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Author: Cheherazada González Aguayo Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
This thesis deals with the solution of the Laplace and heat equations on complicated domains. The approach follows the idea of the fictitious domain method, in which a larger (simpler) domain is introduced with the idea of avoiding the use of meshes that resolve the geometry. The first part of the thesis is dedicated to propose and analyse a new stabilised finite element method for the heat equation. The analysis, not available to date, is based on the introduction of a new projected initial condition that satisfies the boundary conditions of the original problem weakly. This allows us to prove inconditional stability and optimal convergence of the solution, thus avoiding the restriction linking the time discretisation and mesh width parameters present in previous references. In the second part of this thesis the methodology has been adapted and extended to cover the case in which the problem at hand is posed in a domain containing several inclusions of small size. For this case, the usual fictitious domain approach is no longer applicable, and then a new method that compensates for the lack of stability of the original one is proposed, analysed and tested numerically. The numerical analysis has been carried out for the steady state case, but its applicability to time dependent problems is sketched and shown by means of numerical experiments.
Author: Cheherazada González Aguayo Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
This thesis deals with the solution of the Laplace and heat equations on complicated domains. The approach follows the idea of the fictitious domain method, in which a larger (simpler) domain is introduced with the idea of avoiding the use of meshes that resolve the geometry. The first part of the thesis is dedicated to propose and analyse a new stabilised finite element method for the heat equation. The analysis, not available to date, is based on the introduction of a new projected initial condition that satisfies the boundary conditions of the original problem weakly. This allows us to prove inconditional stability and optimal convergence of the solution, thus avoiding the restriction linking the time discretisation and mesh width parameters present in previous references. In the second part of this thesis the methodology has been adapted and extended to cover the case in which the problem at hand is posed in a domain containing several inclusions of small size. For this case, the usual fictitious domain approach is no longer applicable, and then a new method that compensates for the lack of stability of the original one is proposed, analysed and tested numerically. The numerical analysis has been carried out for the steady state case, but its applicability to time dependent problems is sketched and shown by means of numerical experiments.
Author: Stéphane P. A. Bordas Publisher: Springer ISBN: 3319714317 Category : Mathematics Languages : en Pages : 371
Book Description
This book provides a snapshot of the state of the art of the rapidly evolving field of integration of geometric data in finite element computations. The contributions to this volume, based on research presented at the UCL workshop on the topic in January 2016, include three review papers on core topics such as fictitious domain methods for elasticity, trace finite element methods for partial differential equations defined on surfaces, and Nitsche’s method for contact problems. Five chapters present original research articles on related theoretical topics, including Lagrange multiplier methods, interface problems, bulk-surface coupling, and approximation of partial differential equations on moving domains. Finally, two chapters discuss advanced applications such as crack propagation or flow in fractured poroelastic media. This is the first volume that provides a comprehensive overview of the field of unfitted finite element methods, including recent techniques such as cutFEM, traceFEM, ghost penalty, and augmented Lagrangian techniques. It is aimed at researchers in applied mathematics, scientific computing or computational engineering.
Author: Giulio Ventura Publisher: Springer ISBN: 3319412469 Category : Technology & Engineering Languages : en Pages : 272
Book Description
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: Wolfgang Bangerth Publisher: Springer Science & Business Media ISBN: 9783764370091 Category : Mathematics Languages : en Pages : 222
Book Description
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.
Author: Michel Krizek Publisher: Routledge ISBN: 1351448609 Category : Mathematics Languages : en Pages : 370
Book Description
""Based on the proceedings of the first conference on superconvergence held recently at the University of Jyvaskyla, Finland. Presents reviewed papers focusing on superconvergence phenomena in the finite element method. Surveys for the first time all known superconvergence techniques, including their proofs.
Author: Alexandre Ern Publisher: Springer Nature ISBN: 3030569233 Category : Mathematics Languages : en Pages : 491
Book Description
This book is the second volume of a three-part textbook suitable for graduate coursework, professional engineering and academic research. It is also appropriate for graduate flipped classes. Each volume is divided into short chapters. Each chapter can be covered in one teaching unit and includes exercises as well as solutions available from a dedicated website. The salient ideas can be addressed during lecture, with the rest of the content assigned as reading material. To engage the reader, the text combines examples, basic ideas, rigorous proofs, and pointers to the literature to enhance scientific literacy. Volume II is divided into 32 chapters plus one appendix. The first part of the volume focuses on the approximation of elliptic and mixed PDEs, beginning with fundamental results on well-posed weak formulations and their approximation by the Galerkin method. The material covered includes key results such as the BNB theorem based on inf-sup conditions, Céa's and Strang's lemmas, and the duality argument by Aubin and Nitsche. Important implementation aspects regarding quadratures, linear algebra, and assembling are also covered. The remainder of Volume II focuses on PDEs where a coercivity property is available. It investigates conforming and nonconforming approximation techniques (Galerkin, boundary penalty, Crouzeix—Raviart, discontinuous Galerkin, hybrid high-order methods). These techniques are applied to elliptic PDEs (diffusion, elasticity, the Helmholtz problem, Maxwell's equations), eigenvalue problems for elliptic PDEs, and PDEs in mixed form (Darcy and Stokes flows). Finally, the appendix addresses fundamental results on the surjectivity, bijectivity, and coercivity of linear operators in Banach spaces.
Author: Cheherazada González Aguayo
Author: Cheherazada González Aguayo
Author: Stéphane P. A. Bordas
Author: Erik Burman
Author: Erik Burman
Author: Giulio Ventura
Author: Wolfgang Bangerth
Author: Michel Krizek
Author: Alexandre Ern
Author: Korinna Rosin
Author: Sven Groß