A Priori Error Estimates for Finite Element Methods Based on Discontinuous Approximation Spaces for Elliptic Problems

By Beatrice M. Riviere, Mary F. Wheeler, Vivette Girault. SIAM Journal on Numerical Analysis, 2001

413 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/0c4bc7caf6af539a8ce5168ee72f699559516ba4

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Citations in the corpus

A Combined Mixed Finite Element and Discontinuous Galerkin Method for Miscible Displacement Problem in Porous Media (2002, methodology): “The discontinuous Galerkin method gained even greater popularity recently for at least four reasons [10, 11, 12, 13]: 1) the flexibility inherent to it allows more general meshes construction and degree of non uniformity than permitted by the more conventional finite element method; 2) it also conse”
Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case (2007, background): “Furthermore, DG approximations of elliptic equations have become increasingly popular in recent years [11, 15, 20, 28, 36, 37] (see in particular [4] for a unified analysis and [2] for the spectral analysis).”
Discontinuous Galerkin methods for coupled flow and reactive transport problems (2005, background): “of flow d oupling the unily, the r in order ces for flow and t; ecently cal dif- Primal discontinuous Galerkin methods with interior penalty are proposed to solve the coupled system and reactive transport in porous media, which arises from many applications includingmiscible displacement an…”
A parameter-free variational coupling approach for trimmed isogeometric thin shells (2017, methodology): “In the context of the non-symmetric interior penalty discontinous Galerkin method [55,57], penalty-free nonsymmetric Nitsche formulations have been reported to lead to oscillations near interfaces [77].”
Estimation of penalty parameters for symmetric interior penalty Galerkin methods (2007, methodology): “Another related discontinuous Galerkin method is the non-symmetric interior penalty Galerkin (NIPG) method [21] [15]: this method differs from the SIPG method by only one sign: the symmetrizing term is added instead of being subtracted.”
Discontinuous Galerkin methods for reactive transport in porous media (2003, methodology): “DG for elliptic problems has been analyzed by Rivière, Wheeler and Girault [76] for the non-symmetric case with and without penalty.”
Anisotropic and dynamic mesh adaptation for discontinuous Galerkin methods applied to reactive transport (2006, methodology): “DG has recently been applied to flow and transport problems in porous media [37,28,52,41,43], and an optimal convergence in L(H) was demonstrated for flow and transport problems [28,30,36,38].”
Weak imposition of Dirichlet boundary conditions in fluid mechanics (2007, methodology): “For example, the case of γ = −1, C b = 0 corresponds to the method of Baumann and Oden [7]; γ = −1, C eb > 0 is the Non-symmetric Interior Penalty Galerkin (NIPG) method [43]; and γ = 1, C eb > 0 is the Symmetric Interior Penalty Galerkin (SIPG) method [50].”
Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: Motivation, formulation, and numerical examples (2008, methodology): “This analysis studied the effect of stabilization on the performance for the Babuška-Zlámal method [4], the IP method [23], the LDG method [13] and NIPG methods [30].”
Forty Years of the Crouzeix‐Raviart element (2015, methodology): “61) implies the interpolation error is not affected by the hanging nodes, a unique feature of the WOPSIP method that is not shared by other discontinuous finite element methods (such as the ones in [5, 66]).”
A posteriori error estimation and dynamic adaptivity for symmetric discontinuous Galerkin approximations of reactive transport problems (2006, methodology): “It has been found that DG methods have optimal convergence in L(2)(H(1)) for both flow and transport problems [41,42,48].”
The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements (2016, neutral): “Such an interpolant uI only exists for bases of polynomial degree p ≥ 2 [58, 59], so that their analysis does not hold for linear basis functions.”
On Discontinuous Galerkin Methods for Elliptic Problems with Discontinuous Coefficients (2003, methodology): “1) is discretized by the discontinuous Galerkin (DG) method (see [1, 7]) across ∂Ωi while in each Ωi the discretization is given by the conforming finite element method (FEM) on triangular elements.”
Discontinuous Galerkin methods for flow and transport problems in porous media (2001, methodology): “The continuous in time Discontinuous Galerkin (DG) [ 6 ] method is given by the map”