Hessian recovery for finite element methods

Guo, Hailong; Zhang, Zhimin; Zhao, Ren

doi:10.1090/mcom/3186

Cited by 23 publications

(24 citation statements)

References 31 publications

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“…It is a robust and high accuracy approach for recovering gradient on mildly unstructured meshes. This idea has been used to develop a Hessian recovery technique in a recent paper [23]. In this paper, we show the possibility of generalizing the idea to problems on manifolds.…”

Section: Parametric Polynomialmentioning

confidence: 92%

Parametric Polynomial Preserving Recovery on Manifolds

Dong¹,

Guo²

2017

Preprint

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This paper investigates gradient recovery schemes for data defined on discretized manifolds. The proposed method, parametric polynomial preserving recovery (PPPR), does not require the tangent spaces of the exact manifolds, and they have been assumed for some significant gradient recovery methods in the literature. Another advantage of PPPR is that superconvergence is guaranteed without the symmetric condition which has been asked in the existing techniques. There is also numerical evidence that the superconvergence by PPPR is high curvature stable, which distinguishes itself from the others. As an application, we show its capability of constructing an asymptotically exact a posteriori error estimator. Several numerical examples on two-dimensional surfaces are presented to support the theoretical results and comparisons with existing methods are documented.

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Section: Parametric Polynomialmentioning

confidence: 92%

Parametric Polynomial Preserving Recovery on Manifolds

Dong¹,

Guo²

2017

Preprint

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“…▪ Next, we want to mimic the proof of Lemma 2 for the continuous setting in order to show well-posedness of our discrete scheme. However, due to the jump terms on the right-hand sides of the estimates (27) and (28) the proof of the coercivity of the bilinear form a h will fail if is too small, see (11). To this end, stabilization terms in the discrete scheme are needed and we define…”

Section: Proofmentioning

confidence: 99%

Error estimation for second‐order partial differential equations in nonvariational form

Blechschmidt

Herzog

Winkler

2020

Numerical Methods Partial

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Second-order partial differential equations (PDEs) in nondivergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton-Jacobi-Bellman equations in the context of stochastic optimal control, or as the linearization of fully nonlinear second-order PDEs. The nondivergence form in these problems is natural. If the coefficients of the diffusion matrix are not differentiable, the problem cannot be transformed into the more convenient variational form. We investigate tailored nonconforming finite element approximations of second-order PDEs in nondivergence form, utilizing finite-element Hessian recovery strategies to approximate second derivatives in the equation. We study both approximations with continuous and discontinuous trial functions. Of particular interest are a priori and a posteriori error estimates as well as adaptive finite element methods. In numerical experiments our method is compared with other approaches known from the literature. KEYWORDS finite element error estimates, PDEs in nondivergence form This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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“…In [13], the ZZ patch recovery is applied to the recovered values of stresses (gradient) at integration points. This method can use the same patch for stress and stress gradient.…”

Section: Double Zz Patch Recoverymentioning

confidence: 99%

“…This method can use the same patch for stress and stress gradient. However, in [13] numerical proofs have been given that this method is less effective than the method described in the following section (Sect. 3.3)…”

Section: Double Zz Patch Recoverymentioning

confidence: 99%

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A study on the stress gradient reconstruction in finite elements problems with application of radial basis function networks

2019

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The recovery of the stress gradient in finite elements problems is a widely discussed topic with many applications in the design process. The stress gradient is related to the second derivative (Hessian) of the nodal displacements and numerical techniques are required for its calculation. Particular difficulties are encountered in the reconstruction of the stress gradient in the boundary regions of the domain. This is of particular concern in most applications, especially in mechanical components, where the maximum values of stresses are often located in these regions and the stress gradient has a strong influence on the fatigue life of the component.This paper presents a comparison between some already published, partially modified, recovery techniques and a different approach based on radial basis function networks. The aim of the paper is to compare the performances of the different approaches for a number of element types with particular focus on the boundary regions. Some examples of mechanical interest are considered.

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Hessian recovery for finite element methods

Cited by 23 publications

References 31 publications

Parametric Polynomial Preserving Recovery on Manifolds

Parametric Polynomial Preserving Recovery on Manifolds

Error estimation for second‐order partial differential equations in nonvariational form

A study on the stress gradient reconstruction in finite elements problems with application of radial basis function networks

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