1,967 research outputs found
Stabilized Schemes for the Hydrostatic Stokes Equations
Some new stable finite element (FE) schemes are presented for the hydrostatic Stokes
system or primitive equations of the ocean. It is known that the stability of the mixed formulation ap-
proximation for primitive equations requires the well-known LadyzhenskayaâBabuËskaâBrezzi condi-
tion related to the Stokes problem and an extra inf-sup condition relating the pressure and the vertical
velocity.
The main goal of this paper is to avoid this extra condition by adding a residual stabilizing term to the
vertical momentum equation. Then, the stability for Stokes-stable FE combinations is extended to
the primitive equations and some error estimates are provided using TaylorâHood P2 âP1 or miniele-
ment (P1 +bubble)âP1 FE approximations, showing the optimal convergence rate in the P2 âP1 case.
These results are also extended to the anisotropic (nonhydrostatic) problem. On the other hand,
by adding another residual term to the continuity equation, a better approximation of the vertical
derivative of pressure is obtained. In this case, stability and error estimates including this better
approximation are deduced, where optimal convergence rate is deduced in the (P 1 +bubble)âP1 case.
Finally, some numerical experiments are presented supporting previous results
Stabilized schemes for the hydrostatic Stokes equations
Some new stable finite element (FE) schemes are presented for the hydrostatic Stokes system or primitive equations of the ocean. It is known that the stability of the mixed formulation approximation for primitive equations requires the well-known LadyzhenskayaâBabuskaâBrezzi condition related to the Stokes problem and an extra inf-sup condition relating the pressure and the vertical velocity [F. GuillĂ©n-GonzĂĄlez and J. R. RodrĂguez-GalvĂĄn, Numer. Math., 130 (2015), pp. 225â256].
The main goal of this paper is to avoid this extra condition by adding a residual stabilizing term to the vertical momentum equation. Then, the stability for Stokes-stable FE combinations is extended to the primitive equations and some error estimates are provided using TaylorâHood P2âP1 or minielement (P1 +bubble)âP1 FE approximations, showing the optimal convergence rate in the P2âP1 case. These results are also extended to the anisotropic (nonhydrostatic) problem. On the other hand, by adding another residual term to the continuity equation, a better approximation of the vertical derivative of pressure is obtained. In this case, stability and error estimates including this better approximation are deduced, where optimal convergence rate is deduced in the (P1 +bubble)âP1 case. Finally, some numerical experiments are presented supporting previous results.Ministerio de EconomĂa y Competitivida
On the stability of bubble functions and a stabilized mixed finite element formulation for the Stokes problem
In this paper we investigate the relationship between stabilized and enriched
finite element formulations for the Stokes problem. We also present a new
stabilized mixed formulation for which the stability parameter is derived
purely by the method of weighted residuals. This new formulation allows equal
order interpolation for the velocity and pressure fields. Finally, we show by
counterexample that a direct equivalence between subgrid-based stabilized
finite element methods and Galerkin methods enriched by bubble functions cannot
be constructed for quadrilateral and hexahedral elements using standard bubble
functions.Comment: 25 pages, 13 figures (The previous version was compiled by mistake
with the wrong style file, the current one uses amsart, and there is no
difference in the text or the figures
A Review of Element-Based Galerkin Methods for Numerical Weather Prediction: Finite Elements, Spectral Elements, and Discontinuous Galerkin
Numerical weather prediction (NWP) is in a period of transition. As resolutions increase, global models are moving towards fully nonhydrostatic dynamical cores, with the local and global models using the same governing equations; therefore we have reached a point where it will be necessary to use a single model for both applications. The new dynamical cores at the heart of these unified models are designed to scale efficiently on clusters with hundreds of thousands or even millions of CPU cores and GPUs. Operational and research NWP codes currently use a wide range of numerical methods: finite differences, spectral transform, finite volumes and, increasingly, finite/spectral elements and discontinuous Galerkin, which constitute element-based Galerkin (EBG) methods.Due to their important role in this transition, will EBGs be the dominant power behind NWP in the next 10 years, or will they just be one of many methods to choose from? One decade after the review of numerical methods for atmospheric modeling by Steppeler et al. (Meteorol Atmos Phys 82:287â301, 2003), this review discusses EBG methods as a viable numerical approach for the next-generation NWP models. One well-known weakness of EBG methods is the generation of unphysical oscillations in advection-dominated flows; special attention is hence devoted to dissipation-based stabilization methods. Since EBGs are geometrically flexible and allow both conforming and non-conforming meshes, as well as grid adaptivity, this review is concluded with a short overview of how mesh generation and dynamic mesh refinement are becoming as important for atmospheric modeling as they have been for engineering applications for many years.The authors would like to thank Prof. Eugenio Oñate (U. PolitĂšcnica de Catalunya) for his invitation to submit this review article. They are also thankful to Prof. Dale Durran (U. Washington), Dr. Tommaso Benacchio (Met Office), and Dr. Matias Avila (BSC-CNS) for their comments and corrections, as well as
insightful discussion with Sam Watson, Consulting Software Engineer (Exa Corp.) Most of the contribution to this article by the first author stems from his Ph.D. thesis carried out at the Barcelona Supercomputing Center (BSCCNS) and Universitat PolitĂšcnica de Catalunya, Spain, supported by a BSC-CNS student grant, by Iberdrola EnergĂas Renovables, and by grant N62909-09-1-4083 of the Office of Naval Research Global. At NPS, SM, AM, MK, and FXG were supported by the Office of Naval Research through program element PE-0602435N, the Air Force Office of Scientific Research through the Computational Mathematics program, and the National Science Foundation (Division of Mathematical Sciences) through program element 121670. The scalability studies of the
atmospheric model NUMA that are presented in this paper used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. SM, MK, and AM are grateful to the National Research Council of the National Academies.Peer ReviewedPostprint (author's final draft
Finite element approximation of 3D non-hydrostatic turbulent coastal ocean flows using a LES model
In this paper we present a stabilized finite element method for three-dimensional,
non-hydrostatic, turbulent coastal ocean flows. The model we have developed, named
HELIKE, incorporates also surface wind stress, bottom friction, Coriolis forces and
several closure models for both the horizontal and the vertical turbulent eddy vis-
cosity coefficients. Unstructured meshes are employed so that complex geometries
can be accurately approximated, and implicit time stepping allows to use large time
steps. Numerical results are presented in various test cases, in which comparisons
between different turbulence models are provided
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