Convergence rates for adaptive finite elements

Gaspoz, Fernando D.; Morín, Pedro

doi:10.1093/imanum/drn039

Cited by 36 publications

(46 citation statements)

References 24 publications

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“…Applying 1 ν! ∂ ν at y = 0, for any ν = 0, gives 16) which shows by recursion that ∆t ν ∈ L 2 (D) for all ν ∈ F. Integrating against ∆t ν , and applying Young's inequality we find that, for any ε > 0,…”

Section: Space-parameter Approximationmentioning

confidence: 85%

“…In the case of Taylor series (or more general polynomial series where the φ ν are uniformly bounded by 1 over U ), taking V = L ∞ (U, V ), we control the first term by 16) and the second term by…”

Section: Space-parameter Approximationmentioning

confidence: 99%

“…(4.46) Second, the approximation property (3.9) remains valid for the space X = K k θ+1 (D) with the same rate t = k−1 2 as for X = H k (D), if we use Lagrange finite element spaces (V n ) n>0 of degree at least k − 1 and with appropriate mesh refinement near the corners of D (see [5,16], [1, Eqn. (0.3)] and the references there).…”

Section: )mentioning

confidence: 99%

“…Taking ε small enough, so thatθ := (1 + ε)θ τ < 1, and using the convexity of x → |x| τ , we obtain 16) for some fixed C > 1. Integrating and summing over |ν| = n thus gives…”

Section: Towards Space-parameter Adaptivitymentioning

confidence: 99%

See 3 more Smart Citations

Fully Discrete Approximation of Parametric and Stochastic Elliptic PDEs

Bachmayr¹,

Cohen²,

Dũng³

et al. 2017

SIAM J. Numer. Anal.

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It has recently been demonstrated that locality of spatial supports in the parametrization of coefficients in elliptic PDEs can lead to improved convergence rates of sparse polynomial expansions of the corresponding parameter-dependent solutions. These results by themselves do not yield practically realizable approximations, since they do not cover the approximation of the arising expansion coefficients, which are functions of the spatial variable. In this work, we study the combined spatial and parametric approximability for elliptic PDEs with affine or lognormal parametrizations of the diffusion coefficients and corresponding Taylor, Jacobi, and Hermite expansions, to obtain fully discrete approximations. Our analysis yields convergence rates of the fully discrete approximation in terms of the total number of degrees of freedom. The main vehicle consists in ℓ p summability results for the coefficient sequences measured in higher-order Hilbertian Sobolev norms. We also discuss similar results for nonHilbertian Sobolev norms which arise naturally when using adaptive spatial discretizations.

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Section: Space-parameter Approximationmentioning

confidence: 85%

Section: Space-parameter Approximationmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

“…Taking ε small enough, so thatθ := (1 + ε)θ τ < 1, and using the convexity of x → |x| τ , we obtain 16) for some fixed C > 1. Integrating and summing over |ν| = n thus gives…”

Section: Towards Space-parameter Adaptivitymentioning

confidence: 99%

See 2 more Smart Citations

Fully Discrete Approximation of Parametric and Stochastic Elliptic PDEs

Bachmayr¹,

Cohen²,

Dũng³

et al. 2017

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

show abstract

“…Of course, more sophisticated and effective adaptive processes can be used, eg, Van der Zee, Babuška and Rheinboldt, Becker and Rannacher, Giles and Süli, and Gaspoz and Morin. [51][52][53][54][55] The adaptive process gives us an updated coarse mesh, which constitutes the initial mesh for the subsequent adaptive iteration. We repeat this process until the required precision is reached.…”

Section: P-adaptive Algorithmmentioning

confidence: 99%

Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

Darrigrand

Rodríguez-Rozas

Muga

et al. 2017

Numerical Meth Engineering

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SummaryIn goal-oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element-wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal-oriented adaptivity.While the method can be applied to a variety of problems, we focus here on two-and three-dimensional (2-D and 3-D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p-adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection-dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging-while-drilling problem to illustrate the applicability of the proposed method.

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New error estimates of nonconforming mixed finite element methods for the Stokes problem

Mao

Zhang

2013

Math. Meth. Appl. Sci.

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In this paper, we revisit the classical error estimates of nonconforming Crouzeix–Raviart type finite elements for the Stokes equations. By introducing some quasi‐interpolation operators and using the special properties of these nonconforming elements, it is proved that their consistency errors can be bounded by their approximation errors together with a high‐order term, especially which can be of arbitrary order provided that f in the right‐hand side is piecewise smooth enough. Furthermore, we show an interesting result that both in the energy norm and L2 norm the consistency errors are dominated by the approximation errors of their finite element spaces. As byproducts, we derive the error estimates in both energy and L2 norms under the regularity assumption (u,p) ∈ H1 + s(Ω) × Hs(Ω) with any s ∈ (0,1], which fills the gap in the a priori error estimate of these nonconforming elements with low regularity (bold-italicuMathClass-punc,p)MathClass-rel∈boldH1MathClass-bin+s(Ω)MathClass-bin×Hs(Ω)MathClass-punc,0MathClass-rel

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Convergence rates for adaptive finite elements

Cited by 36 publications

References 24 publications

Fully Discrete Approximation of Parametric and Stochastic Elliptic PDEs

Fully Discrete Approximation of Parametric and Stochastic Elliptic PDEs

Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

New error estimates of nonconforming mixed finite element methods for the Stokes problem

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