Galerkin Spectral Approximation Research Articles

A New Legendre Spectral Galerkin and Pseudo-Spectral Approximations for Fractional Initial Value Problems

We extend the application of the Galerkin method for treating the multiterm fractional differential equations (FDEs) subject to initial conditions. A new shifted Legendre-Galerkin basis is constructed which satisfies exactly the homogeneous initial conditions by expanding the unknown variable using a new polynomial basis of functions which is built upon the shifted Legendre polynomials. A new spectral collocation approximation based on the Gauss-Lobatto quadrature nodes of shifted Legendre polynomials is investigated for solving the nonlinear multiterm FDEs. The main advantage of this approximation is that the solution is expanding by a truncated series of Legendre-Galerkin basis functions. Illustrative examples are presented to ensure the high accuracy and effectiveness of the proposed algorithms are discussed.

Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers' law and beyond

We prove a Kramers-type law for metastable transition times for a class of one-dimensional parabolic stochastic partial differential equations (SPDEs) with bistable potential. The expected transition time between local minima of the potential energy depends exponentially on the energy barrier to overcome, with an explicit prefactor related to functional determinants. Our results cover situations where the functional determinants vanish owing to a bifurcation, thereby rigorously proving the results of formal computations announced in [Berglund and Gentz, J. Phys. A 42:052001 (2009)]. The proofs rely on a spectral Galerkin approximation of the SPDE by a finite-dimensional system, and on a potential-theoretic approach to the computation of transition times in finite dimension.

A Legendre Galerkin spectral method for optimal control problems

This paper considers the Legendre Galerkin spectral approximation for the unconstrained optimal control problems. The authors derive a posteriori error estimate for the spectral approximation scheme of optimal control problem. By choosing the appropriate basis functions, the stiff matrix of the discretization equations is sparse. And the authors use the Fast Legendre Transform to improve the efficiency of this method. Two numerical experiments demonstrating our theoretical results are presented.

Uniform-in-time error estimates for spectral Galerkin approximations of a mass diffusion model

Uniform-in-time error estimates for spectral Galerkin approximations of a mass diffusion model

A Legendre–Galerkin Spectral Method for Optimal Control Problems Governed by Stokes Equations

In this paper, we study the Legendre–Galerkin spectral approximation of distributed optimal control problems governed by Stokes equations. We show that the discretized control problems satisfy the well-known Babuška–Brezzi conditions by choosing an appropriate pair of discretization spaces for the velocity and the pressure. Constructing suitable base functions of the discretization spaces leads to sparse coefficient matrices. We first derive a priori error estimates in both $H^1$ and $L^2$ norms for the Legendre–Galerkin approximation of the unconstrained control problems. Then both a priori and a posteriori error estimates are obtained for control problems with the constraints of an integral type, thanks to the higher regularity of the optimal control. Finally, some illustrative numerical examples are presented to demonstrate the error estimates.

A Factorization of the Spectral Galerkin System for Parameterized Matrix Equations: Derivation and Applications

Recent work has explored solver strategies for the linear system of equations arising from a spectral Galerkin approximation of the solution of PDEs with parameterized (or stochastic) inputs. We consider the related problem of a matrix equation whose matrix and right hand side depend on a set of parameters (e.g. a PDE with stochastic inputs semidiscretized in space) and examine the linear system arising from a similar Galerkin approximation of the solution. We derive a useful factorization of this system of equations, which yields bounds on the eigenvalues, clues to preconditioning, and a flexible implementation method for a wide array of problems. We complement this analysis with (i) a numerical study of preconditioners on a standard elliptic PDE test problem and (ii) a fluids application using existing CFD codes; the MATLAB codes used in the numerical studies are available online.

A spectral fictitious domain method with internal forcing for solving elliptic PDEs

A fictitious domain method is presented for solving elliptic partial differential equations using Galerkin spectral approximation. The fictitious domain approach consists in immersing the original domain into a larger and geometrically simpler one in order to avoid the use of boundary fitted or unstructured meshes. In the present study, boundary constraints are enforced using Lagrange multipliers and the novel aspect is that the Lagrange multipliers are associated with smooth forcing functions, compactly supported inside the fictitious domain. This allows the accuracy of the spectral method to be preserved, unlike the classical discrete Lagrange multipliers method, in which the forcing is defined on the boundaries. In order to have a robust and efficient method, equations for the Lagrange multipliers are solved directly with an influence matrix technique. Using a Fourier–Chebyshev approximation, the high-order accuracy of the method is demonstrated on one- and two-dimensional elliptic problems of second- and fourth-order. The principle of the method is general and can be applied to solve elliptic problems using any high order variational approximation.

A uniform error estimate in time for spectral Galerkin approximations of the magneto‐micropolar fluid equations

AbstractWe consider Galerkin approximations for the equations modeling the motion of an incompressible magneto‐micropolar fluid in a bounded domain. We derive an optimal uniform in time error bound in the H1 and L2 ‐norms for the velocity. This is done without explicit assumption of exponential stability for a class of solutions corresponding to decaying external force fields. Our study is done for no‐slip boundary conditions, but the results obtained are easily extended to the case of periodic boundary conditions. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 689–706, 2012

Multivariate modified Fourier series and application to boundary value problems

In this paper we analyse the approximation-theoretic properties of modified Fourier series in Cartesian product domains with coefficients from both full and hyperbolic cross index sets. We show that the number of expansion coefficients may be reduced significantly whilst retaining comparable error estimates. In doing so we extend the univariate results of Iserles, Norsett and S. Olver. We then demonstrate that these series can be used in the spectral-Galerkin approximation of second order Neumann boundary value problems, which offers some advantages over standard Chebyshev or Legendre polynomial discretizations.

Solar pond modeling with density and viscosity dependent on temperature and salinity

The paper presents a 2D numerical model where the behavior of a salt gradient solar pond (SGSP) is described in terms of temperature, salt concentration and velocity with the fluid density and viscosity dependent on temperature and salt concentration. The discretization of the governing equations is based on the respective weak formulations. The rectangular geometry allows for spectral type Galerkin approximations for which the essential homogeneous boundary conditions can easily be imposed. Taking into account the variation of density and viscosity with temperature and salinity improved the agreement between the numerical and the experimental results.

Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift

This paper is concerned with the analysis and implementation of spectral Galerkin methods for a class of Fokker-Planck equations that arises from the kinetic theory of dilute polymers. A relevant feature of the class of equations under consideration from the viewpoint of mathematical analysis and numerical approximation is the presence of an unbounded drift coefficient, involving a smooth convex potential U that is equal to +∞ along the boundary ∂D of the computational domain D. Using a symmetrization of the differential operator based on the Maxwellian M corresponding to U, which vanishes along ∂D, we remove the unbounded drift coefficient at the expense of introducing a degeneracy, through M, in the principal part of the operator. The general class of admissible potentials considered includes the FENE (finitely extendible nonlinear elastic) model. We show the existence of weak solutions to the initial-boundary-value problem, and develop a fully-discrete spectral Galerkin method for such degenerate Fokker-Planck equations that exhibits optimal-order convergence in the Maxwellian-weighted H1 norm on D. In the case of the FENE model, we also discuss variants of these analytical results when the Fokker-Planck equation is subjected to an alternative class of transformations proposed by Chauvière and Lozinski; these map the original Fokker-Planck operator with an unbounded drift coefficient into Fokker-Planck operators with unbounded drift and reaction coefficients, that have improved coercivity properties in comparison with the original operator. The analytical results are illustrated by numerical experiments for the FENE model in two space dimensions.

A posteriori error estimates of spectral method for optimal control problems governed by parabolic equations

In this paper, we investigate the Legendre Galerkin spectral approximation of quadratic optimal control problems governed by parabolic equations. A spectral approximation scheme for the parabolic optimal control problem is presented. We obtain a posteriori error estimates of the approximated solutions for both the state and the control.

A Legendre–Galerkin Spectral Method for Optimal Control Problems Governed by Elliptic Equations

In this paper, we study the Legendre–Galerkin spectral approximation for a constrained optimal control problem. We first derive a priori error estimates for the spectral approximation of optimal control problems. Then a posteriori error estimates are obtained for both the state and the control approximation. A preconditioning projection algorithm is proposed with some numerical tests.

Analysis of a Spectral-Galerkin Approximation to the Helmholtz Equation in Exterior Domains

An error analysis is presented for the spectral-Galerkin method to the Helmholtz equation in 2- and 3-dimensional exterior domains. The problem in unbounded domains is first reduced to a problem on a bounded domain via the Dirichlet-to-Neumann operator, and then a spectral-Galerkin method is employed to approximate the reduced problem. The error analysis is based on exploring delicate asymptotic behaviors of the Hankel functions and on deriving a priori estimates with explicit dependence on the wave number for both the continuous and the discrete problems. Explicit error bounds with respect to the wave number are derived, and some illustrative numerical examples are also presented.

Spectral approximation of Takens–Bogdanov point of the Navier–Stokes equations for the flow between two concentric rotating spheres

The paper studies a spectral Galerkin approximation problem of Takens–Bogdanov point of the Navier–Stokes equations for the flow between two concentric rotating spheres. The author constructs the extended system and its spectral Galerkin approximate system of Takens–Bogdanov point for the Navier–Stokes equations, and prove the regularity of solution of the extended system; the existence and the convergence of solutions of the spectral Galerkin approximate system are discussed. The spectral Galerkin approximation of Takens–Bogdanov point of the Navier–Stokes equations in two concentric rotating spheres is given, and the error estimates is presented.

On the Large Eddy Simulation of the Taylor–Green vortex

In this paper we analyze the role of the Taylor–Green vortex in the study of some Large Eddy Simulation models. More precisely, we consider such a particular flow in connection with the Gradient model and the Rational model. In particular, we study: 1) the mechanism of small scale generation; 2) the symmetries of the flow and their role in the Spectral Galerkin approximation; 3) the connection between pressure fluctuations and quadratic mean velocity.

Spectral Approximation of the Helmholtz Equation with High Wave Numbers

A complete error analysis is performed for the spectral-Galerkin approximation of a model Helmholtz equation with high wave numbers. The analysis presented in this paper does not rely on the explicit knowledge of continuous/discrete Green's functions and does not require any mesh condition to be satisfied. Furthermore, new error estimates are also established for multi-dimensional radial and spherical symmetric domains. Illustrative numerical results in agreement with the theoretical analysis are presented.

Spectral galerkin approximation of couette-taylor flow

Axisymmetric Couette-Taylor flow between two concentric rotating cylinders was simulated numerically by the spectral method. First, stream function form of the Navier-Stokes equations which homogeneous boundary condition was given by introducing Couette flow. Second, the analytical expressions of the eigenfunction of the Stokes operator in the cylindrical gap region were given and its orthogonality was proved. The estimates of growth rate of the eigenvalue were presented. Finally, spectral Galerkin approximation of Couette-Taylor flow was discussed by introducing eigenfunctions of Stokes operator as basis of finite dimensional approximate subspaces. The existence, uniquence and convergence of spectral Galerkin approximation of nonsingular solution for the steady-state Navier-Stokes equations are proved. Moreover, the error estimates are given. Numerical result is presented.

Application of the discontinuous spectral Galerkin method to groundwater flow

The discontinuous spectral Galerkin method uses a finite-element discretization of the groundwater flow domain with basis functions of arbitrary order in each element. The independent choice of the basis functions in each element permits discontinuities in transmissivity in the flow domain. This formulation is shown to be of high order accuracy and particularly suitable for accurately calculating the flow field in porous media. Simulations are presented in terms of streamlines in a bidimensional aquifer, and compared with the solution calculated with a standard finite-element method and a mixed finite-element method. Numerical simulations show that the discontinuous spectral Galerkin approximation is more efficient than the standard finite-element method (in computing fluxes and streamlines/pathlines) for a given accuracy, and it is more accurate on a given grid. On the other hand the mixed finite-element method ensures the continuity of the fluxes at the cell boundaries and it is particular efficient in representing complicated flow fields with few mesh points. Simulations show that the mixed finite-element method is superior to the discontinuous spectral Galerkin method producing accurate streamlines even if few computational nodes are used. The application of the discontinuous Galerkin method is thus of interest in groundwater problems only when high order and extremely accurate solutions are needed.

Thin film epitaxy with or without slope selection

Two nonlinear diffusion equations for thin film epitaxy, with or without slope selection, are studied in this work. The nonlinearity models the Ehrlich–Schwoebel effect – the kinetic asymmetry in attachment and detachment of adatoms to and from terrace boundaries. Both perturbation analysis and numerical simulation are presented to show that such an atomistic effect is the origin of a nonlinear morphological instability, in a rough-smooth-rough pattern, that has been experimentally observed as transient in an early stage of epitaxial growth on rough surfaces. Initial-boundary-value problems for both equations are proven to be well-posed, and the solution regularity is also obtained. Galerkin spectral approximations are studied to provide both a priori bounds for proving the well-posedness and numerical schemes for simulation. Numerical results are presented to confirm part of the analysis and to explore the difference between the two models on coarsening dynamics.