Nonlinear Galerkin Research Articles

A nonlinear galerkin mixed element method and a posteriori error estimator for the stationary navier-stokes equations

A nonlinear Galerkin mixed element (NGME) method and a posteriori error exstimator based on the method are established for the stationary Navier-Stokes equations. The existence and error estimates of the NGME solution are first discussed, and then a posteriori error estimator based on the NGME method is derived.

A nonlinear Galerkin/Petrov-least squares mixed element method for the stationary Navier-Stokes equations

A nonlinear Galerkin/Petrov-least squares mixed element (NGPLSME) method for the stationary Navier-Stokes equations is presented and analyzed. The scheme is that Petrov-least squares forms of residuals are added to the nonlinear Galerkin mixed element method so that it is stable for any combination of discrete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition. The existence, uniqueness and convergence (at optimal rate) of the NGPLSME solution is proved in the case of sufficient viscosity (or small data).

Dual-weighted-residual and nonlinear Galerkin methods in the simulation of the aeroelastic response of windturbines

Dual-weighted-residual and nonlinear Galerkin methods in the simulation of the aeroelastic response of windturbines

Nonlinear Galerkin method for reaction–diffusion systems admitting invariant regions

The article presents an analysis of the nonlinear Galerkin method applied to a system of reaction–diffusion equations. If the system admits a bounded invariant region, it is possible to demonstrate the convergence of the approximate solutions to the weak solution of the system. The proof is based on the compactness technique. It is performed for arbitrary ratio of dimensions of the approximation space and of the correction space used in the nonlinear Galerkin method. This fact, generalizing the previously published results, is important for the practical use of the method and allows optimization of the CPU-time consumption of the algorithm. The method is applied to the well-known Brusselator system for which we present an overview of the computational results and our experience with the numerical method used.

A Nonlinear Galerkin Method for the Shallow-Water Equations on Periodic Domains

A nonlinear Galerkin method for the shallow-water equations is developed, based on spectral transforms. The scheme is compared to a pseudo-spectral Galerkin method. Our numerical results indicate that the nonlinear scheme has the potential advantage of providing similar accuracy at a lower cost than the Galerkin method. The nonlinear method has also less restrictive stability conditions.

Stability and convergence of optimum spectral non‐linear Galerkin methods

AbstractOur objective in this article is to present some numerical schemes for the approximation of the 2‐D Navier–Stokes equations with periodic boundary conditions, and to study the stability and convergence of the schemes. Spatial discretization can be performed by either the spectral Galerkin method or the optimum spectral non‐linear Galerkin method; time discretization is done by the Euler scheme and a two‐step scheme. Our results show that under the same convergence rate the optimum spectral non‐linear Galerkin method is superior to the usual Galerkin methods. Finally, numerical example is provided and supports our results. Copyright © 2001 John Wiley & Sons, Ltd.

Methods for dimension reduction and their application in nonlinear dynamics

We compare linear and nonlinear Galerkin methods in their efficiency to reduce infinite dimensional systems, described by partial differential equations, to low dimensional systems of ordinary differential equations, both concerning the effort in their application and the accuracy of the resulting reduced system. Important questions like the choice of the form of the ansatz functions (modes), the choice of the number m of modes and, finally, the construction of the reduced system are addressed. For the latter point, both the linear or standard Galerkin method making use of the Karhunen Loeve (proper orthogonal decomposition) ansatz functions and the nonlinear Galerkin method, using approximate inertial manifold theory, are used. In addition, also the post-processing Galerkin method is compared with the other approaches.

Efficient methods using high accuracy approximate inertial manifolds

Summary. We extend the idea of the post-processing Galerkin method, in the context of dissipative evolution equations, to the nonlinear Galerkin, the filtered Galerkin, and the filtered nonlinear Galerkin methods. In general, the post-processing algorithm takes advantage of the fact that the error committed in the lower modes of the nonlinear Galerkin method (and Galerkin method), for approximating smooth, bounded solutions, is much smaller than the total error of the method. In each case, an improvement in accuracy is obtained by post-processing these more accurate lower modes with an appropriately chosen, highly accurate, approximate inertial manifold (AIM). We present numerical experiments that support the theoretical improvements in accuracy. Both the theory and computations are presented in the framework of a two dimensional reaction-diffusion system with polynomial nonlinearity. However, the algorithm is very general and can be implemented for other dissipative evolution systems. The computations clearly show the post-processed filtered Galerkin method to be the most efficient method.

Numerical criterion for the stabilization of steady states of the Navier-Stokes equations

This paper introduces an explicit numerical criterion for the stabilization of steady state solutions of the Navier-Stokes equations (NSE) with linear feedback control. Given a linear feedback controller that stabilizes a steady state solution to the closed-loop standard Galerkin (or nonlinear Galerkin) NSE discretization, it is shown that, if the number of modes involved in the computation is large enough, this controller stabilizes a nearby steady state of the closed-loop NSE. We provide an explicit estimate, in terms of the physical parameters, for the number of modes required in order for this criterion to hold. Moreover, we provide an estimate for the distance between the stabilized numberical steady state and the actually stabilized steady state of the closed-loop Navier-Stokes equations. More accurate approximation procedures, based on the concept of postprocessing the Galerkin results, are also presented. Allthe criterion conditions are imposed on the computed numerical solution, and no a priori knowledge is required about the steady state solution of the full PDE. This criterion holds for a large class of unbounded linear feedback operators and can be slightly modified to include certain nonlinear parabolic systems such as reaction-diffusion systems.

The Generalized Nonlinear Galerkin Method Applied to Hydrodynamics of the Open Sea

In this paper we present new numerical algorithms based on a generalized nonlinear Galerkin method in order to solve coastal and oceanic circulation problems. The equations system is based on the primitive equations of the ocean under Boussinesq and hydrostatic approximations. These equations are transformed using, at the same time, the classical σ transformation and an original homogenization of the boundary conditions. We use a well adapted special basis to apply the usual Galerkin method and the nonlinear Galerkin method. This basis is built on a modelization of the energetic transfers through the different scales of flow. Two approaches are proposed to solve the continuity equation: the (nonlinear) Galerkin method and the method of the characteristics. We present the advantages and drawbacks of both methods.

A postprocessed Galerkin method with Chebyshev or Legendre polynomials

We present an approximate-inertial-manifold-based postprocess to enhance Chebyshev or Legendre spectral Galerkin methods. We prove that the postprocess improves the order of convergence of the Galerkin solution, yielding the same accuracy as the nonlinear Galerkin method. Numerical experiments show that the new method is computationally more efficient than Galerkin and nonlinear Galerkin methods. New approximation results for Chebyshev polynomials are presented.

Geometrical non-linear analysis of a simply-supported cross V-shaped folded plate roof

In this paper, the equation of the middle surface of a simply-supported cross V-shaped folded plate roof is established by means of the inclined coordinate system and generalized functions—sign function and step function. Then, the non-linear analysis for this simply-supported roof subjected to uniform load and point load is given by using the non-linear theory of shallow shell and the non-linear Galerkin's method. Finally, an illustrative example of this roof is presented. This example indicates that the largest stress of this roof is much less then that of the four pitch roof of same dimensions. Therefore, a lot of construction material could be saved by using this proposed roof.

Wave suppression by nonlinear finite-dimensional control

Korteweg–de Vries–Burgers (KdVB) and Kuramoto–Sivashinsky (KS) equations are two nonlinear partial differential equations (PDEs) which can adequately describe motion of waves in a variety of fluid flow processes. We synthesize nonlinear low-dimensional output feedback controllers for the KdVB and KS equations that enhance convergence rate and achieve stabilization to spatially uniform steady states, respectively. The approach used for controller synthesis employs nonlinear Galerkin's method to derive low-dimensional approximations of the PDEs, which are subsequently used for controller synthesis via geometric control methods. The controllers use measurements obtained by point sensors and are implemented through point control actuators. The performance of the proposed controllers is successfully tested through simulations.

A modified nonlinear Galerkin method for the viscoelastic fluid motion equations

In this article we first provide a priori estimates of the solution for the nonstationary two-dimensional viscoelastic fluid motion equations with periodic boundary condition. We then present an modified nonlinear Galerkin method for solving such equations. By comparing the convergence rates of the proposed method with the standard Galerkin method, we conclude that the modified nonlinear Galerkin method is better than the standard Galerkin method because the former can save a large amount of computational work and maintain the convergence rate of the latter.

THE NONLINEAR GALERKIN METHOD APPLIED TO SHALLOW WATER EQUATIONS

In this work, we present some numerical approximations for a shallow water problem with a depth-mean velocity formulation and we give, where possible, an error bound. To prove the existence of solutions, we build a sequence of approximated solutions with the Galerkin method for the momentum equation and solve the continuity equation with the method of the characteristics. This leads to an expensive natural numerical scheme. Then, in order to reduce the CPU time, we present other numerical approximations based on the linear or nonlinear Galerkin method.

Resolution to a three-dimensional physical oceanographic problem using the non-linear Galerkin method

This paper presents the numerical results concerning the adaptation of the non-linear Galerkin method to three-dimensional geophysical fluid equations. This method was developed by Marion and Temam to solve the Navier–Stokes two-dimensional equations. It allows a substantial decrease in calculation costs due to the application of an appropriate treatment to each mode based on its position in the spectrum. The large scales involved in the study of geophysical flow require that the earth's rotational effects and the existence of a high degree of stratification be taken into account. These phenomena play an important role in the distribution of the energy spectrum. It is shown here that the non-linear Galerkin method is very well-suited to the treatment of these phenomena. First, the method for the particular situation of a rigid-lid with a flat bottom is validated, for which the functional basis used is particularly well-adapted. Then the more general case of a domain exhibiting variable bathymetry is presented, which necessitates the use of the transformation σ, thus providing a study domain with a cylindrical configuration. Copyright © 1999 John Wiley & Sons, Ltd.

The post-processed Galerkin method applied to non-linear shell vibrations

We present the results of a computational study of the post-processed Galerkin methods put forward by Garcia-Archilla et al. applied to the non-linear von Karman equations governing the dynamic response of a thin cylindrical panel periodically forced by a transverse point load. We spatially discretize the shell using finite differences to produce a large system of ordinary diff erential equations (ODEs). By analogy with spectral non-linear Galerkin methods we split this large system into a 'slowly' contracting subsystem and a 'quickly' contracting subsystem. We then compare the accuracy and efficiency of (i) ignoring the dynamics of the 'quick' system (analogous to a traditional spectral Galerkin truncation and sometimes referred to as 'subspace dynamics' in the finite element community when applied to numerical eigenvectors), (ii) slaving the dynamics of the quick system to the slow system during numerical integration (analogous to a non-linear Galerkin method), and (iii) ignoring the influence of the dynamics of the quick system on the evolution of the slow system until we require some output, when we 'lift' the variables from the slow system to the quick using the same slaving rule as in (ii). This corresponds to the postprocessing of Garcia-Archilla et al. We find that method (iii) produces essentially the same accuracy as method (ii) but requires only the computational power of method (i) and is thus more efficient than either. In contrast with spectral methods, this type of finite-diff erence technique can be applied to irregularly shaped domains. We feel that post-processing of this form is a valuable method that can be implemented in computational schemes for a wide variety of partial differential equations (PDEs) of practical importance.

The spectral method for the flow between two concentric rotating spheres, part II: The full discrete nonlinear Legendre-Galerkin method

The nonlinear Galerkin methods are numerical schemes for evolutionary partial differential equations based on the theory of inertial manifolds and approximate inertial manifolds. In this paper, we consider the flow between two concentric rotating spheres, and combine the Legendre-Galerkin spectral methods in Part I together with the nonlinear Galerkin method, then construct the full discrete nonlinear Legendre-Galerkin spectral scheme, and derive the stability conditions and its error estimate.

Geometrically nonlinear analysis of a simply-supported hip-truncated combined parabolical cylinder roof

In this paper, the equation of the middle surface of a simply-supported hip-truncated combined parabolical cylinder roof is established by means of the local inclined coordinate system and step function. Then, the nonlinear analysis for this simply-supported roof subjected to uniform load and point loads is given by using the nonlinear theory of shallow shell and the nonlinear Galerkin method. Finally, an illustrative example of this roof is presented. This example indicates that the largest stress of this roof is much less than that of the hip-truncated four pitch roof of same dimensions.

Nonlinear Galerkin methods based on the concept of determining modes for the magnetohydrodynamic equations

A new nonlinear Galerkin method based on an approximate inertial manifold and on a finite number of determining modes has been implemented for the three-dimensional magnetohydrodynamic equations. It is found that these active modes, which regulate the dynamics of the system, are not necessarily those with the smallest wavenumbers but those containing most of the time-averaged enstrophy. It is demonstrated that owing to the reduced number of equations the modified nonlinear Galerkin method, which includes only these determining modes in a truncation, improves the computational efficiency in comparison with the traditional Galerkin method.