B-spline based finite-element method for the stationary quasi-geostrophic equations of the ocean

Kim, Tae Yeon; Iliescu, Traian; Fried, Eliot

doi:10.1016/j.cma.2014.12.024

Cited by 24 publications

(13 citation statements)

References 27 publications

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“…Several numerical tests, commonly employed in the geophysical literature, showed the accuracy of the finite element discretization and illustrated the theoretical estimates. Other recent developments of FEM for QGE include discontinuous Galerkin formulation using C 0 elements [62] and B-splines [69][70][71][72][73]. In particular, an adaptive refinement algorithm for B-splines finite element approximation was presented in [71] for the streamfunction formulation.…”

Section: Finite Element Methods For the Qgementioning

confidence: 99%

Reduced Order Models for the Quasi-Geostrophic Equations: A Brief Survey

Mou

Wang

Wells

et al. 2020

Fluids

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Reduced order models (ROMs) are computational models whose dimension is significantly lower than those obtained through classical numerical discretizations (e.g., finite element, finite difference, finite volume, or spectral methods). Thus, ROMs have been used to accelerate numerical simulations of many query problems, e.g., uncertainty quantification, control, and shape optimization. Projection-based ROMs have been particularly successful in the numerical simulation of fluid flows. In this brief survey, we summarize some recent ROM developments for the quasi-geostrophic equations (QGE) (also known as the barotropic vorticity equations), which are a simplified model for geophysical flows in which rotation plays a central role, such as wind-driven ocean circulation in mid-latitude ocean basins. Since the QGE represent a practical compromise between efficient numerical simulations of ocean flows and accurate representations of large scale ocean dynamics, these equations have often been used in the testing of new numerical methods for ocean flows. ROMs have also been tested on the QGE for various settings in order to understand their potential in efficient numerical simulations of ocean flows. In this paper, we survey the ROMs developed for the QGE in order to understand their potential in efficient numerical simulations of more complex ocean flows: We explain how classical numerical methods for the QGE are used to generate the ROM basis functions, we outline the main steps in the construction of projection-based ROMs (with a particular focus on the under-resolved regime, when the closure problem needs to be addressed), we illustrate the ROMs in the numerical simulation of the QGE for various settings, and we present several potential future research avenues in the ROM exploration of the QGE and more complex models of geophysical flows.

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Section: Finite Element Methods For the Qgementioning

confidence: 99%

Reduced Order Models for the Quasi-Geostrophic Equations: A Brief Survey

Mou

Wang

Wells

et al. 2020

Fluids

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“…Due to the numerical success obtained with (6.3) (and spaces defined therein) in [Strazzullo et al, 2017], we shall continue to use (6.3) in this work. Stabilization procedures could also be used, such as in [Kim et al, 2015].…”

Section: Linearized Quasi-geostrophic Equation On the Atlantic Oceanmentioning

confidence: 99%

A weighted POD-reduction approach for parametrized PDE-constrained Optimal Control Problems with random inputs and applications to environmental sciences

Carere¹,

Strazzullo²,

Ballarin³

et al. 2021

Preprint

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Reduced basis approximations of Optimal Control Problems (OCPs) governed by steady partial differential equations (PDEs) with random parametric inputs are analyzed and constructed. Such approximations are based on a Reduced Order Model, which in this work is constructed using the method of weighted Proper Orthogonal Decomposition. This Reduced Order Model then is used to efficiently compute the reduced basis approximation for any outcome of the random parameter. We demonstrate that such OCPs are well-posed by applying the adjoint approach, which also works in the presence of admissibility constraints and in the case of non linear-quadratic OCPs, and thus is more general than the conventional Lagrangian approach. We also show that a step in the construction of these Reduced Order Models, known as the aggregation step, is not fundamental and can in principle be skipped for noncoercive problems, leading to a cheaper online phase. Numerical applications in three scenarios from environmental science are considered, in which the governing PDE is steady and the control is distributed. Various parameter distributions are taken, and several implementations of the weighted Proper Orthogonal Decomposition are compared by choosing different quadrature rules.

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“…Several numerical tests, commonly employed in the geophysical literature, showed the accuracy of the finite element discretization and illustrated the theoretical estimates. Other recent developments of FEM for QGE include discontinuous Galerkin formulation using C 0 elements [44] and Bsplines [45,41,5,43,87]. In particular, an adaptive refinement algorithm for B-splines finite element approximation was presented in [5] for the streamfunction formulation.…”

Section: Finite Elementmentioning

confidence: 99%

Data-driven correction reduced order models for the quasi-geostrophic equations: a numerical investigation

Mou

Liu

Wells

et al. 2020

International Journal of Computational Fluid Dynamics

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This paper investigates the recently introduced data-driven correction reduced order model (DDC-ROM) in the numerical simulation of the quasi-geostrophic equations. The DDC-ROM uses available data to model the correction term that is generally used to represent the missing information in low-dimensional ROMs. Physical constraints are added to the DDC-ROM to create the constrained data-driven correction reduced order model (CDDC-ROM) in order to further improve its accuracy and stability. Finally, the DDC-ROM is tested on time intervals that are longer than the time interval over which it was trained. The numerical investigation shows that, for low-dimensional ROMs, both the DDC-ROM and CDDC-ROM perform better than the standard Galerkin ROM (G-ROM) and the CDDC-ROM provides the best results.(CM, HL,TI)

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B-spline based finite-element method for the stationary quasi-geostrophic equations of the ocean

Cited by 24 publications

References 27 publications

Reduced Order Models for the Quasi-Geostrophic Equations: A Brief Survey

Reduced Order Models for the Quasi-Geostrophic Equations: A Brief Survey

A weighted POD-reduction approach for parametrized PDE-constrained Optimal Control Problems with random inputs and applications to environmental sciences

Data-driven correction reduced order models for the quasi-geostrophic equations: a numerical investigation

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