Adaptive variational multiscale method for the Stokes equations

Song, Lina; Hou, Yanren; Zheng, Haibiao

doi:10.1002/fld.3716

Cited by 9 publications

(12 citation statements)

References 17 publications

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“…This procedure for adaptive mesh refinement is similar to the proposed by other authors, see, for instance, other works . In the literature, other methodologies or procedures are considered to generate adapted meshes.…”

Section: Adaptive Mesh Refinementmentioning

confidence: 91%

“…This procedure for adaptive mesh refinement is similar to the proposed by other authors, see, for instance, other works. 9,15,16,54 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.…”

Section: Adaptive Mesh Refinementmentioning

confidence: 99%

“…Similarly, Larsson et al also developed bounds for the velocity error solving two subproblems but without the need of computing hybrid fluxes. Recently, variational multiscale theory has been employed by Song et al to estimate the error and generate adapted meshes. A local Dirichlet problem is solved both to obtain the stabilized term and to estimate the error that is employed in the mesh refinement process.…”

Section: Introductionmentioning

confidence: 99%

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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: Adaptive Mesh Refinementmentioning

confidence: 91%

Section: Adaptive Mesh Refinementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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“…In the past decades, a great number of researchers have developed the error estimation techniques (e.g. see [1,4,5,15,16,[19][20][21][22][23][24]). …”

Section: Introductionmentioning

confidence: 99%

Adaptive dimension splitting algorithm for three-dimensional elliptic equations

Wei

Hou

Song

2014

International Journal of Computer Mathematics

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An adaptive dimension splitting algorithm for three-dimensional (3D) elliptic equations is presented in this paper. We propose residual and recovery-based error estimators with respect to X − Y plane direction and Z direction, respectively, and construct the corresponding adaptive algorithm. Two-sided bounds of the estimators guarantee the efficiency and reliability of such error estimators. Numerical examples verify their efficiency both in estimating the error and in refining the mesh adaptively. This algorithm can be compared with or even better than the 3D adaptive finite element method with tetrahedral elements in some cases. What is more, our new algorithm involves only two-dimensional mesh and one-dimensional mesh in the process of refining mesh adaptively, and it can be implemented in parallel.

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“…The former estimator employs a patch of elements to estimate the error. For elliptic equations, a general overview can be shown in the papers [38,15,43,37,39,34,42,33]. On the other hand, the element residual estimators only require the information inside the element and the element boundary to obtain a local error estimate.…”

Section: Introductionmentioning

confidence: 99%

A posteriori pointwise error computation for 2-D transport equations based on the variational multiscale method

Irisarri

Hauke

2016

Computer Methods in Applied Mechanics and Engineering

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This article presents a general framework to estimate the pointwise error of linear partial differential equations. The error estimator is based on the variational multiscale theory, in which the error is decomposed in two components according to the nature of the residuals: element interior residuals and interelement jumps. The relationship between the residuals (coarse scales) and the error components (fine scales) is established, yielding to a very simple model. In particular, the pointwise error is modeled as a linear combination of bubble functions and Green's functions. If residual-free bubbles and the classical Green's function are employed, the technology leads to an exact explicit method for the pointwise error. If bubble functions and free-space Green's functions are employed, then a local projection problem must be solved within each element and a global boundary integral equation must be solved on the domain boundary. As a consequence, this gives a model for the so-called fine-scale Green's functions. The numerical error is studied for the standard Galerkin and SUPG methods with application to the heat equation, the reaction-diffusion equation and the convection-diffusion equation. Numerical results show that stabilized methods minimize the propagation of pollution errors, which stay mostly locally.

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Adaptive variational multiscale method for the Stokes equations

Cited by 9 publications

References 17 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

Adaptive dimension splitting algorithm for three-dimensional elliptic equations

A posteriori pointwise error computation for 2-D transport equations based on the variational multiscale method

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