Journal ArticleDOI
Adaptive finite element methods for parabolic problems II: optimal error estimates in L ∞ L 2 and L ∞ L ∞
Kenneth Eriksson,Claes Johnson +1 more
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Optimal error estimates are derived for a complete discretization of linear parabolic problems using space–time finite elements based on the orthogonality of the Galerkin procedure and the use of strong stability estimates.Abstract:
Optimal error estimates are derived for a complete discretization of linear parabolic problems using space–time finite elements. The discretization is done first in time using the discontinuous Galerkin method and then in space using the standard Galerkin method. The underlying partitions in time and space need not be quasi uniform and the partition in space may be changed from time step to time step. The error bounds show, in particular, that the error may be controlled globally in time on a given tolerance level by controlling the discretization error on each individual time step on the same (given) level, i.e., without error accumulation effects. The derivation of the estimates is based on the orthogonality of the Galerkin procedure and the use of strong stability estimates. The particular and precise form of these error estimates makes it possible to design efficient adaptive methods with reliable automatic error control for parabolic problems in the norms under consideration.read more
Citations
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Book
A Posteriori Error Estimation in Finite Element Analysis
Mark Ainsworth,J. Tinsley Oden +1 more
TL;DR: In this paper, a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics is presented, focusing on methods for linear elliptic boundary value problems.
Journal ArticleDOI
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
Bernardo Cockburn,Chi-Wang Shu +1 more
TL;DR: It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case and in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations.
Journal ArticleDOI
An optimal control approach to a posteriori error estimation in finite element methods
Roland Becker,Rolf Rannacher +1 more
TL;DR: The ‘dual-weighted-residual method’ is introduced initially within an abstract functional analytic setting, and is then developed in detail for several model situations featuring the characteristic properties of elliptic, parabolic and hyperbolic problems.
Book ChapterDOI
The Development of Discontinuous Galerkin Methods
TL;DR: An overview of the evolution of the discontinuous Galerkin methods since their introduction in 1973 by Reed and Hill, in the framework of neutron transport, until their most recent developments is presented.
Journal ArticleDOI
Introduction to Adaptive Methods for Differential Equations
TL;DR: The Differential Calculus can be solved by a common method (Gottfried Wilhelm von Leibniz, 1646-1719) as mentioned in this paper, which is known as the Differential Algorithm of this calculus.
References
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Book
The Finite Element Method for Elliptic Problems
Philippe G. Ciarlet,J. T. Oden +1 more
TL;DR: The finite element method has been applied to a variety of nonlinear problems, e.g., Elliptic boundary value problems as discussed by the authors, plate problems, and second-order problems.
Book
Galerkin Finite Element Methods for Parabolic Problems
TL;DR: The standard Galerkin method is based on more general approximations of the elliptic problem as discussed by the authors, and is used to solve problems in algebraic systems at the time level.
Journal ArticleDOI
Adaptive finite element methods for parabolic problems. I.: a linear model problem
Kenneth Eriksson,Claes Johnson +1 more
TL;DR: In this paper, an adaptive finite element method for parabolic problems is presented and analyzed for choosing the space of a finite element in the space a.k.a. the optimal solution.
Journal ArticleDOI
Adaptive finite element methods for parabolic problems IV: nonlinear problems
Kenneth Eriksson,Claes Johnson +1 more
TL;DR: A posteriori error estimates are proved, corresponding adaptive algorithms are designed, and some numerical results are presented on adaptive finite element methods for parabolic problems to a class of nonlinear scalar problems.
Journal ArticleDOI
Time discretization of parabolic problems by the discontinuous Galerkin method
TL;DR: Analyse dans un contexte general de la methode de Galerkin discontinue pour la discretisation temporelle de problemes de type parabolique as mentioned in this paper.