The Mimetic Finite Difference Method For Elliptic Problems

Author : Lourenco Beirao da Veiga,Konstantin Lipnikov,Gianmarco Manzini
Publisher : Springer
Page : 399 pages
File Size : 44,5 Mb
Release : 2014-05-22
Category : Mathematics
ISBN : 9783319026633

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The Mimetic Finite Difference Method for Elliptic Problems by Lourenco Beirao da Veiga,Konstantin Lipnikov,Gianmarco Manzini Pdf

This book describes the theoretical and computational aspects of the mimetic finite difference method for a wide class of multidimensional elliptic problems, which includes diffusion, advection-diffusion, Stokes, elasticity, magnetostatics and plate bending problems. The modern mimetic discretization technology developed in part by the Authors allows one to solve these equations on unstructured polygonal, polyhedral and generalized polyhedral meshes. The book provides a practical guide for those scientists and engineers that are interested in the computational properties of the mimetic finite difference method such as the accuracy, stability, robustness, and efficiency. Many examples are provided to help the reader to understand and implement this method. This monograph also provides the essential background material and describes basic mathematical tools required to develop further the mimetic discretization technology and to extend it to various applications.

Author : Bernd Heinrich
Publisher : Birkhauser
Page : 216 pages
File Size : 48,6 Mb
Release : 1987
Category : Boundary value problems
ISBN : PSU:000014221729

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Finite Difference Methods on Irregular Networks by Bernd Heinrich Pdf

Author : Gabriel R. Barrenechea,Franco Brezzi,Andrea Cangiani,Emmanuil H. Georgoulis
Publisher : Springer
Page : 433 pages
File Size : 40,5 Mb
Release : 2016-10-03
Category : Computers
ISBN : 9783319416403

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Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations by Gabriel R. Barrenechea,Franco Brezzi,Andrea Cangiani,Emmanuil H. Georgoulis Pdf

This volume contains contributed survey papers from the main speakers at the LMS/EPSRC Symposium “Building bridges: connections and challenges in modern approaches to numerical partial differential equations”. This meeting took place in July 8-16, 2014, and its main purpose was to gather specialists in emerging areas of numerical PDEs, and explore the connections between the different approaches. The type of contributions ranges from the theoretical foundations of these new techniques, to the applications of them, to new general frameworks and unified approaches that can cover one, or more than one, of these emerging techniques.

Author : Zi-cai Li
Publisher : World Scientific
Page : 280 pages
File Size : 48,9 Mb
Release : 1990-12-27
Category : Mathematics
ISBN : 9789814506809

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Numerical Methods For Elliptic Problems With Singularities: Boundary Mtds And Nonconforming Combinatn by Zi-cai Li Pdf

This book presents two kinds of numerical methods for solving elliptic boundary value problems with singularities. Part I gives the boundary methods which use analytic and singular expansions, and Part II the nonconforming methods combining finite element methods (FEM) (or finite difference methods (FDM)) and singular (or analytic) expansions. The advantage of these methods over the standard FEM and FDM is that they can cope with complicated geometrical boundaries and boundary conditions as well as singularity. Therefore, accurate numerical solutions near singularities can be obtained. The description of methods, error bounds, stability analysis and numerical experiments are provided for the typical problems with angular, interface and infinity singularities. However, the approximate techniques and coupling strategy given can be applied to solving other PDE and engineering problems with singularities as well. This book is derived from the author's Ph. D. thesis which won the 1987 best doctoral dissertation award given by the Canadian Applied Mathematics Society.

Author : Clément Cancès,Pascal Omnes
Publisher : Springer
Page : 559 pages
File Size : 43,9 Mb
Release : 2017-05-22
Category : Mathematics
ISBN : 9783319573946

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Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems by Clément Cancès,Pascal Omnes Pdf

This book is the second volume of proceedings of the 8th conference on "Finite Volumes for Complex Applications" (Lille, June 2017). It includes reviewed contributions reporting successful applications in the fields of fluid dynamics, computational geosciences, structural analysis, nuclear physics, semiconductor theory and other topics. The finite volume method in its various forms is a space discretization technique for partial differential equations based on the fundamental physical principle of conservation, and recent decades have brought significant advances in the theoretical understanding of the method. Many finite volume methods preserve further qualitative or asymptotic properties, including maximum principles, dissipativity, monotone decay of free energy, and asymptotic stability. 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 l evel. This structural approach to the discretization of partial differential equations becomes particularly important for multiphysics and multiscale applications. The book is useful for researchers, PhD and master’s level students in numerical analysis, scientific computing and related fields such as partial differential equations, as well as for engineers working in numerical modeling and simulations.

Author : Giulio Ventura,Elena Benvenuti
Publisher : Springer
Page : 269 pages
File Size : 46,9 Mb
Release : 2016-08-24
Category : Technology & Engineering
ISBN : 9783319412467

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Advances in Discretization Methods by Giulio Ventura,Elena Benvenuti Pdf

This book gathers selected contributions on emerging research work presented at the International Conference eXtended Discretization MethodS (X-DMS), held in Ferrara in September 2015. It highlights the most relevant advances made at the international level in the context of expanding classical discretization methods, like finite elements, to the numerical analysis of a variety of physical problems. The improvements are intended to achieve higher computational efficiency and to account for special features of the solution directly in the approximation space and/or in the discretization procedure. The methods described include, among others, partition of unity methods (meshfree, XFEM, GFEM), virtual element methods, fictitious domain methods, and special techniques for static and evolving interfaces. The uniting feature of all contributions is the direct link between computational methodologies and their application to different engineering areas.

Author : Jürgen Fuhrmann,Mario Ohlberger,Christian Rohde
Publisher : Springer
Page : 450 pages
File Size : 48,7 Mb
Release : 2014-05-12
Category : Mathematics
ISBN : 9783319056845

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Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects by Jürgen Fuhrmann,Mario Ohlberger,Christian Rohde Pdf

The first volume of the proceedings of the 7th conference on "Finite Volumes for Complex Applications" (Berlin, June 2014) covers topics that include convergence and stability analysis, as well as investigations of these methods from the point of view of compatibility with physical principles. It collects together the focused invited papers, as well as the reviewed contributions from internationally leading researchers in the field of analysis of finite volume and related methods. Altogether, a rather comprehensive overview is given of the state of the art in the field. The finite volume method in its various forms is a space 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 preserve further qualitative or asymptotic properties, including maximum principles, dissipativity, monotone decay of free energy, and asymptotic stability. 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. Researchers, PhD and masters level students in numerical analysis, scientific computing and related fields such as partial differential equations will find this volume useful, as will engineers working in numerical modeling and simulations.

Author : Paola F. Antonietti,Lourenço Beirão da Veiga,Gianmarco Manzini
Publisher : Springer Nature
Page : 621 pages
File Size : 53,6 Mb
Release : 2022-10-08
Category : Mathematics
ISBN : 9783030953195

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The Virtual Element Method and its Applications by Paola F. Antonietti,Lourenço Beirão da Veiga,Gianmarco Manzini Pdf

The purpose of this book is to present the current state of the art of the Virtual Element Method (VEM) by collecting contributions from many of the most active researchers in this field and covering a broad range of topics: from the mathematical foundation to real life computational applications. The book is naturally divided into three parts. The first part of the book presents recent advances in theoretical and computational aspects of VEMs, discussing the generality of the meshes suitable to the VEM, the implementation of the VEM for linear and nonlinear PDEs, and the construction of discrete hessian complexes. The second part of the volume discusses Virtual Element discretization of paradigmatic linear and non-linear partial differential problems from computational mechanics, fluid dynamics, and wave propagation phenomena. Finally, the third part contains challenging applications such as the modeling of materials with fractures, magneto-hydrodynamics phenomena and contact solid mechanics. The book is intended for graduate students and researchers in mathematics and engineering fields, interested in learning novel numerical techniques for the solution of partial differential equations. It may as well serve as useful reference material for numerical analysts practitioners of the field.

Author : Jian Li,Xiaolin Lin,Zhangxing Chen
Publisher : Springer Nature
Page : 129 pages
File Size : 46,8 Mb
Release : 2022-01-20
Category : Science
ISBN : 9783030946364

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Finite Volume Methods for the Incompressible Navier–Stokes Equations by Jian Li,Xiaolin Lin,Zhangxing Chen Pdf

The book aims to provide a comprehensive understanding of the most recent developments in finite volume methods. Its focus is on the development and analysis of these methods for the two- and three-dimensional Navier-Stokes equations, supported by extensive numerical results. It covers the most used lower-order finite element pairs, with well-posedness and optimal analysis for these finite volume methods.The authors have attempted to make this book self-contained by offering complete proofs and theoretical results. While most of the material presented has been taught by the authors in a number of institutions over the past several years, they also include several updated theoretical results for the finite volume methods for the incompressible Navier-Stokes equations. This book is primarily developed to address research needs for students and academic and industrial researchers. It is particularly valuable as a research reference in the fields of engineering, mathematics, physics, and computer sciences.

Author : Jaroslav Fořt,Jiří Fürst,Jan Halama,Raphaèle Herbin,Florence Hubert
Publisher : Springer Science & Business Media
Page : 1003 pages
File Size : 45,9 Mb
Release : 2011-07-21
Category : Mathematics
ISBN : 9783642206719

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Finite Volumes for Complex Applications VI Problems & Perspectives by Jaroslav Fořt,Jiří Fürst,Jan Halama,Raphaèle Herbin,Florence Hubert Pdf

Finite volume methods are used for various applications in fluid dynamics, magnetohydrodynamics, structural analysis or nuclear physics. A closer look reveals many interesting phenomena and mathematical or numerical difficulties, such as true error analysis and adaptivity, modelling of multi-phase phenomena or fitting problems, stiff terms in convection/diffusion equations and sources. To overcome existing problems and to find solution methods for future applications requires many efforts and always new developments. The goal of The International Symposium on Finite Volumes for Complex Applications VI is to bring together mathematicians, physicists and engineers dealing with Finite Volume Techniques in a wide context. This book, divided in two volumes, brings a critical look at the subject (new ideas, limits or drawbacks of methods, theoretical as well as applied topics).

Author : Daniele Antonio Di Pietro,Luca Formaggia,Roland Masson
Publisher : Springer Nature
Page : 342 pages
File Size : 53,6 Mb
Release : 2021-06-14
Category : Mathematics
ISBN : 9783030693633

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Polyhedral Methods in Geosciences by Daniele Antonio Di Pietro,Luca Formaggia,Roland Masson Pdf

The last few years have witnessed a surge in the development and usage of discretization methods supporting general meshes in geoscience applications. The need for general polyhedral meshes in this context can arise in several situations, including the modelling of petroleum reservoirs and basins, CO2 and nuclear storage sites, etc. In the above and other situations, classical discretization methods are either not viable or require ad hoc modifications that add to the implementation complexity. Discretization methods able to operate on polyhedral meshes and possibly delivering arbitrary-order approximations constitute in this context a veritable technological jump. The goal of this monograph is to establish a state-of-the-art reference on polyhedral methods for geoscience applications by gathering contributions from top-level research groups working on this topic. This book is addressed to graduate students and researchers wishing to deepen their knowledge of advanced numerical methods with a focus on geoscience applications, as well as practitioners of the field.

Author : Jérôme Droniou,Robert Eymard,Thierry Gallouët,Cindy Guichard,Raphaèle Herbin
Publisher : Springer
Page : 497 pages
File Size : 44,5 Mb
Release : 2018-07-31
Category : Mathematics
ISBN : 9783319790428

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The Gradient Discretisation Method by Jérôme Droniou,Robert Eymard,Thierry Gallouët,Cindy Guichard,Raphaèle Herbin Pdf

This monograph presents the Gradient Discretisation Method (GDM), which is a unified convergence analysis framework for numerical methods for elliptic and parabolic partial differential equations. The results obtained by the GDM cover both stationary and transient models; error estimates are provided for linear (and some non-linear) equations, and convergence is established for a wide range of fully non-linear models (e.g. Leray–Lions equations and degenerate parabolic equations such as the Stefan or Richards models). The GDM applies to a diverse range of methods, both classical (conforming, non-conforming, mixed finite elements, discontinuous Galerkin) and modern (mimetic finite differences, hybrid and mixed finite volume, MPFA-O finite volume), some of which can be built on very general meshes.span style="" ms="" mincho";mso-bidi-font-family:="" the="" core="" properties="" and="" analytical="" tools="" required="" to="" work="" within="" gdm="" are="" stressed,="" it="" is="" shown="" that="" scheme="" convergence="" can="" often="" be="" established="" by="" verifying="" a="" small="" number="" of="" properties.="" scope="" some="" featured="" techniques="" results,="" such="" as="" time-space="" compactness="" theorems="" (discrete="" aubin–simon,="" discontinuous="" ascoli–arzela),="" goes="" beyond="" gdm,="" making="" them="" potentially="" applicable="" numerical="" schemes="" not="" (yet)="" known="" fit="" into="" this="" framework.span style="font-family:" ms="" mincho";mso-bidi-font-family:="" this="" monograph="" is="" intended="" for="" graduate="" students,="" researchers="" and="" experts="" in="" the="" field="" of="" numerical="" analysis="" partial="" differential="" equations./ppiiiiibr/i/i/i/i/i/p

Author : Daniele Antonio Di Pietro,Alexandre Ern,Luca Formaggia
Publisher : Springer
Page : 312 pages
File Size : 51,7 Mb
Release : 2018-10-12
Category : Mathematics
ISBN : 9783319946764

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Numerical Methods for PDEs by Daniele Antonio Di Pietro,Alexandre Ern,Luca Formaggia Pdf

This volume gathers contributions from participants of the Introductory School and the IHP thematic quarter on Numerical Methods for PDE, held in 2016 in Cargese (Corsica) and Paris, providing an opportunity to disseminate the latest results and envisage fresh challenges in traditional and new application fields. Numerical analysis applied to the approximate solution of PDEs is a key discipline in applied mathematics, and over the last few years, several new paradigms have appeared, leading to entire new families of discretization methods and solution algorithms. This book is intended for researchers in the field.

Author : Assyr Abdulle,Simone Deparis,Daniel Kressner,Fabio Nobile,Marco Picasso
Publisher : Springer
Page : 810 pages
File Size : 43,8 Mb
Release : 2014-11-25
Category : Computers
ISBN : 9783319107059

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Numerical Mathematics and Advanced Applications - ENUMATH 2013 by Assyr Abdulle,Simone Deparis,Daniel Kressner,Fabio Nobile,Marco Picasso Pdf

This book gathers a selection of invited and contributed lectures from the European Conference on Numerical Mathematics and Advanced Applications (ENUMATH) held in Lausanne, Switzerland, August 26-30, 2013. It provides an overview of recent developments in numerical analysis, computational mathematics and applications from leading experts in the field. New results on finite element methods, multiscale methods, numerical linear algebra and discretization techniques for fluid mechanics and optics are presented. As such, the book offers a valuable resource for a wide range of readers looking for a state-of-the-art overview of advanced techniques, algorithms and results in numerical mathematics and scientific computing.

Author : Andrea Cangiani,Zhaonan Dong,Emmanuil H. Georgoulis,Paul Houston
Publisher : Springer
Page : 131 pages
File Size : 41,5 Mb
Release : 2017-11-27
Category : Mathematics
ISBN : 9783319676739

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hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes by Andrea Cangiani,Zhaonan Dong,Emmanuil H. Georgoulis,Paul Houston Pdf

Over the last few decades discontinuous Galerkin finite element methods (DGFEMs) have been witnessed tremendous interest as a computational framework for the numerical solution of partial differential equations. Their success is due to their extreme versatility in the design of the underlying meshes and local basis functions, while retaining key features of both (classical) finite element and finite volume methods. Somewhat surprisingly, DGFEMs on general tessellations consisting of polygonal (in 2D) or polyhedral (in 3D) element shapes have received little attention within the literature, despite the potential computational advantages. This volume introduces the basic principles of hp-version (i.e., locally varying mesh-size and polynomial order) DGFEMs over meshes consisting of polygonal or polyhedral element shapes, presents their error analysis, and includes an extensive collection of numerical experiments. The extreme flexibility provided by the locally variable elemen t-shapes, element-sizes, and element-orders is shown to deliver substantial computational gains in several practical scenarios.