An Analysis of a Finite Element Method for Convection-Diffusion Problems. Part I. Quasi-Optimality PDF Download
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Author: W. G. Szymczak Publisher: ISBN: Category : Languages : en Pages : 61
Book Description
A detailed analysis is performed for a finite element method applied to the general one-dimensional convection diffusion problem. Piecewise polynomials are used for the trial space. The test space is formed by locally projecting L-spline basis functions onto upwinded polynomials. The error is measured in the LP mesh dependent norm. The method is proven to be quasi-optimal (yielding nearly the best approximation from the trial space), provided that the input data is piecewise smooth. This assumption is usually observed in practice. These results are used to establish a posteriori error estimates and an adaptive mesh refinement strategy in Part II of this series (35). (Author).
Author: W. G. Szymczak Publisher: ISBN: Category : Languages : en Pages : 61
Book Description
A detailed analysis is performed for a finite element method applied to the general one-dimensional convection diffusion problem. Piecewise polynomials are used for the trial space. The test space is formed by locally projecting L-spline basis functions onto upwinded polynomials. The error is measured in the LP mesh dependent norm. The method is proven to be quasi-optimal (yielding nearly the best approximation from the trial space), provided that the input data is piecewise smooth. This assumption is usually observed in practice. These results are used to establish a posteriori error estimates and an adaptive mesh refinement strategy in Part II of this series (35). (Author).
Author: W. G. Szymczak Publisher: ISBN: Category : Languages : en Pages : 51
Book Description
A posteriori error estimates are derived for the finite element method presented in Part I. These estimates are proven to have the property that the effectivity index theta = (error estimate/true error) converges to one as the maximum mesh size goes to zero. An adaptive mesh refinement strategy is based on equilibriating local error indicators whose sum comprises the global error estimate. Numerical results show that theta is nearly one even on coarse meshes, and that optimal meshes are created by the adaptive procedure. The successful solution of a non linear problem-modelling flow through an expanding duct, makes evident the robustness of the method. (Author).
Author: I. Babuska Publisher: ISBN: Category : Languages : en Pages : 44
Book Description
This paper analyzes the finite element method applied to a convection diffusion model problem. Linear elements are used for the trial space. The error is measured in a norm closely related to the L sup P norm. When the test space is composed of linear elements with parabolic upwinding, the method is shown to be optimal when the input data is piecewise smooth -- a condition which is usually observed in practice. Without these smoothness assumptions, the method is shown to be non-optimal, even if the class of test spaces is extended to include any elements which have a shape independent of the mesh size. (Author).
Author: Institute of Mathematics and Its Applications Publisher: Oxford University Press, USA ISBN: Category : Mathematics Languages : en Pages : 208
Book Description
Combining theoretical insights with practical applications, this stimulating collection provides a state-of-the-art survey of the finite element method, one of the most powerful tools available for the solution of physical problems. Written by leading experts, this volume consider such topics as parabolic Galerkin methods, nonconforming elements, the treatment of singularities in elliptic boundary value problems, and conforming methods for self-adjount elliptic problems. This will be an invaluable basic reference for computational mathematicians and engineers who use finite element methods in academic or industrial research.
Author: Martin Stynes Publisher: American Mathematical Soc. ISBN: 1470448688 Category : Mathematics Languages : en Pages : 168
Book Description
Many physical problems involve diffusive and convective (transport) processes. When diffusion dominates convection, standard numerical methods work satisfactorily. But when convection dominates diffusion, the standard methods become unstable, and special techniques are needed to compute accurate numerical approximations of the unknown solution. This convection-dominated regime is the focus of the book. After discussing at length the nature of solutions to convection-dominated convection-diffusion problems, the authors motivate and design numerical methods that are particularly suited to this class of problems. At first they examine finite-difference methods for two-point boundary value problems, as their analysis requires little theoretical background. Upwinding, artificial diffusion, uniformly convergent methods, and Shishkin meshes are some of the topics presented. Throughout, the authors are concerned with the accuracy of solutions when the diffusion coefficient is close to zero. Later in the book they concentrate on finite element methods for problems posed in one and two dimensions. This lucid yet thorough account of convection-dominated convection-diffusion problems and how to solve them numerically is meant for beginning graduate students, and it includes a large number of exercises. An up-to-date bibliography provides the reader with further reading.
Author: W. G. Szymczak
Author: W. G. Szymczak
Author: W. G. Szymczak
Author: 
Author: I. Babuska
Author: Thomas J. R. Hughes
Author: Institute of Mathematics and Its Applications
Author: Martin Stynes
Author: Chalmers University of Technology. Dept. of Computer Sciences
Author: John Tinsley Oden
Author: Vladimir V. Akimov