Galerkin method ≈ Galerkin method
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Model Reduction for Flow Analysis and Control Open
Advances in experimental techniques and the ever-increasing fidelity of numerical simulations have led to an abundance of data describing fluid flows. This review discusses a range of techniques for analyzing such data, with the aim of ext…
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Model reduction for nonlinear dynamical systems using deep convolutional autoencoders. Open
Nearly all model-reduction techniques project the governing equations onto a linear subspace of the original state space. Such subspaces are typically computed using methods such as balanced truncation, rational interpolation, the reduced-…
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Constrained sparse Galerkin regression Open
The sparse identification of nonlinear dynamics (SINDy) is a recently proposed data-driven modelling framework that uses sparse regression techniques to identify nonlinear low-order models. With the goal of low-order models of a fluid flow…
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Effect of variable thermal conductivity and viscosity on Casson nanofluid flow with convective heating and velocity slip Open
This work investigates the effects of combined variable viscosity and thermal conductivity, nonlinear radiation and non-Darcian porous medium on a boundary layer MHD Casson nanofluid flow over a vertical flat plate with convective heating …
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Variational Physics-Informed Neural Networks For Solving Partial Differential Equations Open
Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of points in the domain of interest. Here, we develop a P…
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Analysis of L1-Galerkin FEMs for Time-Fractional Nonlinear Parabolic Problems Open
This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of $L1$-Galerkin finite element methods. The analysis of $L1$ methods for time-fractional nonlinear problems is limited mainly due …
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Introduction to the Finite Element Method Open
There are mainly two techniques that are used to discretize elliptic problems: the finite difference method based upon the strong formulation of the problem (and therefore on classical solutions), and the finite element method based upon t…
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Global cracking elements: A novel tool for Galerkin‐based approaches simulating quasi‐brittle fracture Open
Summary Following the so‐called cracking elements method (CEM), we propose a novel Galerkin‐based numerical approach for simulating quasi‐brittle fracture, named global cracking elements method (GCEM). For this purpose the formulation of t…
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POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder Open
Vortex shedding around circular cylinders is a well known and studied phenomenon that appears in many engineering fields. A Reduced Order Model (ROM) of the incompressible ow around a circular cylinder is presented in this work. The ROM is…
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Thetis coastal ocean model: discontinuous Galerkin discretization for the three-dimensional hydrostatic equations Open
Unstructured grid ocean models are advantageous for simulating the coastal ocean and river–estuary–plume systems. However, unstructured grid models tend to be diffusive and/or computationally expensive, which limits their applicability to …
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Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models Open
As a first step, variational formulations and governing equations with boundary conditions are derived for a pair of Euler–Bernoulli beam bending models following a simplified version of Mindlin’s strain gradient elasticity theory of form …
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Control Space Reduction and Real-Time Accurate Modeling of Continuum Manipulators Using Ritz and Ritz–Galerkin Methods Open
To address the challenges with real-time accurate modeling of multi-segment continuum manipulators in the presence of significant external and body loads, we introduce a novel series solution for variable-curvature Cosserat rod static and …
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Effect of Heat source on an unsteady MHD free convection flow of Casson fluid past a vertical oscillating plate in porous medium using finite element analysis Open
The present research paper aims to study the effect Heat source on an MHD Casson fluid through a vertical fluctuating porous plate. Dimensional non-linear coupled differential equations transformed into dimensional less by introducing simi…
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Numerical Analysis of Elliptic Hemivariational Inequalities Open
This paper is devoted to a study of the numerical solution of elliptic hemivariational inequalities with or without convex constraints by the finite element method. For a general family of elliptic hemivariational inequalities that facilit…
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Fast Matrix-Free Evaluation of Discontinuous Galerkin Finite Element Operators Open
We present an algorithmic framework for matrix-free evaluation of discontinuous Galerkin finite element operators. It relies on fast quadrature with sum factorization on quadrilateral and hexahedral meshes, targeting general weak forms of …
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On the Vibrations and Stability of Moving Viscoelastic Axially Functionally Graded Nanobeams Open
In this article, size-dependent vibrations and the stability of moving viscoelastic axially functionally graded (AFG) nanobeams were investigated numerically and analytically, aiming at the stability enhancement of translating nanosystems.…
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Analysis and numerical solution of SEIR epidemic model of measles with non-integer time fractional derivatives by using Laplace Adomian Decomposition Method Open
SEIR epidemic model which represents the direct transmission of infectious disease are considered to control the measles disease for infected population. The Caputo fractional derivative operator of order α∈(0,1] is employed to obtain the …
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A semi-analytical analysis of electro-thermo-hydrodynamic stability in dielectric nanofluids using Buongiorno’s mathematical model together with more realistic boundary conditions Open
The main aim of the present analysis is to examine the electroconvection phenomenon that takes place in a dielectric nanofluid under the influence of a perpendicularly applied alternating electric field. In this investigation, we assume th…
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Incompressible fluid problems on embedded surfaces: Modeling and variational formulations Open
Governing equations of motion for a viscous incompressible material surface are derived from the balance laws of continuum mechanics. The surface is treated as a time-dependent smooth orientable manifold of codimension one in an ambient Eu…
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Galerkin approximation of dynamical quantities using trajectory data Open
Understanding chemical mechanisms requires estimating dynamical statistics such as expected hitting times, reaction rates, and committors. Here, we present a general framework for calculating these dynamical quantities by approximating bou…
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Space--Time Least-Squares Petrov--Galerkin Projection for Nonlinear Model Reduction Open
This work proposes a space--time least-squares Petrov--Galerkin (ST-LSPG) projection method for model reduction of nonlinear dynamical systems. In contrast to typical nonlinear model-reduction methods that first apply (Petrov--)Galerkin pr…
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Physics-constrained, low-dimensional models for magnetohydrodynamics: First-principles and data-driven approaches Open
Plasmas are highly nonlinear and multiscale, motivating a hierarchy of models to understand and describe their behavior. However, there is a scarcity of plasma models of lower fidelity than magnetohydrodynamics (MHD), although these reduce…
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Arbitrary-Lagrangian–Eulerian Discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes Open
We present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may al…
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Subdiffusion with a time-dependent coefficient: Analysis and numerical solution Open
In this work, a complete error analysis is presented for fully discrete solutions of the subdiffusion equation with a time-dependent diffusion coefficient, obtained by the Galerkin finite element method with conforming piecewise linear fin…
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Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method Open
We present a novel, cell-local shock detector for use with discontinuous Galerkin (DG) methods. The output of this detector is a reliably scaled, element-wise smoothness estimate which is suited as a control input to a shock capture mechan…
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Nonintrusive reduced order modeling framework for quasigeostrophic turbulence Open
In this study, we present a nonintrusive reduced order modeling (ROM) framework for large-scale quasistationary systems. The framework proposed herein exploits the time series prediction capability of long short-term memory (LSTM) recurren…
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Galerkin Finite Element Approximations for Stochastic Space-Time Fractional Wave Equations Open
The traditional wave equation models wave propagation in an ideal medium. To characterize wave propagation in inhomogeneous media with frequency-dependent power-law attenuation, the space-time fractional wave equation is needed; further in…
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Progress in analytical methods to predict and control azimuthal combustion instability modes in annular chambers Open
Longitudinal low-frequency thermoacoustic unstable modes in combustion chambers have been intensively studied experimentally, numerically, and theoretically, leading to significant progress in both understanding and controlling these acous…
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High Order ADER Schemes for Continuum Mechanics Open
In this paper we first review the development of high order ADER finite\nvolume and ADER discontinuous Galerkin schemes on fixed and moving meshes,\nsince their introduction in 1999 by Toro et al. We show the modern variant of\nADER based …
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Discontinuous Galerkin Approximation of Flows in Fractured Porous Media on Polytopic Grids Open
We present a numerical approximation of Darcy's flow through a fractured porous medium which employs discontinuous Galerkin methods on polytopic grids. For simplicity, we analyze the case of a single fracture represented by a (d 1)-dimensi…