A Posteriori Error Estimation Based on Potential and Flux Reconstruction for the Heat Equation

Ern, Alexandre; Vohralı́k, Martin

doi:10.1137/090759008

Cited by 66 publications

(72 citation statements)

References 37 publications

(28 reference statements)

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“…, N }, represent the degrees of freedom in the space V h . Following [29,31,11], our a posteriori error estimates rely on the concept of the diffusive flux reconstruction θ h ∈ V h ; following [10], we also introduce a convective flux reconstruction w h ∈ V h .…”

Section: Diffusive and Convective Flux Reconstructionsmentioning

confidence: 99%

“…In a posteriori estimates for finite element, vertex-centered finite volume, or mixed finite element methods, cf. [25,11], data oscillation estimators of the form m n f − f n D appear. The other terms of the present estimators η n DOQ,D are related to the non-variational general nature of (4.5) (or of (3.1)).…”

Section: A Posteriori Error Estimatementioning

confidence: 99%

“…There is, however, no distinction between the space and time errors. Using the triangle inequality, we have, as in [11], for the example of η n DCF,D (t),…”

Section: Distinguishing the Space And Time Error Contributionsmentioning

confidence: 99%

“…Following [10,11], using definitions (2.1) and (2.6), estimate (5.2), and the Cauchy-Schwarz inequality, we readily have…”

Section: Proof Of Theorem 44 Let Us First Bound the Termmentioning

confidence: 99%

“…Firstly, we derive fully and easily computable estimates, not featuring any undetermined constants. We achieve this by introducing H(div, Ω)-conforming locally conservative diffusive and convective flux reconstructions, following [29,31,10,11]. Consequently, the estimates allow for actual error control in unsteady convection-diffusion-reaction problems.…”

Section: Introductionmentioning

confidence: 99%

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A posteriori error estimates for combined finite volume–finite element discretizations of reactive transport equations on nonmatching grids

Hilhorst¹,

Vohralı́k²

2011

Computer Methods in Applied Mechanics and Engineering

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We derive fully computable a posteriori error estimates for vertex-centered finite volume-type discretizations of transient convection-diffusion-reaction equations. Our estimates enable actual control of the error measured either in the energy norm or in the energy norm augmented by a dual norm of the skew-symmetric part of the differential operator. Lower bounds, global-in-space but local-in-time, are also derived. These lower bounds are fully robust with respect to convection or reaction dominance and the final simulation time in the augmented norm setting. On the basis of the derived estimates, we propose an adaptive algorithm which enables to automatically achieve a user-given relative precision. Moreover, this algorithm leads to optimal efficiency as it balances the time and space error contributions. As an example, we apply our estimates to the combined finite volume-finite element scheme, including such features as use of mass lumping for the time evolution or reaction terms, of upwind weighting for the convection term, and discretization on nonmatching meshes possibly containing nonconvex and non-star-shaped elements. Numerical experiments illustrate the theoretical developments.

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Section: Diffusive and Convective Flux Reconstructionsmentioning

confidence: 99%

Section: A Posteriori Error Estimatementioning

confidence: 99%

“…There is, however, no distinction between the space and time errors. Using the triangle inequality, we have, as in [11], for the example of η n DCF,D (t),…”

Section: Distinguishing the Space And Time Error Contributionsmentioning

confidence: 99%

“…Following [10,11], using definitions (2.1) and (2.6), estimate (5.2), and the Cauchy-Schwarz inequality, we readily have…”

Section: Proof Of Theorem 44 Let Us First Bound the Termmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A posteriori error estimates for combined finite volume–finite element discretizations of reactive transport equations on nonmatching grids

Hilhorst¹,

Vohralı́k²

2011

Computer Methods in Applied Mechanics and Engineering

Self Cite

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Towards simplified and optimized a posteriori error estimation using PGD reduced models

Allier

Chamoin

2017

Numerical Meth Engineering

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Summary The paper deals with the use of model order reduction within a posteriori error estimation procedures in the context of the finite element method. More specifically, it focuses on the constitutive relation error concept, which has been widely used over the last 40 years for FEM verification of computational mechanics models. A technical key‐point when using constitutive relation error is the construction of admissible fields, and we propose here to use the proper generalized decomposition to facilitate this task. In addition to making the implementation into commercial FE software easier, it is shown that the use of proper generalized decomposition enables to optimize the verification procedure and to get both accurate and reasonably expensive upper bounds on the discretization error. Numerical illustrations are presented to assess the performance of the proposed approach.

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A face‐centred finite volume method for second‐order elliptic problems

Sevilla

Giacomini

Huerta

2018

Numerical Meth Engineering

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Summary This work proposes a novel finite volume paradigm, ie, the face‐centred finite volume (FCFV) method. Contrary to the popular vertex and cell‐centred finite volume methods, the novel FCFV defines the solution on the mesh faces (edges in two dimensions) to construct locally conservative numerical schemes. The idea of the FCFV method stems from a hybridisable discontinuous Galerkin formulation with constant degree of approximation, and thus inheriting the convergence properties of the classical hybridisable discontinuous Galerkin. The resulting FCFV features a global problem in terms of a piecewise constant function defined on the faces of the mesh. The solution and its gradient in each element are then recovered by solving a set of independent element‐by‐element problems. The mathematical formulation of FCFV for Poisson and Stokes equation is derived, and numerical evidence of optimal convergence in two dimensions and three dimensions is provided. Numerical examples are presented to illustrate the accuracy, efficiency, and robustness of the proposed methodology. The results show that, contrary to other finite volume methods, the accuracy of the FCFV method is not sensitive to mesh distortion and stretching. In addition, the FCFV method shows its better performance, accuracy, and robustness using simplicial elements, facilitating its application to problems involving complex geometries in three dimensions.

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A Posteriori Error Estimation Based on Potential and Flux Reconstruction for the Heat Equation

Cited by 66 publications

References 37 publications

A posteriori error estimates for combined finite volume–finite element discretizations of reactive transport equations on nonmatching grids

A posteriori error estimates for combined finite volume–finite element discretizations of reactive transport equations on nonmatching grids

Towards simplified and optimized a posteriori error estimation using PGD reduced models

A face‐centred finite volume method for second‐order elliptic problems

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