Continuous and Discontinuous Galerkin Finite Element Methods for Stochastic Differential Equations PDF Download
Are you looking for read ebook online? Search for your book and save it on your Kindle device, PC, phones or tablets. Download Continuous and Discontinuous Galerkin Finite Element Methods for Stochastic Differential Equations PDF full book. Access full book title Continuous and Discontinuous Galerkin Finite Element Methods for Stochastic Differential Equations by Bryan P. Johnson. Download full books in PDF and EPUB format.
Author: Bryan P. Johnson
Publisher:
ISBN: 9781321664003
Category : Differential equations
Languages : en
Pages : 122
Get Book Here
Book Description
In this thesis, we develop high order numerical methods for strong solution of Itô stochastic differential equations (SDEs). We first construct an approximate deterministic ODE with a random coefficient on each element using the Wong-Zakai approximation theorem. Since the resulting equation converges to the solution of the corresponding Stratonovich SDE, we apply a transformation to the drift term to obtain an ODE which converges to the solution of the original SDE. The corrected equation is then discretized using the standard continuous and discontinuous finite element methods for deterministic ODEs. Our methods are demonstrated to be strongly convergent, accurate, and computationally efficient. More precisely, numerical evidence demonstrate that our proposed continuous and discontinuous finite element methods, respectively, have strong convergence order of p/2 and p, when p-degree piecewise polynomials are used. Several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method.
Author: Bryan P. Johnson
Publisher:
ISBN: 9781321664003
Category : Differential equations
Languages : en
Pages : 122
Get Book Here
Book Description
In this thesis, we develop high order numerical methods for strong solution of Itô stochastic differential equations (SDEs). We first construct an approximate deterministic ODE with a random coefficient on each element using the Wong-Zakai approximation theorem. Since the resulting equation converges to the solution of the corresponding Stratonovich SDE, we apply a transformation to the drift term to obtain an ODE which converges to the solution of the original SDE. The corrected equation is then discretized using the standard continuous and discontinuous finite element methods for deterministic ODEs. Our methods are demonstrated to be strongly convergent, accurate, and computationally efficient. More precisely, numerical evidence demonstrate that our proposed continuous and discontinuous finite element methods, respectively, have strong convergence order of p/2 and p, when p-degree piecewise polynomials are used. Several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method.
Author: Bernardo Cockburn
Publisher: Springer Science & Business Media
ISBN: 3642597211
Category : Mathematics
Languages : en
Pages : 468
Get Book Here
Book Description
A class of finite element methods, the Discontinuous Galerkin Methods (DGM), has been under rapid development recently and has found its use very quickly in such diverse applications as aeroacoustics, semi-conductor device simula tion, turbomachinery, turbulent flows, materials processing, MHD and plasma simulations, and image processing. While there has been a lot of interest from mathematicians, physicists and engineers in DGM, only scattered information is available and there has been no prior effort in organizing and publishing the existing volume of knowledge on this subject. In May 24-26, 1999 we organized in Newport (Rhode Island, USA), the first international symposium on DGM with equal emphasis on the theory, numerical implementation, and applications. Eighteen invited speakers, lead ers in the field, and thirty-two contributors presented various aspects and addressed open issues on DGM. In this volume we include forty-nine papers presented in the Symposium as well as a survey paper written by the organiz ers. All papers were peer-reviewed. A summary of these papers is included in the survey paper, which also provides a historical perspective of the evolution of DGM and its relation to other numerical methods. We hope this volume will become a major reference in this topic. It is intended for students and researchers who work in theory and application of numerical solution of convection dominated partial differential equations. The papers were written with the assumption that the reader has some knowledge of classical finite elements and finite volume methods.
Author: Vít Dolejší
Publisher: Springer
ISBN: 3319192671
Category : Mathematics
Languages : en
Pages : 575
Get Book Here
Book Description
The subject of the book is the mathematical theory of the discontinuous Galerkin method (DGM), which is a relatively new technique for the numerical solution of partial differential equations. The book is concerned with the DGM developed for elliptic and parabolic equations and its applications to the numerical simulation of compressible flow. It deals with the theoretical as well as practical aspects of the DGM and treats the basic concepts and ideas of the DGM, as well as the latest significant findings and achievements in this area. The main benefit for readers and the book’s uniqueness lie in the fact that it is sufficiently detailed, extensive and mathematically precise, while at the same time providing a comprehensible guide through a wide spectrum of discontinuous Galerkin techniques and a survey of the latest efficient, accurate and robust discontinuous Galerkin schemes for the solution of compressible flow.
Author: Xiaobing Feng
Publisher: Springer Science & Business Media
ISBN: 3319018183
Category : Mathematics
Languages : en
Pages : 289
Get Book Here
Book Description
The field of discontinuous Galerkin finite element methods has attracted considerable recent attention from scholars in the applied sciences and engineering. This volume brings together scholars working in this area, each representing a particular theme or direction of current research. Derived from the 2012 Barrett Lectures at the University of Tennessee, the papers reflect the state of the field today and point toward possibilities for future inquiry. The longer survey lectures, delivered by Franco Brezzi and Chi-Wang Shu, respectively, focus on theoretical aspects of discontinuous Galerkin methods for elliptic and evolution problems. Other papers apply DG methods to cases involving radiative transport equations, error estimates, and time-discrete higher order ALE functions, among other areas. Combining focused case studies with longer sections of expository discussion, this book will be an indispensable reference for researchers and students working with discontinuous Galerkin finite element methods and its applications.
Author: Xiaobing Feng
Publisher:
ISBN: 9783319018195
Category :
Languages : en
Pages : 292
Get Book Here
Book Description
Author: Vidar Thomee
Publisher: Springer Science & Business Media
ISBN: 3662033593
Category : Mathematics
Languages : en
Pages : 310
Get Book Here
Book Description
My purpose in this monograph is to present an essentially self-contained account of the mathematical theory of Galerkin finite element methods as applied to parabolic partial differential equations. The emphases and selection of topics reflects my own involvement in the field over the past 25 years, and my ambition has been to stress ideas and methods of analysis rather than to describe the most general and farreaching results possible. Since the formulation and analysis of Galerkin finite element methods for parabolic problems are generally based on ideas and results from the corresponding theory for stationary elliptic problems, such material is often included in the presentation. The basis of this work is my earlier text entitled Galerkin Finite Element Methods for Parabolic Problems, Springer Lecture Notes in Mathematics, No. 1054, from 1984. This has been out of print for several years, and I have felt a need and been encouraged by colleagues and friends to publish an updated version. In doing so I have included most of the contents of the 14 chapters of the earlier work in an updated and revised form, and added four new chapters, on semigroup methods, on multistep schemes, on incomplete iterative solution of the linear algebraic systems at the time levels, and on semilinear equations. The old chapters on fully discrete methods have been reworked by first treating the time discretization of an abstract differential equation in a Hilbert space setting, and the chapter on the discontinuous Galerkin method has been completely rewritten.
Author: Garret Dan Vo
Publisher:
ISBN:
Category : Differential equations, Parabolic
Languages : en
Pages : 208
Get Book Here
Book Description
A number of different discretization techniques and algorithms have been developed for approximating the solution of parabolic partial differential equations. A standard approach, especially for applications that involve complex geometries, is the classic continuous Galerkin finite element method. This approach has a strong theoretical foundation and has been widely and successfully applied to this category of differential equations. One challenging sub-category of problems, however, are equations that include an advection term that is large relative to the second-order, diffusive term. For these advection dominated problems, the continuous Galerkin finite element method can become unstable and yield highly inaccurate results. An alternative to the continuous Galerkin finite element method is the discontinuous Galerkin finite element method, and, through the use of a numerical flux term used in deriving the weak form, the discontinuous approach has the potential to be much more stable in highly advective problems. However, the discontinuous Galerkin finite element method also has significantly more degrees-of-freedom due to the replication of nodes along element edges and vertices. The work presented here compares the computational cost, stability, and accuracy (when possible) of continuous and discontinuous Galerkin finite element methods for four different test problems, including the advection-diffusion equation, viscous Burgers' equation, and the Turing pattern formation equation system. The comparison is performed using as much shared code as possible between the two algorithms and direct, iterative, and multilevel linear solvers. The results show that, for implicit time stepping, the continuous Galerkin finite element method is typically 5-20 times less computationally expensive than the discontinuous Galerkin finite element method using the same finite element mesh and element order. However, the discontinuous Galerkin finite element method is significantly more stable than the continuous Galerkin finite element method for advection dominated problems and is able to obtain accurate approximate solutions for cases where the classic, un-stabilized continuous Galerkin finite element method fails.
Author: Jan S. Hesthaven
Publisher: Springer Science & Business Media
ISBN: 0387720677
Category : Mathematics
Languages : en
Pages : 502
Get Book Here
Book Description
This book offers an introduction to the key ideas, basic analysis, and efficient implementation of discontinuous Galerkin finite element methods (DG-FEM) for the solution of partial differential equations. It covers all key theoretical results, including an overview of relevant results from approximation theory, convergence theory for numerical PDE’s, and orthogonal polynomials. Through embedded Matlab codes, coverage discusses and implements the algorithms for a number of classic systems of PDE’s: Maxwell’s equations, Euler equations, incompressible Navier-Stokes equations, and Poisson- and Helmholtz equations.
Author: Haihang You
Publisher:
ISBN:
Category :
Languages : en
Pages : 112
Get Book Here
Book Description
The Discontinuous Galerkin Method is one variant of the Finite Element Methods for solving partial differential equations, which was first introduced by Reed and Hill in 1970's [27]. Discontinuous Galerkin Method (DGFEM) differs from the standard Galerkin FEM that continuity constraints are not imposed on the inter-element boundaries. It results in a solution which is composed of totally piecewise discontinuous functions. The absence of continuity constraints on the inter-element boundaries implies that DG method has a great deal of flexibility at the cost of increasing the number of degrees of freedom. This flexibility is the source of many but not all of the advantages of the DGFEM method over the Continuous Galerkin (CGFEM) method that uses spaces of continuous piecewise polynomial functions and other "less standard" methods such as nonconforming methods. As DGFEM method leads to bigger system to solve, theoretical and practical approaches to speed it up are our main focus in this dissertation. This research aims at designing and building an adaptive discontinuous Galerkin finite element method to solve partial differential equations with fast time for desired accuracy on modern architecture.
Author: Vidar Thomée
Publisher: Springer Science & Business Media
ISBN: 9783540632368
Category :
Languages : en
Pages : 320
Get Book Here
Book Description
