Adaptivity and Error Estimation for the Finite Element Method Applied to Convection Diffusion Problems

Szymczak, William G.; Babuvška, I.

doi:10.1137/0721059

Cited by 38 publications

(20 citation statements)

References 22 publications

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“…The bound (2.9) follows directly from the de"nition of the norms and the CauchySchwarz inequality. (2.10), (2.11) follow immediately from Lemma 2.6 of Szymczak & Babu0 ska (1984).…”

Section: Variational Formulationmentioning

confidence: 99%

“…The crucial ingredient in the work of Szymczak (1982) and Szymczak & Babu0 ska (1984) (similarly to Christie et al (1978), Hemker (1977)) was the construction of suitable, upwinded test functions by asymptotic analysis of the elemental adjoint problem. The generalization of this asymptotic analysis (and the techniques used by Christie et al (1978), Hemker (1977)) to high-order elements and higher dimensions is not straightforward.…”

Section: Problem Formulationmentioning

confidence: 99%

“…This leads us to a formulation which is situated`between' the weak one based on H 1 0 × H 1 0 and the very weak one based on L 2 × H 2 ∩ H 1 0 . We present Sobolev spaces with mesh-dependent norms introduced by Szymczak & Babu0 ska (1984). For a collection of nodes…”

Section: Variational Formulationmentioning

confidence: 99%

“…We base our hp FEM on the variational framework of Szymczak (1982) and Szymczak & Babu0 ska (1984), where the h-version FEM was analysed and optimal convergence rates, uniform in ε, were shown. The crucial ingredient in the work of Szymczak (1982) and Szymczak & Babu0 ska (1984) (similarly to Christie et al (1978), Hemker (1977)) was the construction of suitable, upwinded test functions by asymptotic analysis of the elemental adjoint problem.…”

Section: Problem Formulationmentioning

confidence: 99%

“…If = 1, we simply write S p 0 (T ). As test space we choose, following Szymczak & Babu0 ska (1984), the space of L * -splines of degree p de"ned by…”

Section: The "Nite Element Spacesmentioning

confidence: 99%

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An hp finite element method for convection-diffusion problems in one dimension

Melenk

Schwab

1999

IMA Journal of Numerical Analysis

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We analyse an hp FEM for convection-diffusion problems. Stability is achieved by suitably upwinded test functions, generalizing the classical α-quadratically upwinded and the Hemker test-functions for piecewise linear trial spaces (see, e.g., Morton 1995 Numerical Solutions of Convection-Diffusion Problems, Oxford: Oxford University Press, and the references therein). The method is proved to be stable independently of the viscosity. Further, the stability is shown to depend only weakly on the spectral order. We show how suf"ciently accurate, approximate upwinded test functions can be computed on each element by a local least-squares FEM. Under the assumption of analyticity of the input data, we prove robust exponential convergence of the method. Numerical experiments con"rm our convergence estimates and show robust exponential convergence of the hp FEM even for viscosities of the order of the machine precision, i.e., for the limiting transport problem.

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“…The bound (2.9) follows directly from the de"nition of the norms and the CauchySchwarz inequality. (2.10), (2.11) follow immediately from Lemma 2.6 of Szymczak & Babu0 ska (1984).…”

Section: Variational Formulationmentioning

confidence: 99%

Section: Problem Formulationmentioning

confidence: 99%

Section: Variational Formulationmentioning

confidence: 99%

Section: Problem Formulationmentioning

confidence: 99%

“…If = 1, we simply write S p 0 (T ). As test space we choose, following Szymczak & Babu0 ska (1984), the space of L * -splines of degree p de"ned by…”

Section: The "Nite Element Spacesmentioning

confidence: 99%

See 3 more Smart Citations

An hp finite element method for convection-diffusion problems in one dimension

Melenk

Schwab

1999

IMA Journal of Numerical Analysis

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Uniform superconvergence analysis of Ciarlet‐Raviart scheme for Bi‐wave singular perturbation problem

Shi

2018

Math Methods in App Sciences

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Uniform superconvergence analysis of the Ciarlet‐Raviart mixed finite element scheme is discussed for solving the fourth‐order Bi‐wave singular perturbation problem (SPP) by the bilinear element. Firstly, the existence and uniqueness of the approximation solution are proved. Secondly, with the help of the special characters of this element, uniform superclose result of order O(h2) for the original variable in H1 norm and uniform optimal order estimate of order O(h2) for the intermediate variable in L2 norm are deduced with respect to the real perturbation parameter δ appearing in the considered problem. Furthermore, the global uniform superconvergent estimate is obtained through the interpolated postprocessing approach. Finally, some numerical results are provided to verify the theoretical analysis. Here, h denotes the mesh size.

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Error Estimates for the Approximate Solution of Linear Fixed Point Equations

Lippold

1990

Z Angew Math Mech

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The defects which are caused by the solutions of approximate equations in the original equation provide an effective tool for the asymptotically exact estimation of approximation errors. Using these defects may be justified without uniformity restrictions on the underlying mesh sequences. As an example, asymptotically exact error estimates are investigated for the numerical solution of a one‐dimensional linear diffusion‐convection equation by means of finite elements.

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Adaptivity and Error Estimation for the Finite Element Method Applied to Convection Diffusion Problems

Cited by 38 publications

References 22 publications

An hp finite element method for convection-diffusion problems in one dimension

An hp finite element method for convection-diffusion problems in one dimension

Uniform superconvergence analysis of Ciarlet‐Raviart scheme for Bi‐wave singular perturbation problem

Error Estimates for the Approximate Solution of Linear Fixed Point Equations

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