Interpolation Error Bounds for Curvilinear Finite Elements and Their Implications on Adaptive Mesh Refinement

Moxey, David; Sastry, Shankar Prasad; Kirby, Robert M.

doi:10.1007/s10915-018-0795-6

Cited by 15 publications

(5 citation statements)

References 15 publications

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“…Returning to the physical DG weak form, Eq. ( 13), as discussed in [43,67], there is no claim that the physical flux has a polynomial basis function expansion for curvilinear elements. We term the scheme "nonconservative" because it does not recover the definition of the reference divergence operator in Eq.…”

Section: Dg -Non-conservative Strong Formmentioning

confidence: 99%

“…As highlighted by the SBP community [41,42,40], discrete integration by parts is not satisfied in the physical space for curvilinear coordinates. This is due to the physical flux never explicitly being represented by an interpolating polynomial in the physical space [43]. This distinction, to the authors' knowledge, has not been investigated within the ESFR and DG communities [44,45,46,47].…”

Section: Introductionmentioning

confidence: 98%

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Provably Stable Flux Reconstruction High-Order Methods on Curvilinear Elements

Cicchino,

Fernández,

Nadarajah

et al. 2021

Preprint

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Provably stable flux reconstruction (FR) schemes are derived for partial differential equations cast in curvilinear coordinates. Specifically, energy stable flux reconstruction (ESFR) schemes are considered as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements.Additionally, split forms are examined as they enable the development of energy stability proofs. The first critical step proves, that in curvilinear coordinates, the discontinuous Galerkin (DG) conservative and nonconservative forms are inherently different-even under exact integration and analytically exact metric terms. This analysis demonstrates that the split form is essential to developing provably stable DG schemes on curvilinear coordinates and motivates the construction of metric dependent ESFR correction functions in each element. Furthermore, the provably stable FR schemes differ from schemes in the literature that only apply the ESFR correction functions to surface terms or on the conservative form, and instead incorporate the ESFR correction functions on the full split form of the equations. It is demonstrated that the scheme is divergent when the correction functions are only used for surface reconstruction in curvilinear coordinates. We numerically verify the stability claims for our proposed FR split forms and compare them to ESFR schemes in the literature. Lastly, the newly proposed provably stable FR schemes are shown to obtain optimal orders ⋆

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Section: Dg -Non-conservative Strong Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Provably Stable Flux Reconstruction High-Order Methods on Curvilinear Elements

Cicchino,

Fernández,

Nadarajah

et al. 2021

Preprint

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show abstract

“…The metric field's own intrinsic curvature may derive from any error estimate, be it boundary approximation error [37,38] or an interpolation error estimate. So far, interpolation error estimates on high-order elements are limited to isotropy ([39] in L 2 and [40] in L 1 norms) or require that the curvature of the element be bounded, essentially establishing a range where it may be considered linear [41].…”

Section: Introductionmentioning

confidence: 99%

P² Cavity Operator and Riemannian Curved Edge Length Optimization: a Path to High-Order Mesh Adaptation

Rochery

Loseille

2021

AIAA Scitech 2021 Forum

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We present a new P 2 extension of the P 1 cavity operator used as the basis for topological modification in 3D metric based mesh adaptation, with notable success in strongly anisotropic industrial cases of CFD. The P 2 operator inherits the P 1 cavity operator's robustness -mesh validity is guaranteed at all times -and manages to recover a metric field's inherent curvature through a Riemannian edge length optimization algorithm. This generic approach allows it to tackle a variety of problems, which are defined only by the input metric field such as the classic problem of surface approximation -through a geometric error surface metric propagated to the volume -or unit mesh construction such as for interpolation error minimization through high-order L p error estimates.Consistence with the log-euclidian metric interpolation scheme used in P 1 adaptation is obtained by a rigorous formulation of the optimization problem. This guarantees full compliance of the operator with the general adaptation process, by accurately measuring Riemannian edge lengths.Particular stress was put on the performance of the operator because of its central role in anisotropic mesh adaptation. All curving operations are carried out locally: this is in opposition with global approaches, be they optimization or PDE based. The optimization itself is carried out by an inhouse solver tailored to the problem at hand. As a result, the added cost is strictly linear.Numerical results illustrating the P 2 cavity operator's ability to recover curvature, be it surface curvature extended to boundary layers or metric field induced curvature of the volume, will be presented through cases representative of real-world geometries encountered in CFD. Finally, the operator's ability to handle rather large cases (10M elements) in minutes will be demonstrated.

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“…Lebesgue constant depends only on the grid nodes. Many publications have been devoted to it [5][6][7][8][9][10][11][12][13]. In paper [15] the results for the Lebesgue constants and the behavior of the Lebesgue functions in view of the optimal interpolation points are given.…”

Section: Introductionmentioning

confidence: 99%

Approximation by the Third-Order Splines on Uniform and Non-uniform Grids and Image Processing

Бурова¹,

Muzafarova

Narbutovskikh³

2020

Wseas Transactions on Mathematics

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This work is one of a series of papers that is devoted to the further investigation of polynomial splines and trigonometric splines of the third order approximation. Polynomial basis splines are better known and therefore more commonly used. However, the use of trigonometric basis splines often provides a smaller approximation error. In some cases, the use of the trigonometric approximations is preferable to the polynomial approximations. Here we continue to compare these two types of approximation. The Lebesgue functions and constants are discussed for the polynomial splines and the trigonometric splines. The examples of the applications of the splines to image enlargement are given.

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Interpolation Error Bounds for Curvilinear Finite Elements and Their Implications on Adaptive Mesh Refinement

Abstract: The version presented here may differ from the published version. If citing, you are advised to consult the published version for pagination, volume/issue and date of publication Noname manuscript No.

Cited by 15 publications

References 15 publications

Provably Stable Flux Reconstruction High-Order Methods on Curvilinear Elements

Provably Stable Flux Reconstruction High-Order Methods on Curvilinear Elements

P² Cavity Operator and Riemannian Curved Edge Length Optimization: a Path to High-Order Mesh Adaptation

Approximation by the Third-Order Splines on Uniform and Non-uniform Grids and Image Processing

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Interpolation Error Bounds for Curvilinear Finite Elements and Their Implications on Adaptive Mesh Refinement

Abstract: The version presented here may differ from the published version. If citing, you are advised to consult the published version for pagination, volume/issue and date of publication Noname manuscript No.

Cited by 15 publications

References 15 publications

Provably Stable Flux Reconstruction High-Order Methods on Curvilinear Elements

Provably Stable Flux Reconstruction High-Order Methods on Curvilinear Elements

P2 Cavity Operator and Riemannian Curved Edge Length Optimization: a Path to High-Order Mesh Adaptation

Approximation by the Third-Order Splines on Uniform and Non-uniform Grids and Image Processing

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Product

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P² Cavity Operator and Riemannian Curved Edge Length Optimization: a Path to High-Order Mesh Adaptation