A two-energies principle for the biharmonic equation and an<i>a posteriori</i>error estimator for an interior penalty discontinuous Galerkin approximation

Braess, Dietrich; Hoppe, Ronald H. W.; Linsenmann, Christopher

doi:10.1051/m2an/2016074

Cited by 9 publications

(15 citation statements)

References 50 publications

(65 reference statements)

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“…The principle was reformulated several times in order to obtain error estimates by a postprocessing also when nonconforming finite elements are involved. The principle was formulated for problems of fourth order in [29] and used for computing a posteriori error bonds in [10]. It is based on the fact that there is no duality gap between the minimum problem…”

Section: A Two-energies Principle For the Biharmonic Equationmentioning

confidence: 99%

“…To this end, an equilibrated moment tensor σ eq h will be constructed. As was pointed out in [10], we usually get two additional terms in a posteriori error estimates.…”

Section: 2mentioning

confidence: 99%

“…An interpolation by a Hsieh-Clough-Tocher element, by an element of the TUBA family [2] or by another H 2 -function u conf implies an additional term |u h − u conf | 2,h . This term does not spoil the efficiency, since it can be bounded by terms of residual a posteriori error estimates that are known to be efficient [10,13].…”

Section: 2mentioning

confidence: 99%

“…in distributional sense is our first aim and the main task of this section. The computation will be done explicitly by a local postprocessing, but Theorem 3.1 indicates already that it is done on a different basis than for the IPDG method in [10]. We want to find σ eq h such that div div σ eq h , v can be evaluated for less smooth v ∈ H 2 0 + V 0 h .…”

Section: Equilibrationmentioning

confidence: 99%

“…Several a posteriori estimates of residual type can be found in the literature [7,13,21,22,37]. Recently an a posteriori error estimate has been established by the two-energies principle (hypercircle method) for the full discontinuous interior penalty (IPDG) method [10], where the finite elements for the u-variable are not even H 1 -conforming.…”

Section: Introductionmentioning

confidence: 99%

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An equilibration-based a posteriori error bound for the biharmonic equation and two finite element methods

Braess

Pechstein

Schöberl

2019

IMA Journal of Numerical Analysis

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We develop an a posteriori error estimator for the Interior Penalty Discontinuous Galerkin approximation of the biharmonic equation with continuous finite elements. The error bound is based on the two-energies principle and requires the computation of an equilibrated moment tensor. The natural space for the moment tensor consists of symmetric tensor fields with continuous normal-normal components. It is known from the Hellan-Herrmann-Johnson (HHJ) mixed formulation. We propose a construction that is totally local. The procedure can also be applied to the original HHJ formulation, which directly provides an equilibrated moment tensor.

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Section: A Two-energies Principle For the Biharmonic Equationmentioning

confidence: 99%

“…To this end, an equilibrated moment tensor σ eq h will be constructed. As was pointed out in [10], we usually get two additional terms in a posteriori error estimates.…”

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Equilibrationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An equilibration-based a posteriori error bound for the biharmonic equation and two finite element methods

Braess

Pechstein

Schöberl

2019

IMA Journal of Numerical Analysis

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A C⁰ interior penalty discontinuous Galerkin method for fourth‐order total variation flow. I: Derivation of the method and numerical results

Bhandari

Hoppe

Kumar

2019

Numerical Methods Partial

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We consider the numerical solution of a fourth-order total variation flow problem representing surface relaxation below the roughening temperature. Based on a regularization and scaling of the nonlinear fourth-order parabolic equation, we perform an implicit discretization in time and a C 0 Interior Penalty Discontinuous Galerkin (C 0 IPDG) discretization in space. The C 0 IPDG approximation can be derived from a mixed formulation involving numerical flux functions where an appropriate choice of the flux functions allows to eliminate the discrete dual variable. The fully discrete problem can be interpreted as a parameter dependent nonlinear system with the discrete time as a parameter. It is solved by a predictor corrector continuation strategy featuring an adaptive choice of the time step sizes. A documentation of numerical results is provided illustrating the performance of the C 0 IPDG method and the predictor corrector continuation strategy. The existence and uniqueness of a solution of the C 0 IPDG method will be shown in the second part of this paper. KEYWORDS C 0 interior penalty discontinuous Galerkin method, fourth-order total variation flow, surface relaxation

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Numerical solution of a phase field model for polycrystallization processes in binary mixtures

Hoppe

Winkle

2019

Comput. Visual Sci.

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We consider the numerical solution of a phase field model for polycrystallization in the solidification of binary mixtures in a domain Ω ⊂ R 2 . The model is based on a free energy in terms of three order parameters: the local orientation Θ of the crystals, the local crystallinity φ, and the concentration c of one of the components of the binary mixture. The equations of motion are given by an initial-boundary value problem for a coupled system of partial differential equations consisting of a regularized second order total variation flow in Θ, an L 2 gradient flow in φ, and a W 1,2 (Ω) * gradient flow in c. Based on an implicit discretization in time by the backward Euler scheme, we suggest a splitting method such that the three semidiscretized equations can be solved separately and prove existence of a solution. As far as the discretization in space is concerned, the fourth order Cahn-Hilliard type equation in c is taken care of by a C 0 Interior Penalty Discontinuous Galerkin approximation which has the advantage that the same finite element space can be used as well for the spatial discretization of the equations in Θ and φ. The fully discretized equations represent parameter dependent nonlinear algebraic systems with the discrete time as a parameter. They are solved by a predictor corrector continuation strategy featuring an adaptive choice of the time-step. Numerical results illustrate the performance of the suggested numerical method.

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A two-energies principle for the biharmonic equation and ana posteriorierror estimator for an interior penalty discontinuous Galerkin approximation

Cited by 9 publications

References 50 publications

An equilibration-based a posteriori error bound for the biharmonic equation and two finite element methods

An equilibration-based a posteriori error bound for the biharmonic equation and two finite element methods

A C⁰ interior penalty discontinuous Galerkin method for fourth‐order total variation flow. I: Derivation of the method and numerical results

Numerical solution of a phase field model for polycrystallization processes in binary mixtures

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A two-energies principle for the biharmonic equation and ana posteriorierror estimator for an interior penalty discontinuous Galerkin approximation

Cited by 9 publications

References 50 publications

An equilibration-based a posteriori error bound for the biharmonic equation and two finite element methods

An equilibration-based a posteriori error bound for the biharmonic equation and two finite element methods

A C0 interior penalty discontinuous Galerkin method for fourth‐order total variation flow. I: Derivation of the method and numerical results

Numerical solution of a phase field model for polycrystallization processes in binary mixtures

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A C⁰ interior penalty discontinuous Galerkin method for fourth‐order total variation flow. I: Derivation of the method and numerical results