Nonlinear Galerkin Methods

DOI10.1137/0726063zbMath0683.65083OpenAlexW2023743841MaRDI QIDQ4733339

Martine Marion, Roger M. Temam

Publication date: 1989

Published in: SIAM Journal on Numerical Analysis (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1137/0726063

zbMATH Keywords

convergenceNavier-Stokes equationsnonlinear Galerkin methodKuramoto-Sivashinsky equationintegration methodlong-term integrationnonlinear projection

Mathematics Subject Classification ID

Iterative procedures involving nonlinear operators (47J25) Nonlinear differential equations in abstract spaces (34G20) Navier-Stokes equations (35Q30) Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs (65M60) Numerical solutions to equations with nonlinear operators (65J15)

Related Items (only showing first 100 items - show all)

A holistic finite difference approach models linear dynamics consistently ⋮ THE NONLINEAR GALERKIN METHOD APPLIED TO SHALLOW WATER EQUATIONS ⋮ Finite-dimensional behavior in dissipative partial differential equations ⋮ The post-processed Galerkin method applied to non-linear shell vibrations ⋮ Nonlinear galerkin methods using hierarchical almost-orthogonal finite elements bases ⋮ The Generalized Nonlinear Galerkin Method Applied to Hydrodynamics of the Open Sea ⋮ A postprocessing mixed finite element method for the Navier–Stokes equations ⋮ Finite element multilevel approximation of a function and applications ⋮ Stability Analysis of the Nonlinear Galerkin Method ⋮ Nonlinear Galerkin finite element method applied to the system of reaction-diffusion equations in one space dimension ⋮ A partial integral equation (PIE) representation of coupled linear PDEs and scalable stability analysis using LMIs ⋮ Optimizing Oblique Projections for Nonlinear Systems using Trajectories ⋮ Approximate inertial manifolds for the pattern formation Cahn-Hilliard equation ⋮ A nonlinear adaptative multiresolution method in finite differences with incremental unknowns ⋮ Unnamed Item ⋮ A linear discrete scheme for the Ginzburg–Landau equation ⋮ Conservative model reduction for finite-volume models ⋮ Reduced and bifurcation analysis of intrinsically bursting neuron model ⋮ Sum-of-squares approach to feedback control of laminar wake flows ⋮ Uniform-in-Time Error Estimates for the Postprocessing Galerkin Method Applied to a Data Assimilation Algorithm ⋮ Linearized Stability of Partial Differential Equations with Application to Stabilization of the Kuramoto--Sivashinsky Equation ⋮ On the Rate of Convergence of the Nonlinear Galerkin Methods ⋮ Numerical investigation on global dynamics for nonlinear stochastic heat conduction via global random attractors theory ⋮ Nonlinear dynamics of mechanical systems under complicated forcing ⋮ On a construction of approximate inertial manifolds for second order in time evolution equations ⋮ An intrinsic stabilization scheme for proper orthogonal decomposition based low-dimensional models ⋮ Stability and convergence of the spectral Galerkin method for the Cahn-Hilliard equation ⋮ Fourier spectral approximation to long-time behavior of three dimensional Ginzburg-Landau type equation ⋮ Stability and convergence of optimum spectral non-linear Galerkin methods ⋮ A family of approximate inertial manifolds for a Van der Pol/FitzHugh–Nagumo perturbation problem ⋮ Numerical solutions of the Navier-Stokes equations using wavelet-like incremental unknowns ⋮ Fully discrete nonlinear galerkin methods for Kuramoto-Sivashinsky equation and their error estimates ⋮ A sparse grid method for the Navier-Stokes equations based on hyperbolic cross ⋮ Incremental unknowns method and compact schemes ⋮ Stability and convergence of the reform postprocessing Galerkin method ⋮ The application of modern numerical methods to the neutron transport equation ⋮ Holistic discretization ensures fidelity to Burgers' equation ⋮ A dynamical multi-level scheme for the Burgers equation: Wavelet and hierarchical finite element ⋮ Two-grid finite-element schemes for the transient Navier-Stokes problem ⋮ An AIM and one-step Newton method for the Navier-Stokes equations ⋮ Nonlinear Galerkin method for reaction-diffusion systems admitting invariant regions ⋮ A nonlinear Galerkin method for the shallow-water equations on periodic domains ⋮ Dimensionality reduction using secant-based projection methods: the induced dynamics in projected systems ⋮ A two-level correction method in space and time based on Crank–Nicolson scheme for Navier–Stokes equations ⋮ Approximate inertial manifolds and effective viscosity in turbulent flows ⋮ On bifurcation from steady-state solutions to rotating waves in the Kuramoto-Sivashinsky equation ⋮ Separation of variables in the Stokes problem application to its finite element multiscale approximation ⋮ Analysis and Approximation of a Fractional Cahn--Hilliard Equation ⋮ Induced trajectories and approximate inertial manifolds ⋮ Long time stability and convergence for fully discrete nonlinear galerkin methods ⋮ Nonlinear hybrid procedures and fixed point iterations ⋮ Incremental unknowns on nonuniform meshes ⋮ Résolution des équations de shallow water par la méthode de Galerkin non linéaire ⋮ On the construction of suitable solutions to the Navier-Stokes equations and questions regarding the definition of large eddy simulation ⋮ Bifurcations of finite difference schemes and their approximate inertial forms ⋮ A two-level method in space and time for the Navier-Stokes equations ⋮ An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equations ⋮ A modified nonlinear spectral Galerkin method for the equations of motion arising in the Kelvin–Voigt fluids ⋮ Nonlinear Galerkin method in the finite difference case and wavelet-like incremental unknowns ⋮ Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum ⋮ Multilevel finite volume methods and boundary conditions for geophysical flows ⋮ Full discrete nonlinear Galerkin method for the Navier-Stokes equations ⋮ Implementation of finite element nonlinear Galerkin methods using hierarchical bases ⋮ An invariant manifold approach to nonlinear normal modes of oscillation ⋮ Tangent space correction method for the Galerkin approximation based on two-grid finite element ⋮ Implementation and numerical analysis of the nonlinear Galerkin methods with finite elements discretization ⋮ A space-time formulation and improved spatial reconstruction for the ``divide-and-conquer multiscale method ⋮ Asymptotic construction of nonlinear normal modes for continuous systems ⋮ A finite difference scheme for computing inertial manifolds ⋮ A second order accuracy for a full discretized time-dependent Navier-Stokes equations by a two-grid scheme ⋮ Dynamical multilevel schemes for the solution of evolution equations by hierarchical finite element discretization ⋮ On the effectiveness of the approximate inertial manifold -- a computational study ⋮ The nonlinear Galerkin method: A multiscale method applied to the simulation of homogeneous turbulent flows ⋮ Subgrid modelling and the interaction of small and large wavelength in turbulent flows ⋮ Some closure results for inertial manifolds ⋮ Fourier nonlinear Galerkin approximation for the two-dimensional Navier-Stokes equations ⋮ Inadequacy of linear methods for minimal sensor placement and feature selection in nonlinear systems: a new approach using secants ⋮ A two-level method in time and space for solving the Navier-Stokes equations based on Newton iteration ⋮ Higher-order incremental unknowns, hiearchical basis, and nonlinear dissipative evolutionary equations ⋮ Solution of generalized Stokes problems using hierarchical methods and incremental unknowns ⋮ A finite element method for computing the bifurcation function for semilinear elliptic BVPs ⋮ Stability analysis of nonlinear Galerkin and Galerkin methods for nonlinear evolution equations ⋮ A multilevel method for the resolution of a stochastic weakly damped nonlinear Schrödinger equation ⋮ Galerkin and inertial manifold methods for finite-dimensional approximation of nonlocal equations of nonlinear optics ⋮ Remarks on nonlinear Galerkin method for Kuramoto-Sivashinsky equation ⋮ Stability and convergence analysis of fully discrete Fourier collocation spectral method for 3-D viscous Burgers' equation ⋮ Nonlinear Galerkin finite element methods for fourth-order bi-flux diffusion model with nonlinear reaction term ⋮ A new nonlinear Galerkin finite element method for the computation of reaction diffusion equations ⋮ A nonlinear Galerkin mixed element method and a posteriori error estimator for the stationary Navier-Stokes equations ⋮ Closed-loop fluid flow control using a low dimensional model ⋮ Galerkin v. least-squares Petrov-Galerkin projection in nonlinear model reduction ⋮ A priori analysis of multilevel finite volume approximation of 1D convective Cahn-Hilliard equation ⋮ Strong and weak constraint variational assimilations for reduced order fluid flow modeling ⋮ Finite volume multilevel approximation of the shallow water equations ⋮ Optimal error bounds for two-grid schemes applied to the Navier-Stokes equations ⋮ Order reduction and nonlinear behaviors of a continuous rotor system ⋮ Induced trajectories and approximate inertial manifolds for the Ginzburg- Landau partial differential equation ⋮ IMD based nonlinear Galerkin method. ⋮ Nonlinear Galerkin methods: The finite elements case ⋮ The nonlinear Galerkin method in computational fluid dynamics

This page was built for publication: Nonlinear Galerkin Methods