Download or read book Adaptive Methods in the Finite Element Exterior Calculus Framework written by Adam Mihalik and published by . This book was released on 2014 with total page 100 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis we explore convergence theory for adaptive mixed finite element methods. In particular, we introduce an a posteriori error-indicator, and prove convergence and optimality results for the mixed formulation of the Hodge Laplacian posed on domains of arbitrary dimensionality and topology in R/n. After developing this framework, we introduce a new algorithm and extend our theory and results to problems posed on Euclidean hypersurfaces. We begin by introducing the finite element exterior calculus framework, which is the key tool allowing us to address the convergence proofs in such generality. This introduction focuses on the fundamentals of the well-developed a priori theory and the results needed to extend the core of this theory to problems posed on surfaces. A basic set of results needed to develop adaptivity in this framework is also established. We then introduce an adaptive algorithm, and show convergence using this infrastructure as a tool to generalize existing finite element theory. The algorithm is then shown to be computationally optimal through a series of complexity analysis arguments. Finally, a second algorithm is introduced for problems posed on surfaces, and our original convergence and optimality results are extended using properties of specific geometric maps between surfaces

Download or read book Finite Element Exterior Calculus written by Douglas N. Arnold and published by SIAM. This book was released on 2018-12-12 with total page 126 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational methods to approximate the solution of differential equations play a crucial role in science, engineering, mathematics, and technology. The key processes that govern the physical world?wave propagation, thermodynamics, fluid flow, solid deformation, electricity and magnetism, quantum mechanics, general relativity, and many more?are described by differential equations. We depend on numerical methods for the ability to simulate, explore, predict, and control systems involving these processes. The finite element exterior calculus, or FEEC, is a powerful new theoretical approach to the design and understanding of numerical methods to solve partial differential equations (PDEs). The methods derived with FEEC preserve crucial geometric and topological structures underlying the equations and are among the most successful examples of structure-preserving methods in numerical PDEs. This volume aims to help numerical analysts master the fundamentals of FEEC, including the geometrical and functional analysis preliminaries, quickly and in one place. It is also accessible to mathematicians and students of mathematics from areas other than numerical analysis who are interested in understanding how techniques from geometry and topology play a role in numerical PDEs.

Download or read book Adaptive Finite Element Methods for Differential Equations written by Wolfgang Bangerth and published by Springer Science & Business Media. This book was released on 2003-01-23 with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt: The key issues are a posteriori error estimation and it automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method for goal-oriented error estimation, is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. `Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost. At the end of each chapter some exercises are posed in order to assist the interested reader in better understanding the concepts presented. Solutions and accompanying remarks are given in the Appendix.

Download or read book Finite Element Exterior Calculus with Applications to the Numerical Solution of the Green Naghdi Equations written by Adam Morgan and published by . This book was released on 2018 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of finite element methods for the numerical solution of differential equations is one of the gems of modern mathematics, boasting rigorous analytical foundations as well as unambiguously useful scientific applications. Over the past twenty years, several researchers in scientific computing have realized that concepts from homological algebra and differential topology play a vital role in the theory of finite element methods. Finite element exterior calculus is a theoretical framework created to clarify some of the relationships between finite elements, algebra, geometry, and topology. The goal of this thesis is to provide an introduction to the theory of finite element exterior calculus, and to illustrate some applications of this theory to the design of mixed finite element methods for problems in geophysical fluid dynamics. The presentation is divided into two parts. Part 1 is intended to serve as a self-contained introduction to finite element exterior calculus, with particular emphasis on its topological aspects. Starting from the basics of calculus on manifolds, I go on to describe Sobolev spaces of differential forms and the general theory of Hilbert complexes. Then, I explain how the notion of cohomology connects Hilbert complexes to topology. From there, I discuss the construction of finite element spaces and the proof that special choices of finite element spaces can be used to ensure that the cohomological properties of a particular problem are preserved during discretization. In Part 2, finite element exterior calculus is applied to derive mixed finite element methods for the Green-Naghdi equations (GN). The GN extend the more well-known shallow water equations to the regime of non-infinitesimal aspect ratio, thus allowing for some non-hydrostatic effects. I prove that, using the mixed formulation of the linearized GN, approximations of balanced flows remain steady. Additionally, one of the finite element methods presented for the fully nonlinear GN provably conserves mass, vorticity, and energy at the semi-discrete level. Several computational test cases are presented to assess the practical performance of the numerical methods, including a collision between solitary waves, the motion of solitary waves over variable bottom topography, and the breakdown of an unstable balanced state.

Download or read book Accuracy Estimates and Adaptive Refinements in Finite Element Computations written by Ivo Babuška and published by John Wiley & Sons. This book was released on 1986 with total page 422 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains papers discussing the recent developments in adaptive methods and their applications, an area of finite elements methods applicable to the needs of civil engineering. Topics covered range from an exposition of basic theory and techniques to detailed discussions of specific applications. Adaptive approaches, and the computer assessment of the reliability of the results obtained are also examined.

Download or read book Adaptive Finite Element Methods written by Wenbin Liu and published by Alpha Science International Limited. This book was released on 2012 with total page 197 pages. Available in PDF, EPUB and Kindle. Book excerpt: Summary: "This book emphasizes the discussions of some unique issues from the adaptive finite element approximation of optimal control. The main idea used in the approximation error analysis (both a priori and a posteriori) is to first combine convex analysis and interpolation error estimations of suitable interpolators, which much depend on the structure of the control constraints, to derive the error estimates for the control via the variational inequalities in the optimality conditions, and then to apply the standard techniques to derive the error estimates for the state equations. The need, the framework and the techniques of using multi adaptive meshes in developing efficient numerical algorithms for optimal control have been emphasized throughout the book. The book starts from several typical examples of optimal control problems and then discusses existence and optimality conditions for some optimal control problems. It is believed that these discussions are especially useful for the researchers and students who first entered this area. Then the finite element approximation schemes for several typical optimal control problems are set up, their a priori and a posteriori error estimates are derived following the main idea mentioned, and their computational methods are studied."-- Publisher website, viewed 13th July, 2012.

Download or read book Fast Solvers for Numerical Schemes Based On Finite Element Exterior Calculus written by Lin Zhong and published by . This book was released on 2015 with total page 131 pages. Available in PDF, EPUB and Kindle. Book excerpt: Finite element exterior calculus (FEEC) is a framework to design and understand finite element discretizations for a wide variety of systems of partial differential equations. The applications are already made to the Hodge Laplacian, Maxwell's equations, the equations of elasticity, elliptic eigenvalue problems and etc.. In this thesis, we propose fast solvers for several numerical schemes based on the discretization of this approach and present theoretical analysis. Specifically, in the first part, we propose efficient block diagonal and block triangular preconditioners for solving the discretized linear system of the vector Laplacian by mixed finite element methods. A variable V-cycle multigrid method with the standard point-wise Gauss-Seidel smoother is proved to be a good preconditioner for the Schur complement. The major benefit of our approach is that the point-wise Gauss-Seidel smoother is more algebraic and can be easily implemented as a 'black-box' smoother. The multigrid solver for the Schur complement will be further used to build preconditioners for the original saddle point systems. In the second part, we propose a discretization method for the Darcy-Stokes equations under the framework of FEEC. The discretization is shown to be uniform with respect to the perturbation parameter. A preconditioner for the discrete system is also proposed and shown to be efficient. In the last part, we focus on the stochastic Stokes equations. The stochastic saddle-point linear systems are obtained by using finite element discretization under the framework of FEEC in physical space and generalized polynomial chaos expansion in random space. We prove the existence and uniqueness of the solutions to the continuous problem and its corresponding stochastic Galerkin discretization. Optimal error estimates are also derived. We construct block-diagonal/triangular preconditioners for use with the generalized minimum residual method and the bi-conjugate gradient stabilized method. An optimal multigrid solver is applied to efficiently solve the diagonal blocks that correspond to deterministic discrete Stokes systems. To demonstrate the efficiency and robustness of the discretization methods and proposed preconditioners, various numerical examples also are provided.

Download or read book Adaptive Finite Element Methods for the Helmholtz Equation in Exterior Domains written by James R. Stewart and published by . This book was released on 1995 with total page 376 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Space Time Methods written by Ulrich Langer and published by Walter de Gruyter GmbH & Co KG. This book was released on 2019-09-23 with total page 261 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume provides an introduction to modern space-time discretization methods such as finite and boundary elements and isogeometric analysis for time-dependent initial-boundary value problems of parabolic and hyperbolic type. Particular focus is given on stable formulations, error estimates, adaptivity in space and time, efficient solution algorithms, parallelization of the solution pipeline, and applications in science and engineering.

Download or read book Adaptive Finite Element Solution Algorithm for the Euler Equations written by Richard A. Shapiro and published by Notes on Numerical Fluid Mechanics and Multidisciplinary Design. This book was released on 1991 with total page 188 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on the author's Ph. D. thesis (Massachusetts Institute of Technology).

Download or read book H P Adaptive Methods for Finite Element Analysis of Aerothermal Loads in High speed Flows written by and published by . This book was released on 1993 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Automated Solution of Differential Equations by the Finite Element Method written by Anders Logg and published by Springer Science & Business Media. This book was released on 2012-02-24 with total page 723 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Following are chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Download or read book An Adaptive Mixed Finite Element Method Using the Lagrange Multiplier Technique written by Michael Anthony Gagnon and published by . This book was released on 2009 with total page 52 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: Adaptive methods in finite element analysis are essential tools in the efficient computation and error control of problems that may exhibit singularities. In this paper, we consider solving a boundary value problem which exhibits a singularity at the origin due to both the structure of the domain and the regularity of the exact solution. We introduce a hybrid mixed finite element method using Lagrange Multipliers to initially solve the partial differential equation for the both the flux and displacement. An a posteriori error estimate is then applied both locally and globally to approximate the error in the computed flux with that of the exact flux. Local estimation is the key tool in identifying where the mesh should be refined so that the error in the computed flux is controlled while maintaining efficiency in computation. Finally, we introduce a simple refinement process in order to improve the accuracy in the computed solutions. Numerical experiments are conducted to support the advantages of mesh refinement over a fixed uniform mesh.

Download or read book Convergence of Adaptive Finite Element Methods written by and published by . This book was released on 2005 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Adaptive Methods in Finite Elements written by Ravindran Gopalakrishnan and published by . This book was released on 1989 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs written by Josef Malek and published by SIAM. This book was released on 2014-12-22 with total page 106 pages. Available in PDF, EPUB and Kindle. Book excerpt: Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs is about the interplay between modeling, analysis, discretization, matrix computation, and model reduction. The authors link PDE analysis, functional analysis, and calculus of variations with matrix iterative computation using Krylov subspace methods and address the challenges that arise during formulation of the mathematical model through to efficient numerical solution of the algebraic problem. The book?s central concept, preconditioning of the conjugate gradient method, is traditionally developed algebraically using the preconditioned finite-dimensional algebraic system. In this text, however, preconditioning is connected to the PDE analysis, and the infinite-dimensional formulation of the conjugate gradient method and its discretization and preconditioning are linked together. This text challenges commonly held views, addresses widespread misunderstandings, and formulates thought-provoking open questions for further research.

Download or read book Adaptive Finite Element Methods for Time Dependent Partial Differential Equations written by Slimane Adjerid and published by . This book was released on 1985 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: