Pointwise error estimation and adaptivity for the finite element method using fundamental solutions

Grätsch, Thomas; Hartmann, Friedel

doi:10.1007/s00466-005-0711-4

Cited by 10 publications

(8 citation statements)

References 23 publications

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“…The implicit estimator decomposes the error in two parts: the first component involves solving a local problem set on each element, constituted by a bubble space; and the second term computes the contribution of the inter‐element residuals on the elemental boundaries using the free‐space Green's functions. In other works, we can find similar concepts on the use of Green's functions for error estimation. On the other hand, the explicit estimator predicts the error by multiplying the residuals with the corresponding error time‐scale, τ 's.…”

Section: Introductionmentioning

confidence: 80%

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A posteriori error estimation and adaptivity based on VMS for the Stokes problem

Irisarri

Hauke

2018

Numerical Methods in Fluids

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Summary In this paper, we present residual‐based error estimators applied to the Stokes problem. Implicit and explicit error estimators are developed to predict the error of the numerical solution obtained by a stabilized finite element formulation. Both error estimators arise from the variational multiscale framework, in which the variational form is split into coarse scales (finite element method solution) and fine scales (committed error). This leads to a local problem set on each element in which the error and the residuals are involved. The different ways of linking the error to the residuals produce two error estimators. Local and global error estimates measured in the H1‐seminorm are provided for triangles and quadrilaterals. As an application, using the local error estimates, an adaptive mesh refinement strategy is implemented in order to optimize the computational resources.

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Section: Introductionmentioning

confidence: 80%

“…In the literature, other methodologies or procedures are considered to generate adapted meshes. For example, starting from a initial mesh, other authors split the elements with large errors and merge those with small errors . In this latter case, usually, there appear hanging nodes that must be treated.…”

Section: Adaptive Mesh Refinementmentioning

confidence: 99%

A posteriori error estimation and adaptivity based on VMS for the Stokes problem

Irisarri

Hauke

2018

Numerical Methods in Fluids

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“…Similar enrichment techniques were developed in [19] for elastodynamics problems and in [11] for quasi-static problems.…”

Section: The Adjoint Problem and Its Solutionmentioning

confidence: 91%

“…For steady-state problems, several works have proposed answers [7,28,33,34,38] and some of them provide guaranteed estimates, which is useful for robust design. However, to our knowledge, verification methods developed in dynamics [14,19,20,36,40] do not provide guaranteed local error estimation.…”

Section: Introductionmentioning

confidence: 97%

Guaranteed error bounds on pointwise quantities of interest for transient viscodynamics problems

Waeytens¹,

Chamoin²

2011

Comput Mech

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A new method is developed to obtain guaranteed error bounds on pointwise quantities of interest for linear transient viscodynamics problems. The calculation of strict error bounds is based on the concept of "constitutive relation error" (CRE) and the solution of an adjoint problem. The central and original point of this work is the treatment of the singularity in space and time introduced by the loading of the adjoint problem. Hence, the adjoint solution is decomposed into two parts: (i) an analytical part determined from Green's functions; (ii) a residual part approximated with classical numerical tools (finite element method, Newmark integration scheme). The capabilities and the limits of the proposed approach are analyzed on a 2D example.

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“…example, [21][22][23][24][25]). Several specific estimates, such as the adjoint-weighted residual method [17,20], the energy norm based estimates [26], the Green's function decomposition method [27], the strict-bounding approach based on Lagrangian formulation [28], the CRE-based error estimation [25], have been proposed and applied in solutions of Poisson's equation, linear and non-linear static problems in solid mechanics, eigenvalue problems, time-dependent problems, non-trivial problems of CFD and etc [29,30]. Among the available techniques, the CRE-based error estimation is a one to guarantee strict bounds of quantities.…”

Section: Introductionmentioning

confidence: 99%

Goal-oriented error estimation for beams on elastic foundation with double shear effect

Guo

Zhong

2015

Applied Mathematical Modelling

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a b s t r a c tIn this paper, goal-oriented error estimation for Timoshenko beams on Pasternak foundation, which involves double shear effect, is performed. The constitutive relation error (CRE) estimation is used in finite element analysis (FEA) to acquire strict bounds on quantities of interest. Due to the coupling of the displacement field and the internal force field in the equilibrium equations of the beam, the prolongation condition for construction of the admissible internal force field, a pillar of the CRE estimation, is not directly applicable. To overcome this difficulty, an auxiliary approximate problem whose stress solution enables the CRE estimation to proceed is introduced. Thereafter, strict bounds of outputs for the beam are obtained by dual analysis to which a significant adjunct is the circumvention of shear locking in low-order finite element analysis. Numerical results are presented to validate the strict bounding properties for quantities of beams on elastic foundation and accurate estimation of displacement quantities that is impervious to shear locking.

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Pointwise error estimation and adaptivity for the finite element method using fundamental solutions

Cited by 10 publications

References 23 publications

A posteriori error estimation and adaptivity based on VMS for the Stokes problem

A posteriori error estimation and adaptivity based on VMS for the Stokes problem

Guaranteed error bounds on pointwise quantities of interest for transient viscodynamics problems

Goal-oriented error estimation for beams on elastic foundation with double shear effect

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