Output-based error estimation and mesh adaptation for variational multiscale methods

Granzow, Brian; Shephard, Mark S.; Oberai, Assad A.

doi:10.1016/j.cma.2017.05.008

Cited by 13 publications

(5 citation statements)

References 26 publications

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“…[34][35][36][37] Further, VMS procedure has been employed to develop error estimation and apply mesh adaptation. [38][39][40][41][42][43][44][45] A rigorous and systematic VMS approach was developed by Hughes and Sangalli 3 for the advection-diffusion equation. At first, the fine-scale solution is defined based on an integral operator involving the fine-scale Green's function and the strong-form residual due to the coarse-scale solution.…”

Section: Introductionmentioning

confidence: 99%

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A dynamic variational multiscale method on unstructured meshes for stationary transport problems

Sahni

2023

Numerical Methods in Fluids

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Summary This paper presents a variational multiscale (VMS) based finite element method where the stabilization parameter is computed dynamically. The current dynamic procedure takes in a general structure/form of the stabilization parameter with unknown coefficients and computes them dynamically in a local fashion resulting in a dynamic VMS‐based finite element method. Thus, a static stabilization parameter with pre‐defined coefficients is not needed. A variational Germano identity (VGI) based local procedure suitable for unstructured meshes is developed to perform the dynamic computation in a local fashion. The local VGI based procedure is applied for each interior vertex in the mesh and unknown coefficients are first determined locally at each vertex, and subsequently, for each element a maximum value is taken over the vertices of the element. To make the current procedure practical, a coarser secondary solution is constructed from the primary coarse‐scale solution, which is done locally over a patch of elements around each interior vertex. Further, averaging steps are employed to make the local dynamic procedure robust. Currently, the new dynamic VMS formulation is applied to steady problems governed by the advection‐diffusion and incompressible Navier‐Stokes equations in both 1D and 2D to demonstrate its efficacy and effectiveness.

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Section: Introductionmentioning

confidence: 99%

“…Reduced order modeling has also been introduced using the VMS‐related stabilization approach and has shown benefit in solving flow problems 34‐37 . Further, VMS procedure has been employed to develop error estimation and apply mesh adaptation 38‐45 …”

Section: Introductionmentioning

confidence: 99%

A dynamic variational multiscale method on unstructured meshes for stationary transport problems

Sahni

2023

Numerical Methods in Fluids

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“…for modeling small scales in LES turbulence models [14,24,1,71], and for estimating the numerical error. Application of the latter can be found in many applications, like fluid mechanics [39,40,42,41,43,38,37,54,53,55,6,5,68,73,33,12,13,64], elliptic problems [58,59,52], and elasticity [63,45,8]. A recent application of VMS error estimation to the propagation of error in uncertainty quantification has been published in [25].…”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimation and adaptivity based on VMS for the incompressible Navier–Stokes equations

Irisarri

Hauke

2021

Computer Methods in Applied Mechanics and Engineering

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“…In the context of adaptivity, the a posteriori evaluation of the error of a given quantity of interest is in general exploited in order to improve the accuracy of that particular quantity, see for example [4,12,16,21,27,31,34,40]. Duality-based approaches to a posteriori goal-oriented error estimation are described in [11,17,18,33]. Specific techniques for controlling and bounding the errors in quantities of interest are described in [5,25,38].…”

Section: Introductionmentioning

confidence: 99%

Orthogonality constrained gradient reconstruction for superconvergent linear functionals

Roberto

Chiaramonte

2020

Bit Numer Math

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The post-processing of the solution of variational problems discretized with Galerkin finite element methods is particularly useful for the computation of quantities of interest. Such quantities are generally expressed as linear functionals of the solution and the error of their approximation is bounded by the error of the solution itself. Several a posteriori recovery procedures have been developed over the years to improve the accuracy of post-processed results. Nonetheless such recovery methods usually deteriorate the convergence properties of linear functionals of the solution and, as a consequence, of the quantities of interest as well. The paper develops an enhanced gradient recovery scheme able to both preserve the good qualities of the recovered gradient and increase the accuracy and the convergence rates of linear functionals of the solution.

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Output-based error estimation and mesh adaptation for variational multiscale methods

Cited by 13 publications

References 26 publications

A dynamic variational multiscale method on unstructured meshes for stationary transport problems

A dynamic variational multiscale method on unstructured meshes for stationary transport problems

A posteriori error estimation and adaptivity based on VMS for the incompressible Navier–Stokes equations

Orthogonality constrained gradient reconstruction for superconvergent linear functionals

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