A posteriori error of a discontinuous Galerkin scheme for compressible miscible displacement problems with molecular diffusion and dispersion

Abstract We are concerned with an Interior Penalty Discontinuous Galerkin (IPDG) approximation of the p-Laplace equation and an equilibrated a posteriori error estimator. The IPDG method can be derived from a discretization of the associated minimization problem involving appropriately defined reconstruction operators. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W 1,p norm and relies on the construction of an equilibrated flux in terms of a numerical flux function associated with the mixed formulation of the IPDG approximation. The relationship with a residual-type a posteriori error estimator is established as well. Numerical results illustrate the performance of both estimators.

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Towards A Posteriori Error Estimators for Realistic Problems in Incompressible Miscible Displacement

Numerical Mathematics and Advanced Applications 2011 ◽

10.1007/978-3-642-33134-3_35 ◽

2012 ◽

pp. 323-331

Author(s):

J. Chapman ◽

M. Jensen

Keyword(s):

Posteriori Error ◽

Miscible Displacement ◽

A Posteriori ◽

A Posteriori Error Estimators ◽

A Posteriori Error ◽

Error Estimators ◽

Posteriori Error Estimators

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On the error estimates of a hybridizable discontinuous Galerkin method for second-order elliptic problem with discontinuous coefficients

IMA Journal of Numerical Analysis ◽

10.1093/imanum/drz003 ◽

2019 ◽

Vol 40 (2) ◽

pp. 1577-1600

Author(s):

Gang Chen ◽

Jintao Cui

Keyword(s):

Error Estimates ◽

Discontinuous Galerkin ◽

Elliptic Problem ◽

Posteriori Error ◽

Discontinuous Coefficients ◽

Second Order ◽

A Posteriori ◽

A Posteriori Error ◽

Hybridizable Discontinuous Galerkin ◽

Order Elliptic Problem

Abstract Hybridizable discontinuous Galerkin (HDG) methods retain the main advantages of standard discontinuous Galerkin (DG) methods, including their flexibility in meshing, ease of design and implementation, ease of use within an $hp$-adaptive strategy and preservation of local conservation of physical quantities. Moreover, HDG methods can significantly reduce the number of degrees of freedom, resulting in a substantial reduction of computational cost. In this paper, we study an HDG method for the second-order elliptic problem with discontinuous coefficients. The numerical scheme is proposed on general polygonal and polyhedral meshes with specially designed stabilization parameters. Robust a priori and a posteriori error estimates are derived without a full elliptic regularity assumption. The proposed a posteriori error estimators are proved to be efficient and reliable without a quasi-monotonicity assumption on the diffusion coefficient.

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Divergence-conforming discontinuous Galerkin finite elements for Stokes eigenvalue problems

Numerische Mathematik ◽

10.1007/s00211-019-01095-x ◽

2019 ◽

Vol 144 (3) ◽

pp. 585-614

Author(s):

Joscha Gedicke ◽

Arbaz Khan

Keyword(s):

Discontinuous Galerkin ◽

Eigenvalue Problems ◽

Convergence Rates ◽

A Priori ◽

Posteriori Error ◽

Error Estimator ◽

A Posteriori ◽

A Posteriori Error Estimator ◽

Posteriori Error Estimator ◽

A Posteriori Error

AbstractIn this paper, we present a divergence-conforming discontinuous Galerkin finite element method for Stokes eigenvalue problems. We prove a priori error estimates for the eigenvalue and eigenfunction errors and present a residual based a posteriori error estimator. The a posteriori error estimator is proven to be reliable and (locally) efficient. We finally present some numerical examples that verify the a priori convergence rates and the reliability and efficiency of the residual based a posteriori error estimator.

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