Finite element methods for linear hyperbolic problems

Johnson, Claes; Nävert, Uno; Pitkäranta, Juhani

doi:10.1016/0045-7825(84)90158-0

Cited by 626 publications

(355 citation statements)

References 6 publications

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“…Optimal convergence in h of the error in the streamline derivative can also be derived. These results are similar to the typical estimates for other stabilized methods such as Discontinuous Galerkin (DG) method or the Streamline-Diffusion (SD) method [13]. The right-hand side of these estimates, however, depends on a global Sobolev norm.…”

Section: Introductionsupporting

confidence: 73%

Weighted error estimates of the continuous interior penalty method for singularly perturbed problems

Burman

Guzmán

Leykekhman

2008

IMA Journal of Numerical Analysis

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Abstract.In this paper we analyze local properties of the Continuous Interior Penalty (CIP) Method for a model convection-dominated singularly perturbed convection-diffusion problem. We show weighted a priori error estimates, where the weight function exponentially decays outside the subdomain of interest. This result shows that locally, the CIP method is comparable to the Streamline Diffusion (SD) or the Discontinuous Galerkin (DG) methods.

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Section: Introductionsupporting

confidence: 73%

Weighted error estimates of the continuous interior penalty method for singularly perturbed problems

Burman

Guzmán

Leykekhman

2008

IMA Journal of Numerical Analysis

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“…For this type of hyperbolic problems, instabilities still develop when relying on the standard Galerkin FEM, and for this case, we use either streamline upwinding or discontinuous Galerkin finite elements [Johnson et al, 1984;Brezzi et al, 2004].…”

Section: Fem Discretization and Treatment Of Nonlinearitiesmentioning

confidence: 99%

Dynamically and Observationally Constrained Estimates of Water-Mass Distributions and Ages in the Global Ocean

DeVries

Primeau

2011

Journal of Physical Oceanography

180

202

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[1] Ice flow models used to project the mass balance of ice sheets in Greenland and Antarctica usually rely on the Shallow Ice Approximation (SIA) and the Shallow-Shelf Approximation (SSA), sometimes combined into so-called "hybrid" models. Such models, while computationally efficient, are based on a simplified set of physical assumptions about the mechanical regime of the ice flow, which does not uniformly apply everywhere on the ice sheet/ice shelf system, especially near grounding lines, where rapid changes are taking place at present. Here, we present a new thermomechanical finite element model of ice flow named ISSM (Ice Sheet System Model) that includes higher-order stresses, high spatial resolution capability and data assimilation techniques to better capture ice dynamics and produce realistic simulations of ice sheet flow at the continental scale. ISSM includes several approximations of the momentum balance equations, ranging from the two-dimensional SSA to the three-dimensional full-Stokes formulation. It also relies on a massively parallelized architecture and state-of-the-art scalable tools. ISSM employs data assimilation techniques, at all levels of approximation of the momentum balance equations, to infer basal drag at the ice-bed interface from satellite radar interferometry-derived observations of ice motion. Following a validation of ISSM with standard benchmarks, we present a demonstration of its capability in the case of the Greenland Ice Sheet. We show ISSM is able to simulate the ice flow of an entire ice sheet realistically at a high spatial resolution, with higher-order physics, thereby providing a pathway for improving projections of ice sheet evolution in a warming climate.Citation: Larour, E., H. Seroussi, M. Morlighem, and E. Rignot (2012), Continental scale, high order, high spatial resolution, ice sheet modeling using the Ice Sheet System Model (ISSM),

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“…Error analysis The SUPG method was originally analyzed in Johnson, Nävert and Pitkäranta (1984) and Nävert (1982). In this section an a priori error analysis of Galerkin/least-squares is performed.…”

Section: Remarkmentioning

confidence: 99%

Multiscale and Stabilized Methods

Hughes

Scovazzi

Franca

2004

Encyclopedia of Computational Mechanics

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This article presents an introduction to multiscale and stabilized methods, which represent unified approaches to modeling and numerical solution of fluid dynamic phenomena. Finite element applications are emphasized but the ideas are general and apply to other numerical methods as well. (They have been used in the development of finite difference, finite volume, and spectral methods, in addition to finite‐element methods.) The analytical ideas are first illustrated for time‐harmonic wave‐propagation problems in unbounded fluid domains governed by the Helmholtz equation. This leads to the well‐known Dirichlet‐to‐Neumann formulation. A general treatment of the variational multiscale method in the context of an abstract Dirichlet problem is then presented, which is applicable to advective–diffusive processes and other processes of physical interest. It is shown how the exact theory represents a paradigm for subgrid‐scale models and a posteriori error estimation. Hierarchical p ‐methods and bubble function methods are examined in order to understand and ultimately approximate the ‘fine‐scale Green's function’ that appears in the theory. Relationships among so‐called residual‐free bubbles, element Green's functions, and stabilized methods are exhibited. These ideas are then generalized to a class of nonsymmetric, linear evolution operators formulated in space–time. The variational multiscale method also provides guidelines and inspiration for the development of stabilized methods (e.g. SUPG, GLS, etc.) that have attracted considerable interest and have been extensively utilized in engineering and the physical sciences. An overview of stabilized methods for advective–diffusive equations is presented. A variational multiscale treatment of incompressible viscous flows, including turbulence is also described. This represents an alternative formulation of large eddy simulation (LES), which provides a simplified theoretical framework of LES with potential for improved modeling.

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Finite element methods for linear hyperbolic problems

Cited by 626 publications

References 6 publications

Weighted error estimates of the continuous interior penalty method for singularly perturbed problems

Weighted error estimates of the continuous interior penalty method for singularly perturbed problems

Dynamically and Observationally Constrained Estimates of Water-Mass Distributions and Ages in the Global Ocean

Multiscale and Stabilized Methods

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