<i>A posteriori</i> error analysis for the Morley plate element with general boundary conditions

Veiga, L. Beirão da; Niiranen, Jarkko; Stenberg, Rolf

doi:10.1002/nme.2821

Cited by 25 publications

(9 citation statements)

References 28 publications

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“…Let us return to our preliminary formulation (2). We now know that we have to interpret the interface terms as…”

Section: Variational Formulation and Dpg Methodsmentioning

confidence: 99%

“…Second, verification of numerical accuracy of finite element algorithms is at the hearth of simulation governance, see [37]. This is a serious challenge in practical plate-bending problems where both the geometry and applied loading can be very irregular so that many of the contemporary developments in the finite element modeling of plate problems are devoted to a posteriori error estimation and adaptivity, see, e.g., [9,2,27].…”

Section: Introductionmentioning

confidence: 99%

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An ultraweak formulation of the Kirchhoff–Love plate bending model and DPG approximation

2018

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We develop and analyze an ultraweak variational formulation for a variant of the Kirchhoff-Love plate bending model. Based on this formulation, we introduce a discretization of the discontinuous Petrov-Galerkin type with optimal test functions (DPG). We prove wellposedness of the ultraweak formulation and quasi-optimal convergence of the DPG scheme.The variational formulation and its analysis require tools that control traces and jumps in H 2 (standard Sobolev space of scalar functions) and H(div div) (symmetric tensor functions with L 2 -components whose twice iterated divergence is in L 2 ), and their dualities. These tools are developed in two and three spatial dimensions. One specific result concerns localized traces in a dense subspace of H(div div). They are essential to construct basis functions for an approximation of H(div div).To illustrate the theory we construct basis functions of the lowest order and perform numerical experiments for a smooth and a singular model solution. They confirm the expected convergence behavior of the DPG method both for uniform and adaptively refined meshes.

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“…Let us return to our preliminary formulation (2). We now know that we have to interpret the interface terms as…”

Section: Variational Formulation and Dpg Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An ultraweak formulation of the Kirchhoff–Love plate bending model and DPG approximation

2018

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“…We remark in passing that the decomposition had been originally proposed to construct the residual-based a posteriori error estimate of the nonconforming Morley plate bending element [33], which was then improved in [30] and was extended to the case of general boundary conditions [7]. A different approach [25] was taken in treating the case of a fully discontinuous interior penalty method [24], where the derivation of the reliability bound heavily depends on a suitable recovery operator mapping discontinuous finite element spaces into H 2 0 -conforming spaces composed of high-order versions of the classical Hsieh-Clough-Tochner macro-element defined in [22].…”

Section: Introductionmentioning

confidence: 99%

A Posteriori Error Estimates for Local $C^0$ Discontinuous Galerkin Methods for Kirchhoff Plate Bending Problems

Xu¹

2014

JCM

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We derive some residual-type a posteriori error estimates for the local C 0 discontinuous Galerkin (LCDG) approximations ([31]) of the Kirchhoff bending plate clamped on the boundary. The estimator is both reliable and efficient with respect to the moment-field approximation error in an energy norm. Some numerical experiments are reported to demonstrate theoretical results.Mathematics subject classification: 65N15, 65N30.

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“…in [15,Section 5.9]. Inhomogeneous boundary data can be addressed following [11]; see also Remark 4.10 below.…”

Section: Introductionmentioning

confidence: 99%

Residual-Based A Posteriori Error Estimates for $hp$-Discontinuous Galerkin Discretizations of the Biharmonic Problem

Dong¹,

Mascotto²,

Sutton³

2021

SIAM J. Numer. Anal.

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We introduce a residual-based a posteriori error estimator for a novel hp-version interior penalty discontinuous Galerkin method for the biharmonic problem in two and three dimensions. We prove that the error estimate provides an upper bound and a local lower bound on the error, and that the lower bound is robust to the local mesh size but not the local polynomial degree. The suboptimality in terms of the polynomial degree is fully explicit and grows at most algebraically. Our analysis does not require the existence of a C 1 -conforming piecewise polynomial space and is instead based on an elliptic reconstruction of the discrete solution to the H 2 space and a generalised Helmholtz decomposition of the error. This is the first hp-version error estimator for the biharmonic problem in two and three dimensions. The practical behaviour of the estimator is investigated through numerical examples in two and three dimensions.

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A posteriori error analysis for the Morley plate element with general boundary conditions

Cited by 25 publications

References 28 publications

An ultraweak formulation of the Kirchhoff–Love plate bending model and DPG approximation

An ultraweak formulation of the Kirchhoff–Love plate bending model and DPG approximation

A Posteriori Error Estimates for Local $C^0$ Discontinuous Galerkin Methods for Kirchhoff Plate Bending Problems

Residual-Based A Posteriori Error Estimates for $hp$-Discontinuous Galerkin Discretizations of the Biharmonic Problem

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