Download or read book A Posteriori Error Estimates for Finite Volume Approximations of Elliptic Equations on General Surfaces written by and published by . This book was released on 2009 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we present a residual-based a posteriori error estimate for the finite volume discretization of steady convection- diffusion-reaction equations defined on surfaces in R3, which are often implicitly represented as level sets of smooth functions. Reliability and efficiency of the proposed a posteriori error estimator are rigorously proved. Numerical experiments are also conducted to verify the theoretical results and demonstrate the robustness of the error estimator.

Download or read book Residual Type a Posteriori Error Estimates for Upwinding Finite Volume Approximations of Elliptic Boundary Value Problems written by Lutz Angermann and published by . This book was released on 2010 with total page 36 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A Posteriori Error Analysis Via Duality Theory written by Weimin Han and published by Springer Science & Business Media. This book was released on 2006-07-30 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work provides a posteriori error analysis for mathematical idealizations in modeling boundary value problems, especially those arising in mechanical applications, and for numerical approximations of numerous nonlinear var- tional problems. An error estimate is called a posteriori if the computed solution is used in assessing its accuracy. A posteriori error estimation is central to m- suring, controlling and minimizing errors in modeling and numerical appr- imations. In this book, the main mathematical tool for the developments of a posteriori error estimates is the duality theory of convex analysis, documented in the well-known book by Ekeland and Temam ([49]). The duality theory has been found useful in mathematical programming, mechanics, numerical analysis, etc. The book is divided into six chapters. The first chapter reviews some basic notions and results from functional analysis, boundary value problems, elliptic variational inequalities, and finite element approximations. The most relevant part of the duality theory and convex analysis is briefly reviewed in Chapter 2.

Download or read book Finite Volumes for Complex Applications X Volume 2 Hyperbolic and Related Problems written by Emmanuel Franck and published by Springer Nature. This book was released on 2023-10-12 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume comprises the second part of the proceedings of the 10th International Conference on Finite Volumes for Complex Applications, FVCA, held in Strasbourg, France, during October 30 to November 3, 2023. The Finite Volume method, and several of its variants, is a spatial discretization technique for partial differential equations based on the fundamental physical principle of conservation. Recent decades have brought significant success in the theoretical understanding of the method. Many finite volume methods are also built to preserve some properties of the continuous equations, including maximum principles, dissipativity, monotone decay of the free energy, asymptotic stability, or stationary solutions. Due to these properties, finite volume methods belong to the wider class of compatible discretization methods, which preserve qualitative properties of continuous problems at the discrete level. This structural approach to the discretization of partial differential equations becomes particularly important for multiphysics and multiscale applications. In recent years, the efficient implementation of these methods in numerical software packages, more specifically to be used in supercomputers, has drawn some attention. The first volume contains all invited papers, as well as the contributed papers focusing on finite volume schemes for elliptic and parabolic problems. They include structure-preserving schemes, convergence proofs, and error estimates for problems governed by elliptic and parabolic partial differential equations. This volume is focused on finite volume methods for hyperbolic and related problems, such as methods compatible with the low Mach number limit or able to exactly preserve steady solutions, the development and analysis of high order methods, or the discretization of kinetic equations.

Download or read book A Posteriori Error Estimate for Finite Volume Approximations to Singularly Perturbed Nonlinear Convectiondiffusion Equations written by Mario Ohlberger and published by . This book was released on 1999 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A Posteriori Error Estimates for Vertex Centered Finite Volume Approximations of Convection Diffusionreaction Equations written by Mario Ohlberger and published by . This book was released on 2000 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Balanced A posteriori Error Estimates for Finite Volume Type Discretizations of Convection dominated Elliptic Problems written by Lutz Angermann and published by . This book was released on 1994 with total page 34 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A Posteriori Error Estimation for Finite Element Approximations of Fractional Laplacian Problems and Applications to Poro elasticity written by Raphaël Bulle and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This manuscript is concerned with a posteriori error estimation for the finiteelement discretization of standard and fractional partial differential equationsas well as an application of fractional calculus to the modeling of thehuman meniscus by poro-elasticity equations.In the introduction, we give an overview of the literature about a posteriori errorestimation for finite element methods and about adaptive mesh refinement methods.We also review the literature about fractional partial differential equationsand Caputo's fractional derivative with anomalous diffusion applications.We emphasize on the state-of-the-art of the Bank-Weiser estimator and of aposteriori error estimation for the spectral fractional Laplacian.The rest of the manuscript is organized as follows.The Chapter 1 is concerned with a proof of the reliability of theBank-Weiser estimator for three-dimensional problems discretized with linearLagrange finite elements. This result is an extension of a previous result fromthe literature.In Chapter 2 we present a numerical study of the Bank-Weiserestimator.We provide a novel implementation of the estimator in the FEniCS finiteelement software and working in parallel.We apply our code to a variety of elliptic equations, several differenttwo-dimensional Poisson problems and a three-dimensional linear elasticityproblem.In particular, we use our implementation into an adaptive mesh refinement method anda goal-oriented error estimation method.In addition we provide convergence studies for these methods as well as atimescale study of our error estimation method when performed in parallel.In Chapter 3 we derive a novel a posteriori estimator for theL2 error induced by the finite element discretization of the fractionalLaplacian operator.We provide an implementation of our method in the FEniCS finite elementsoftware.We apply our estimator to an adaptive refinement method for two-dimensional andthree-dimensional fractional Poisson equations.In addition, we provide numerical results on the convergence of this method.In Chapter 4 we present new theoretical results on theconvergence of a rational approximation method with consequences on theapproximation of fractional norms and a priori error estimation of the semi-discretization of the spectral fractional Laplacian.Finally, in Chapter 5 we provide an application of fractionalcalculus to the study of the human meniscus via poro-elasticity equations and the Caputo derivative.

Download or read book Finite Volumes for Complex Applications X Volume 1 Elliptic and Parabolic Problems written by Emmanuel Franck and published by Springer. This book was released on 2023-10-23 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume comprises the first part of the proceedings of the 10th International Conference on Finite Volumes for Complex Applications, FVCA, held in Strasbourg, France, during October 30 to November 3, 2023. The Finite Volume method, and several of its variants, is a spatial discretization technique for partial differential equations based on the fundamental physical principle of conservation. Recent decades have brought significant success in the theoretical understanding of the method. Many finite volume methods are also built to preserve some properties of the continuous equations, including maximum principles, dissipativity, monotone decay of the free energy, asymptotic stability, or stationary solutions. Due to these properties, finite volume methods belong to the wider class of compatible discretization methods, which preserve qualitative properties of continuous problems at the discrete level. This structural approach to the discretization of partial differential equations becomes particularly important for multiphysics and multiscale applications. In recent years, the efficient implementation of these methods in numerical software packages, more specifically to be used in supercomputers, has drawn some attention. This volume contains all invited papers, as well as the contributed papers focusing on finite volume schemes for elliptic and parabolic problems. They include structure-preserving schemes, convergence proofs, and error estimates for problems governed by elliptic and parabolic partial differential equations. The second volume is focused on finite volume methods for hyperbolic and related problems, such as methods compatible with the low Mach number limit or able to exactly preserve steady solutions, the development and analysis of high order methods, or the discretization of kinetic equations.

Download or read book A Review of Posteriori Error Estimation Adaptive Mesh Refinement Techniques written by Rudiger Verfurth and published by Wiley. This book was released on 1996-06-11 with total page 134 pages. Available in PDF, EPUB and Kindle. Book excerpt: Wiley—Teubner Series Advances in Numerical Mathematics Editors Hans Georg Bock Mitchell Luskin Wolfgang Hackbusch Rolf Rannacher A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques Rüdiger Verfürth Ruhr-Universität Bochum, Germany Self-adaptive discretization methods have gained an enormous importance for the numerical solution of partial differential equations which arise in physical and technical applications. The aim of these methods is to obtain a numerical solution within a prescribed tolerance using a minimal amount of work. The main tools utilised are a posteriori error estimators and indicators which are able to give global and local information on the error of the numerical solution, using only the computed numerical solution and known data of the problem. Presenting the most frequently used error estimators which have been developed by various scientists in the last two decades, this book demonstrates that they are all based on the same basic principles. These principles are then used to develop an abstract framework which is able to handle general nonlinear problems. The abstract results are applied to various classes of nonlinear elliptic partial differential equations from, for example, fluid and continuum mechanics, to yield reliable and easily computable error estimators. The book covers stationary problems but omits transient problems, where theory is often still too complex and not yet well developed.

Download or read book Some a Posteriori Error Estimates for Elliptic Partial Differential Equations written by M. R. Phillips and published by . This book was released on 1997 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Applied mechanics reviews written by and published by . This book was released on 1948 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A Posteriori Error Estimation Techniques for Finite Element Methods written by Rüdiger Verfürth and published by Oxford University Press. This book was released on 2013-04-18 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: A posteriori error estimation techniques are fundamental to the efficient numerical solution of PDEs arising in physical and technical applications. This book gives a unified approach to these techniques and guides graduate students, researchers, and practitioners towards understanding, applying and developing self-adaptive discretization methods.

Download or read book A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques written by Rüdiger Verführt and published by Springer. This book was released on 1996-07 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Estimates for Solutions of Elliptic Partial Differential Equations with Explicit Constants and Aspects of the Finite Element Method for Second Order Equations written by Andrew William Cameron and published by . This book was released on 2011 with total page 167 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book A Posteriori Error Estimation for Partial Differential Equations with Random Input Data written by Diane Sylvie Guignard and published by . This book was released on 2016 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mots-clés de l'autrice: PDEs with random inputs ; uncertainty quantification ; a priori and a posteriori error analysis ; finite elements ; perturbation techniques ; stochastic collocation ; elliptic equations ; steady Navier-Stokes equations ; heat equation.

Download or read book A Posteriori Error Control for Stationary Coupled Bulk surface Equations written by Martin Eigel and published by . This book was released on 2015 with total page 1650 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider a system of two coupled elliptic equations, one defined on a bulk domain and the other one on the boundary surface. Problems of this kind of problem are relevant for applications in engineering, chemistry and in biology like e.g. biological signal transduction. For the a posteriori error control of the coupled system, a residual error estimator is derived which takes into account the approximation errors due to the finite element discretisation in space as well as the polyhedral approximation of the surface. An adaptive refinement algorithm controls the overall error. Numerical experiments illustrate the performance of the a posteriori error estimator and the proposed adaptive algorithm with several benchmark examples.