimensional integral equations of Mellin type of means of projection methods and, in particular, by collocation methods. A survey of results on quadrature methods for solving boundary integral equations is presented by Andreas Rathsfeld. Wolfgang Hackbusch and Boris Khoromski present a novel approach for a very efficient treatment of integral operators. Ernst Stephan examines multilevel methods for the h-, p- and hp- versions of the boundary element method, including pre-conditioning techniques. George Hsiao, Olaf Steinbach and Wolfgang Wendland analyze various boundary element methods employed in local discretization schemes.
Error estimates for Galerkin methods for quasilinear par abolic and elliptic differential equations in divergence form
Author: Owe AxelssonNumerical Solution of Partial Differential Equations—II, Synspade 1970
Author: Bert HubbardPublisher: Academic Press
ISBN: 1483262480
Category : Mathematics
Languages : en
Pages : 660
Book Description
Numerical Solution of Partial Differential Equations—II: Synspade 1970 provides information pertinent to the fundamental aspects of partial differential equations. This book covers a variety of topics that range from mathematical numerical analysis to numerical methods applied to problems in mechanics, meteorology, and fluid dynamics. Organized into 18 chapters, this book begins with an overview of the methods of the Rayleigh–Ritz–Galerkin type for the approximation of boundary value problems using spline basis functions and Sobolev spaces. This text then analyzes a special approach aimed at solving elliptical equations. Other chapters consider the approximation theoretic study of special sets of approximating functions. This book discusses as well combining the alternating-direction methods with Galerkin methods to obtain highly efficient procedures for the numerical solution of second order parabolic and hyperbolic problems. The final chapter deals with the results concerning Chebyshev rational approximations of reciprocals of certain entire functions. This book is a valuable resource for mathematicians.