Suli, E. and Houston, P. (1997). Finite element methods for hyperbolic problems: a posteriori error analysis and adaptivity. In: I.S. Duff and G.A. Watson, eds. The State of the Art in Numerical Analysis, Clarendon Press, Oxford, 441--471. 71  Home/Search   Document Details and Download   Summary   Related Articles   Check

....algorithm requires some form of error information to guide the positioning of the degrees of freedom. This is not an issue which will be addressed in this paper however, other than to note that such information may take the form of a formal estimate of the error in a computed solution (as in [1, 4, 34] for example) or else may simply be some form of error indicator based upon derivatives of dependent variables for example (as in [9, 28] For steady problems we will assume that a local error estimate or indicator is available whenever a solution has been computed, and for transient problems ....

E. Suli and P. Houston (1997), Finite Element Methods for Hyperbolic Problems: A Posteriori Error Analysis and Adaptivity. In State of the Art in Numerical Analysis (ed. I. Duff and G.A. Watson), OUP, pp. 441--471.

....in u h n across each edge of Omega e . This is an example of one of the simplest a posteriori estimates for the size of the error j u Gamma u h j H 1( Omega Gamma . Many other, more complex and more general, algorithms have been proposed but these are outside the scope of this work (see [1, 7, 24, 25, 76, 96] for a number of examples however) 2.2 Different Types of Refinements There are two popular classes of refinement algorithm that may be associated with unstructured meshes, namely regeneration schemes and mesh adaptation schemes. Below we describe these schemes briefly. 2.2.1 Regeneration ....

E Suli and P Houston. Finite element methods for hyperbolic problems: A posteriori error analysis and adaptivity. In I Duff and G A Watson, editors, State of the Art in Numerical Analysis, pages 441--471. OUP, 1997.

....the use of the Aubin Nitsche technique to derive error bounds for finite element approximations of model problems such as the convection diffusion equation [1] However, when applied to approximations of hyperbolic p.d.e. s this usually results in error bounds using negative Sobolev norms (e.g. [2]) which have little engineering significance. Therefore, this has generally been of little help to engineers although it has led to practical grid refinement indicators [3, 4] Very recently, however, a promising new approach to error analysis and optimal grid refinement has been introduced by ....

E. Suli and P. Houston. Finite element methods for hyperbolic problems: a posteriori error analysis and adaptivity. Paper presented as an Invited Lecture at the State of the Art in Numerical Analysis Conference, York, Apr, 1996.

.... in [7] and is based on the general methodology developed by Johnson et al. and outlined for example in [3] The idea of estimating the error in a negative norm came to us during a research presentation by Endre Suli who adopts a similar approach for first order hyperbolic systems, see for example [12]. We have chosen a convolution problem here because our main interest is in the numerical solution of viscoelasticity problems for which (1) can serve as a prototype. In these problems a convolution kernel with fading memory is appropriate, and sharp stability estimates can be derived by ....

Endre Suli and Paul Houston. Finite element methods for hyperbolic problems: a posteriori error analysis and adaptivity. Technical report, Numerical Analysis Group, Oxford University Computing Laboratory, OX1 3QD, England, 1996. 96/09.

....in hLemmas 35, 36 and 37 i. 5 Conclusion This appears to be the first time that such adaptive time stepping is possible for robust error control for second kind Volterra problems. The idea of employing a weaker norm to measure the error was prompted by the work of Suli et al. in, for example, [14] who use the same error control paradigm in the context of first order hyperbolic problems. Future work is expected to be concerned with the re formulation of the above problem in terms of internal variables, and also with the dynamic problem and problems of non Fickian diffusion. ....

Endre Suli and Paul Houston. Finite element methods for hyperbolic problems: a posteriori error analysis and adaptivity. Technical report, Numerical Analysis Group, Oxford University Computing Laboratory, OX1 3QD, England, 1996. 96/09.

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Suli, E. and Houston, P. (1997). Finite element methods for hyperbolic problems: a posteriori error analysis and adaptivity. In: I.S. Duff and G.A. Watson, eds. The State of the Art in Numerical Analysis, Clarendon Press, Oxford, 441--471. 71

....C is a computable constant, h is the mesh function and r h is the residual of the partial differential equation. The proof of this a posteriori error bound is based on exploiting a hyperbolic duality argument together with the Galerkin orthogonality of the finite element method, cf. [7, 8]. Section 5 will focus on the more fundamental question of mesh design. To control the global error with respect to a user defined tolerance TOL, the most direct, and indeed the most common approach used in practice is to equidistribute the right hand side of (1.1) over each element in the ....

....element in the computational mesh T h ; i.e. if a mesh is designed such that Ckhr h k L 2 ( TOL= p N 8 2 T h ; 1. 2) where N is the number of elements in the mesh, then clearly the global error measured in the (H 1 0 ( Omega Gamma2 0 norm will be less than the prescribed tolerance, cf. [5, 8]. Clearly, the success of this approach greatly depends on the relationship between the size of the local error (u Gamma u h )j and the size of the local residual r h j . In this paper we show that the local residual on an element only controls a portion of the local error on , referred to as ....

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E. Suli and P. Houston, Finite element methods for hyperbolic problems: a posteriori error analysis and adaptivity. In: I. Duff and G.A. Watson, eds., State of the Art in Numerical Analysis. Oxford University Press, 1997, pp. 441--471.

....paper can equally be applied to problems which lack the regularity needed for an interpolant to exist. In particular, we have derived a posteriori error estimates for the discontinuity capturing Lagrange Galerkin discretisation of multi dimensional scalar (first order) hyperbolic equations, see [23]. 30 ....

E. Suli and P. Houston. Finite element methods for hyperbolic problems: a posteriori error analysis and adaptivity. Paper presented as an Invited Lecture at the State of the Art in Numerical Analysis Conference, York, 1996.

No context found.

E. Suli and P. Houston. Finite element methods for hyperbolic problems: a posteriori error analysis and adaptivity. In I.S. Duff and G.A. Watson, editors, State of the Art in Numerical Analysis. Oxford University Press, 1996.

No context found.

E. Suli and P. Houston, Finite element methods for hyperbolic problems: a posteriori error analysis and adaptivity. Technical Report NA 96/09, Oxford University Computing Laboratory, (1996).

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