Error Analysis of the Immersed Interface Method for Elliptic Problems with an Interface
Author : Rui Hu
Publisher :
Page : 84 pages
File Size : 41,94 MB
Release : 2017
Category :
ISBN :
Author : Rui Hu
Publisher :
Page : 84 pages
File Size : 41,94 MB
Release : 2017
Category :
ISBN :
Author : Zhilin Li
Publisher : SIAM
Page : 348 pages
File Size : 17,44 MB
Release : 2006-01-01
Category : Mathematics
ISBN : 9780898717464
This book provides an introduction to the immersed interface method (IIM), a powerful numerical method for solving interface problems and problems defined on irregular domains for which analytic solutions are rarely available. This book gives a complete description of the IIM, discusses recent progress in the area, and describes numerical methods for a number of classic interface problems. It also contains many numerical examples that can be used as benchmark problems for numerical methods designed for interface problems on irregular domains.
Author : Zhilin Li
Publisher : SIAM
Page : 343 pages
File Size : 20,24 MB
Release : 2006-07-01
Category : Mathematics
ISBN : 0898716098
"This book will be a useful resource for mathematicians, numerical analysts, engineers, graduate students, and anyone who uses numerical methods to solve computational problems, particularly problems with fixed and moving interfaces, free boundary problems, and problems on regular domains."--BOOK JACKET.
Author :
Publisher :
Page : pages
File Size : 13,17 MB
Release : 2004
Category :
ISBN :
The purpose of the research has been to develop a class of new finite-element methods, called immersed-interface finite-element methods, to solve elliptic and elasticity interface problems with homogeneous and non-homogeneous jump conditions. Simple non-body-fitted meshes are used. Single functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface. With such functions, the discontinuities across the interface in the solution and flux are removed; and equivalent elliptic and elasticity interface problems with homogeneous jump conditions are formulated. Special finite-element basis functions are constructed for nodal points near the interface to satisfy the homogeneous jump conditions. Error analysis and numerical tests are presented to demonstrate that such methods have an optimal convergence rate. These methods are designed as an efficient component of the finite-element level-set methodology for fast simulation of interface dynamics that does not require re-meshing. Such simulation has been a powerful numerical approach in understanding material properties, biological processes, and many other important phenomena in science and engineering.
Author : Luca Heltai
Publisher :
Page : 212 pages
File Size : 18,60 MB
Release : 2016
Category :
ISBN :
When solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using a uniform background discretisation, not aligned with the interface itself. Optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface, at the cost of a higher complexity in the implementation. A much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions, as done in immersed boundary methods. These methods are known to have inferior convergence properties, depending on the global regularity of the solution across the interface, when compared to enriched methods. In this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon, restricted to a small neighbourhood of the interface. In particular we show that optimal approximations can be constructed in a natural and inexpensive way, simply by reformulating the problem in a distributionally consistent way, and by resorting to weighted norms when computing the global error of the approximation.
Author : Peter Minev
Publisher : Nova Publishers
Page : 312 pages
File Size : 34,80 MB
Release : 2001
Category : Computers
ISBN : 9781590330272
Scientific Computing & Applications
Author : Alois Kufner
Publisher :
Page : 130 pages
File Size : 24,36 MB
Release : 1985-07-23
Category : Mathematics
ISBN :
A systematic account of the subject, this book deals with properties and applications of the Sobolev spaces with weights, the weight function being dependent on the distance of a point of the definition domain from the boundary of the domain or from its parts. After an introduction of definitions, examples and auxilliary results, it describes the study of properties of Sobolev spaces with power-type weights, and analogous problems for weights of a more general type. The concluding chapter addresses applications of weighted spaces to the solution of the Dirichlet problem for an elliptic linear differential operator.
Author : Dietrich Braess
Publisher : Cambridge University Press
Page : 348 pages
File Size : 44,26 MB
Release : 2007-04-12
Category : Mathematics
ISBN : 113946146X
This definitive introduction to finite element methods was thoroughly updated for this 2007 third edition, which features important material for both research and application of the finite element method. The discussion of saddle-point problems is a highlight of the book and has been elaborated to include many more nonstandard applications. The chapter on applications in elasticity now contains a complete discussion of locking phenomena. The numerical solution of elliptic partial differential equations is an important application of finite elements and the author discusses this subject comprehensively. These equations are treated as variational problems for which the Sobolev spaces are the right framework. Graduate students who do not necessarily have any particular background in differential equations, but require an introduction to finite element methods will find this text invaluable. Specifically, the chapter on finite elements in solid mechanics provides a bridge between mathematics and engineering.
Author : Zhilin Li
Publisher : Cambridge University Press
Page : 305 pages
File Size : 17,36 MB
Release : 2017-11-30
Category : Mathematics
ISBN : 1107163226
A practical and concise guide to finite difference and finite element methods. Well-tested MATLAB® codes are available online.
Author : Rüdiger Verfürth
Publisher : Oxford University Press
Page : 414 pages
File Size : 26,96 MB
Release : 2013-04-18
Category : Mathematics
ISBN : 0199679428
A posteriori error estimation techniques are fundamental to the efficient numerical solution of PDEs arising in physical and technical applications. This book gives a unified approach to these techniques and guides graduate students, researchers, and practitioners towards understanding, applying and developing self-adaptive discretization methods.