A  $C^0$ Interior Penalty Discontinuous Galerkin Method and an equilibrated <i>a posteriori</i> error estimator for a nonlinear fourth order elliptic boundary value problem of $p$-biharmonic type

Hoppe, Ronald H. W.

doi:10.1051/m2an/2022058

A

C^{0}

Interior Penalty Discontinuous Galerkin Method and an equilibrated a posteriori error estimator for a nonlinear fourth order elliptic boundary value problem of

p

-biharmonic type

Hoppe, Ronald H. W.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 6, pp. 2051-2079

Résumé

We consider a C⁰ Interior Penalty Discontinuous Galerkin (C0IPDG) approximation of a nonlinear fourth order elliptic boundary value problem of p-biharmonic type and an equilibrated a posteriori error estimator. The C0IPDG method can be derived from a discretization of the corresponding minimization problem involving a suitably defined reconstruction operator. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W$$ norm in terms of the associated primal and dual energy functionals. It requires the construction of an equilibrated flux and an equilibrated moment tensor based on a three-field formulation of the C0IPDG approximation. The relationship with a residual-type a posteriori error estimator is studied as well. Numerical results illustrate the performance of the suggested approach.

MR

DOI : 10.1051/m2an/2022058

Classification : 35K35, 35K55, 65M60
Keywords: C⁰ Interior Penalty Discontinuous Galerkin approximation, nonlinear fourth order elliptic problem of $$-biharmonic type, $$ error estimation, equilibration

@article{M2AN_2022__56_6_2051_0,
     author = {Hoppe, Ronald H. W.},
     title = {A  $C^0$ {Interior} {Penalty} {Discontinuous} {Galerkin} {Method} and an equilibrated \protect\emph{a posteriori} error estimator for a nonlinear fourth order elliptic boundary value problem of $p$-biharmonic type},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {2051--2079},
     year = {2022},
     publisher = {EDP-Sciences},
     volume = {56},
     number = {6},
     doi = {10.1051/m2an/2022058},
     mrnumber = {4481121},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2022058/}
}

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Hoppe, Ronald H. W. A  $C^0$ Interior Penalty Discontinuous Galerkin Method and an equilibrated a posteriori error estimator for a nonlinear fourth order elliptic boundary value problem of $p$-biharmonic type. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 6, pp. 2051-2079. doi: 10.1051/m2an/2022058

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