A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems

Adjerid, Slimane; Devine, Karen Dragon; Flaherty, Joseph E.; Krivodonova, Lilia

doi:10.1016/s0045-7825(01)00318-8

Cited by 193 publications

(174 citation statements)

References 16 publications

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“…If Ù Ü is the exact solution of (1) and is the size of element (e.g. the radius of the circumsphere of a tetrahedron) it has been proven in [1] that the following result holds 1…”

Section: Isolation Of Discontinuities Using a Smoothness Indicatormentioning

confidence: 99%

Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods

Remacle

Shephard

et al. 2005

Int. J. Numer. Meth. Engng

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SUMMARYAn anisotropic adaptive analysis procedure based on a discontinuous Galerkin finite element discretization and local mesh modification of simplex elements is presented. The procedure is applied to transient 2-and 3-dimensional problems governed by Euler's equation. A smoothness indicator is used to isolate jump features where an aligned mesh metric field in specified. The mesh metric field in smooth portions of the domain is controlled by a Hessian matrix constructed using a variational procedure to calculate the second derivatives. The transient examples included demonstrate the ability of the mesh modification procedures to effectively track evolving interacting features of general shape as they move through a domain.

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“…If Ù Ü is the exact solution of (1) and is the size of element (e.g. the radius of the circumsphere of a tetrahedron) it has been proven in [1] that the following result holds 1…”

Section: Isolation Of Discontinuities Using a Smoothness Indicatormentioning

confidence: 99%

Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods

Remacle

Shephard

et al. 2005

Int. J. Numer. Meth. Engng

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“…Early work on error estimation was conducted [85]. Significant work has been done in adjoint-based error estimation for hp-adaptivity in PDEs using finite element discretizations [28,6,6,179,201,17,80,53,141,211,29,167,4,21,89,90,106,122]. Additional work has been done on finite-volume methods [221,130].…”

Section: Adjoints For Error Estimation Uncertainty Quantificationmentioning

confidence: 99%

“…T_1 T_1 T_1 T_1 T_2 T_2 T_2 T_2 T_3 T_3 T_3 T_3 T_4 T_4 T_4 The first type of operation performed by MultiVector::apply(...) is the parallel/parallel matrix-matrix products performed in the lines 4 The aspect ratio of the number of rows to number of columns in Figure 8.9 is exaggerated in that in a realistic case the number of rows usually numbers in the tens to hundreds of thousands while the number of columns usually number in only the tens. This was done for illustrative purposes.…”

Section: Multivectormentioning

confidence: 99%

Sensitivity technologies for large scale simulation.

Collis¹,

Bartlett²,

Smith³

et al. 2005

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“…In [30,31] Remacle et al present a second order scheme for shallow water equations with anisotropic grid adaptivity on triangular meshes, but without adressing well-balancing and positivity-preserving. Since these concepts offer no reliable error control, a priori as well as a posteriori error estimates have been developed to control the adaptive process, e.g., Bey and Oden [32], Adjerid et al [33], Houston et al [34,35,36], Dedner et al [37], and recently Mavriplis et al [38].…”

Section: Introductionmentioning

confidence: 99%

Multiwavelet-based grid adaptation with discontinuous Galerkin schemes for shallow water equations

Gerhard

Caviedes‐Voullième

Müller

et al. 2015

Journal of Computational Physics

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We provide an adaptive strategy for solving shallow water equations with dynamic grid adaptation including a sparse representation of the bottom topography. A challenge in computing approximate solutions to the shallow water equations including wetting and drying is to achieve the positivity of the water height and the well-balancing of the approximate solution. A key property of our adaptive strategy is that it guarantees that these properties are preserved during the refinement and coarsening steps in the adaptation process.The underlying idea of our adaptive strategy is to perform a multiresolution analysis using multiwavelets on a hierarchy of nested grids. This provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth. Applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients. Furthermore we use the multiresolution analysis of the underlying data as an additional indicator whether the limiter has to be applied on a cell or not. By this the number of cells where the limiter is applied is reduced without spoiling the accuracy of the solution.By means of well-known 1D and 2D benchmark problems, we verify that multiwavelet-based grid adaptation can significantly reduce the computational cost by sparsening the computational grids, while retaining accuracy and keeping well-balancing and positivity.

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A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems

Cited by 193 publications

References 16 publications

Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods

Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods

Sensitivity technologies for large scale simulation.

Multiwavelet-based grid adaptation with discontinuous Galerkin schemes for shallow water equations

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