Finite Element Exterior Calculus

Finite Element Exterior Calculus

Author: Douglas N. Arnold

Publisher: SIAM

Published: 2018-12-12

Total Pages: 126

ISBN-13: 1611975530

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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.


Book Synopsis Finite Element Exterior Calculus by : Douglas N. Arnold

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.

Finite Element Exterior Calculus

Finite Element Exterior Calculus

Author: Douglas N. Arnold

Publisher: SIAM

Published: 2018-12-12

Total Pages: 120

ISBN-13: 1611975549

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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.


Book Synopsis Finite Element Exterior Calculus by : Douglas N. Arnold

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 120 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.

Finite Element Exterior Calculus with Applications to the Numerical Solution of the Green-Naghdi Equations

Finite Element Exterior Calculus with Applications to the Numerical Solution of the Green-Naghdi Equations

Author: Adam Morgan

Publisher:

Published: 2018

Total Pages: 220

ISBN-13:

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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.


Book Synopsis Finite Element Exterior Calculus with Applications to the Numerical Solution of the Green-Naghdi Equations by : Adam Morgan

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.

Fast Solvers for Numerical Schemes Based On Finite Element Exterior Calculus

Fast Solvers for Numerical Schemes Based On Finite Element Exterior Calculus

Author: Lin Zhong

Publisher:

Published: 2015

Total Pages: 131

ISBN-13: 9781339528366

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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.


Book Synopsis Fast Solvers for Numerical Schemes Based On Finite Element Exterior Calculus by : Lin Zhong

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.

Finite Element Methods for Computational Fluid Dynamics

Finite Element Methods for Computational Fluid Dynamics

Author: Dmitri Kuzmin

Publisher: SIAM

Published: 2014-12-18

Total Pages: 321

ISBN-13: 1611973600

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This informal introduction to computational fluid dynamics and practical guide to numerical simulation of transport phenomena covers the derivation of the governing equations, construction of finite element approximations, and qualitative properties of numerical solutions, among other topics. To make the book accessible to readers with diverse interests and backgrounds, the authors begin at a basic level and advance to numerical tools for increasingly difficult flow problems, emphasizing practical implementation rather than mathematical theory.?Finite Element Methods for Computational Fluid Dynamics: A Practical Guide?explains the basics of the finite element method (FEM) in the context of simple model problems, illustrated by numerical examples. It comprehensively reviews stabilization techniques for convection-dominated transport problems, introducing the reader to streamline diffusion methods, Petrov?Galerkin approximations, Taylor?Galerkin schemes, flux-corrected transport algorithms, and other nonlinear high-resolution schemes, and covers Petrov?Galerkin stabilization, classical projection schemes, Schur complement solvers, and the implementation of the k-epsilon turbulence model in its presentation of the FEM for incompressible flow problem. The book also describes the open-source finite element library ELMER, which is recommended as a software development kit for advanced applications in an online component.?


Book Synopsis Finite Element Methods for Computational Fluid Dynamics by : Dmitri Kuzmin

Download or read book Finite Element Methods for Computational Fluid Dynamics written by Dmitri Kuzmin and published by SIAM. This book was released on 2014-12-18 with total page 321 pages. Available in PDF, EPUB and Kindle. Book excerpt: This informal introduction to computational fluid dynamics and practical guide to numerical simulation of transport phenomena covers the derivation of the governing equations, construction of finite element approximations, and qualitative properties of numerical solutions, among other topics. To make the book accessible to readers with diverse interests and backgrounds, the authors begin at a basic level and advance to numerical tools for increasingly difficult flow problems, emphasizing practical implementation rather than mathematical theory.?Finite Element Methods for Computational Fluid Dynamics: A Practical Guide?explains the basics of the finite element method (FEM) in the context of simple model problems, illustrated by numerical examples. It comprehensively reviews stabilization techniques for convection-dominated transport problems, introducing the reader to streamline diffusion methods, Petrov?Galerkin approximations, Taylor?Galerkin schemes, flux-corrected transport algorithms, and other nonlinear high-resolution schemes, and covers Petrov?Galerkin stabilization, classical projection schemes, Schur complement solvers, and the implementation of the k-epsilon turbulence model in its presentation of the FEM for incompressible flow problem. The book also describes the open-source finite element library ELMER, which is recommended as a software development kit for advanced applications in an online component.?

Automated Solution of Differential Equations by the Finite Element Method

Automated Solution of Differential Equations by the Finite Element Method

Author: Anders Logg

Publisher: Springer Science & Business Media

Published: 2012-02-24

Total Pages: 723

ISBN-13: 3642230997

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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.


Book Synopsis Automated Solution of Differential Equations by the Finite Element Method by : Anders Logg

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.

The Finite Element Method: Theory, Implementation, and Applications

The Finite Element Method: Theory, Implementation, and Applications

Author: Mats G. Larson

Publisher: Springer Science & Business Media

Published: 2013-01-13

Total Pages: 403

ISBN-13: 3642332870

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This book gives an introduction to the finite element method as a general computational method for solving partial differential equations approximately. Our approach is mathematical in nature with a strong focus on the underlying mathematical principles, such as approximation properties of piecewise polynomial spaces, and variational formulations of partial differential equations, but with a minimum level of advanced mathematical machinery from functional analysis and partial differential equations. In principle, the material should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed. Throughout the text we emphasize implementation of the involved algorithms, and have therefore mixed mathematical theory with concrete computer code using the numerical software MATLAB is and its PDE-Toolbox. We have also had the ambition to cover some of the most important applications of finite elements and the basic finite element methods developed for those applications, including diffusion and transport phenomena, solid and fluid mechanics, and also electromagnetics.​


Book Synopsis The Finite Element Method: Theory, Implementation, and Applications by : Mats G. Larson

Download or read book The Finite Element Method: Theory, Implementation, and Applications written by Mats G. Larson and published by Springer Science & Business Media. This book was released on 2013-01-13 with total page 403 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives an introduction to the finite element method as a general computational method for solving partial differential equations approximately. Our approach is mathematical in nature with a strong focus on the underlying mathematical principles, such as approximation properties of piecewise polynomial spaces, and variational formulations of partial differential equations, but with a minimum level of advanced mathematical machinery from functional analysis and partial differential equations. In principle, the material should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed. Throughout the text we emphasize implementation of the involved algorithms, and have therefore mixed mathematical theory with concrete computer code using the numerical software MATLAB is and its PDE-Toolbox. We have also had the ambition to cover some of the most important applications of finite elements and the basic finite element methods developed for those applications, including diffusion and transport phenomena, solid and fluid mechanics, and also electromagnetics.​

A Discrete Exterior Calculus Finite Element Method for Solving Two Phase Flow Problems

A Discrete Exterior Calculus Finite Element Method for Solving Two Phase Flow Problems

Author: Peter Klimas

Publisher:

Published: 2009

Total Pages: 416

ISBN-13: 9780494602607

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Book Synopsis A Discrete Exterior Calculus Finite Element Method for Solving Two Phase Flow Problems by : Peter Klimas

Download or read book A Discrete Exterior Calculus Finite Element Method for Solving Two Phase Flow Problems written by Peter Klimas and published by . This book was released on 2009 with total page 416 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Finite Element Methods for Maxwell's Equations

Finite Element Methods for Maxwell's Equations

Author: Peter Monk

Publisher: Clarendon Press

Published: 2003-04-17

Total Pages: 468

ISBN-13: 0191545228

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Since the middle of the last century, computing power has increased sufficiently that the direct numerical approximation of Maxwell's equations is now an increasingly important tool in science and engineering. Parallel to the increasing use of numerical methods in computational electromagnetism there has also been considerable progress in the mathematical understanding of the properties of Maxwell's equations relevant to numerical analysis. The aim of this book is to provide an up to date and sound theoretical foundation for finite element methods in computational electromagnetism. The emphasis is on finite element methods for scattering problems that involve the solution of Maxwell's equations on infinite domains. Suitable variational formulations are developed and justified mathematically. An error analysis of edge finite element methods that are particularly well suited to Maxwell's equations is the main focus of the book. The methods are justified for Lipschitz polyhedral domains that can cause strong singularities in the solution. The book finishes with a short introduction to inverse problems in electromagnetism.


Book Synopsis Finite Element Methods for Maxwell's Equations by : Peter Monk

Download or read book Finite Element Methods for Maxwell's Equations written by Peter Monk and published by Clarendon Press. This book was released on 2003-04-17 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the middle of the last century, computing power has increased sufficiently that the direct numerical approximation of Maxwell's equations is now an increasingly important tool in science and engineering. Parallel to the increasing use of numerical methods in computational electromagnetism there has also been considerable progress in the mathematical understanding of the properties of Maxwell's equations relevant to numerical analysis. The aim of this book is to provide an up to date and sound theoretical foundation for finite element methods in computational electromagnetism. The emphasis is on finite element methods for scattering problems that involve the solution of Maxwell's equations on infinite domains. Suitable variational formulations are developed and justified mathematically. An error analysis of edge finite element methods that are particularly well suited to Maxwell's equations is the main focus of the book. The methods are justified for Lipschitz polyhedral domains that can cause strong singularities in the solution. The book finishes with a short introduction to inverse problems in electromagnetism.

Numerical Solution of Partial Differential Equations by the Finite Element Method

Numerical Solution of Partial Differential Equations by the Finite Element Method

Author: Claes Johnson

Publisher: Courier Corporation

Published: 2012-05-23

Total Pages: 290

ISBN-13: 0486131599

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An accessible introduction to the finite element method for solving numeric problems, this volume offers the keys to an important technique in computational mathematics. Suitable for advanced undergraduate and graduate courses, it outlines clear connections with applications and considers numerous examples from a variety of science- and engineering-related specialties.This text encompasses all varieties of the basic linear partial differential equations, including elliptic, parabolic and hyperbolic problems, as well as stationary and time-dependent problems. Additional topics include finite element methods for integral equations, an introduction to nonlinear problems, and considerations of unique developments of finite element techniques related to parabolic problems, including methods for automatic time step control. The relevant mathematics are expressed in non-technical terms whenever possible, in the interests of keeping the treatment accessible to a majority of students.


Book Synopsis Numerical Solution of Partial Differential Equations by the Finite Element Method by : Claes Johnson

Download or read book Numerical Solution of Partial Differential Equations by the Finite Element Method written by Claes Johnson and published by Courier Corporation. This book was released on 2012-05-23 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: An accessible introduction to the finite element method for solving numeric problems, this volume offers the keys to an important technique in computational mathematics. Suitable for advanced undergraduate and graduate courses, it outlines clear connections with applications and considers numerous examples from a variety of science- and engineering-related specialties.This text encompasses all varieties of the basic linear partial differential equations, including elliptic, parabolic and hyperbolic problems, as well as stationary and time-dependent problems. Additional topics include finite element methods for integral equations, an introduction to nonlinear problems, and considerations of unique developments of finite element techniques related to parabolic problems, including methods for automatic time step control. The relevant mathematics are expressed in non-technical terms whenever possible, in the interests of keeping the treatment accessible to a majority of students.