Galerkin Spectral Approximation Research Articles

Galerkin spectral method for a fourth-order optimal control problem with H1-norm state constraint

In this article, we are concerned with the Galerkin spectral approximation of an H1-norm state-constrained optimal control problem governed by a fourth-order partial differential equation (PDE). In the investigations, the optimality conditions for both the original control problem and its spectral approximation problem are, respectively, established, which are helpful to analyze the solution property of the spectral approximation problem. A priori and a posteriori error estimates of the spectral approximation problem are, therewith, built in detail. Finally, two numerical examples are carried out to validate the theoretical analysis.

Optimal Petrov–Galerkin Spectral Approximation Method for the Fractional Diffusion, Advection, Reaction Equation on a Bounded Interval

In this paper we investigate the numerical approximation of the fractional diffusion, advection, reaction equation on a bounded interval. Recently the explicit form of the solution to this equation was obtained. Using the explicit form of the boundary behavior of the solution and Jacobi polynomials, a Petrov–Galerkin approximation scheme is proposed and analyzed. Numerical experiments are presented which support the theoretical results, and demonstrate the accuracy and optimal convergence of the approximation method.

A spectral Galerkin approximation of optimal control problem governed by fractional advection–diffusion–reaction equations

A spectral Galerkin approximation of a optimal control problem governed by a fractional advection–diffusion–reaction equation with integral fractional Laplacian is investigated in 1D. We first derive a first-order optimality condition and analyze the regularity of the solution based on this optimality condition. We present a spectral Galerkin scheme for the control problem using weighted Jacobi polynomials and prove optimal error estimates of the spectral method for state, adjoint state and control variables. We also propose a fast projected gradient algorithm of quasilinear complexity and present two numerical examples verifying our theoretical findings.

The Convergence of the Legendre–Galerkin Spectral Method for Constructing Atmospheric Acoustic Normal Modes

The asymptotic rate of convergence of the Legendre–Galerkin spectral approximation to an atmospheric acoustic eigenvalue problem is established, as the dimension of the approximating subspace approaches infinity. Convergence is in the [Formula: see text] Sobolev norm and is based on the existing theory [F. Chatelin, Spectral Approximations of Linear Operators (SIAM, 2011)]. The assumption is made that the eigenvalues are simple. Numerical results that help interpret the theory are presented. Eigenvalues corresponding to acoustic modes with smaller [Formula: see text] norms are especially accurately approximated, even with lower dimensioned basis sets of Legendre polynomials. The deficiencies in the potential applications of the theoretical results are noted in connection with the numerical examples.

Absolute continuity and numerical approximation of stochastic Cahn–Hilliard equation with unbounded noise diffusion

In this article, we consider the absolute continuity and numerical approximation of the solution of the stochastic Cahn–Hilliard equation with unbounded noise diffusion. We first obtain the Hölder continuity and Malliavin differentiability of the solution of the stochastic Cahn–Hilliard equation by using the strong convergence of the spectral Gakerkin approximation. Then we prove the existence and strict positivity of the density function of the law of the exact solution for the stochastic Cahn–Hilliard equation with sublinear growth diffusion coefficient, which fills a gap for the existed result when the diffusion coefficient satisfies a growth condition of order 1/3<α<1. To approximate the density function of the exact solution, we propose a full discretization based on the spatial spectral Galerkin approximation and the temporal drift implicit Euler scheme. Furthermore, a general framework for deriving the strong convergence rate of the full discretization is developed based on the variation approach and the factorization method. Consequently, we obtain the sharp mean square convergence rates in both time and space via Sobolev interpolation inequalities and semigroup theories. To the best of our knowledge, this is the first result on the convergence rate of full discretizations for the considered equation.

Convergence rates of approximations of incompressible flows through granular porous media

We consider spectral Galerkin approximations for the strong solutions of a system of incompressible flows through granular porous medium in a bounded domain of $$\mathbb {R}^n, n=2,3$$ . We obtain uniform in time error bounds in the spatial $$L^2$$ and $$H^1$$ -norms for approximations of the velocity. Finally, we present some numerical simulations to verify the good behavior of spectral Galerkin approximations.

An efficient spectral-Galerkin approximation based on dimension reduction scheme for transmission eigenvalues in polar geometries

In this paper, we put forward an efficient spectral-Galerkin approximation in view of dimension reduction scheme for transmission eigenvalue problem in polar geometries. Firstly, we turn the original problem into an equivalent fourth order nonlinear eigenvalue problem. Then the fourth order nonlinear eigenvalue problem is transformed into a coupled fourth order linear eigenvalue system by introducing an auxiliary Poisson equation. Secondly, based on polar coordinate transformation, we further reduce the coupled fourth order linear eigenvalue system to a series of equivalent one-dimensional eigenvalue systems. Thirdly, we derive the essential polar condition and introduce the appropriate weighted Sobolev space according to the polar condition, and establish the weak form and the corresponding discrete form. In addition, by utilizing spectral theory of compact operators, we prove the error estimates of approximation eigenvalues and eigenvectors for each one-dimensional eigenvalue system. Finally, we provide ample numerical experiments, and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results.

Shifted Jacobi spectral-Galerkin method for solving fractional order initial value problems

In this paper, the shifted Jacobi spectral-Galerkin method is introduced to deal with fractional order initial value problems (FIVPs). In the proposed method, the exact solution of the FIVP is approximated by using shifted Jacobi polynomials on each subinterval of the total time. The main advantage of the proposed method is that the rate of convergence in L2-norm depends on the local smoothness of solution. This enables the proposed method to work for nonsmooth solutions. The second merit is that the flexibility between the length of sub-interval and the degree of polynomial significantly enhances the numerical accuracy. We derive the spectral-Galerkin approximation formula for the Volterra integral equation of the second kind which is equivalent to the FIVP. Error analysis in L2-norm for the case α=β=0 in shifted Jacobi polynomials is provided and it justifies the spectral rate of convergence with respect to both the degree and the length of the subinterval when the source function is Lipschitz continuous in the second argument. Numerical illustrations for multi-order, linear, and nonlinear FIVPs are demonstrated to confirm the convergence rate.

Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients

Strong convergence rates for time-discrete numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the literature. Weak convergence rates for time-discrete numerical approximations of such SEEs have been investigated since about 12 years and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for time-discrete numerical approximations of parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117] for details. In the recent article [D. Conus, A. Jentzen & R. Kurniawan, Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients, arXiv:1408.1108] the weak convergence problem emerged from Debussche's article has been solved in the case of spatial spectral Galerkin approximations for semilinear SEEs with nonlinear diffusion coefficient functions. In this article we overcome the problem emerged from Debussche's article in the case of a class of time-discrete Euler-type approximation methods (including exponential and linear-implicit Euler approximations as special cases) and, in particular, we establish essentially sharp weak convergence rates for linear-implicit Euler approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Key ingredients of our approach are applications of a mild It\^o type formula and the use of suitable semilinear integrated counterparts of the time-discrete numerical approximation processes.

Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions

AbstractWe show that if a sequence of piecewise affine linear processes converges in the strong sense with a positive rate to a stochastic process that is strongly Hölder continuous in time, then this sequence converges in the strong sense even with respect to much stronger Hölder norms and the convergence rate is essentially reduced by the Hölder exponent. Our first application hereof establishes pathwise convergence rates for spectral Galerkin approximations of stochastic partial differential equations. Our second application derives strong convergence rates of multilevel Monte Carlo approximations of expectations of Banach-space-valued stochastic processes.

On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients

We develop a perturbation theory for stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs). In particular, we estimate the $L^{p}$-distance between the solution process of an SDE and an arbitrary Itô process, which we view as a perturbation of the solution process of the SDE, by the $L^{q}$-distances of the differences of the local characteristics for suitable $p,q>0$. As one application of the developed perturbation theory, we establish strong convergence rates for numerical approximations of a class of SODEs with nonglobally monotone coefficients. As another application of the developed perturbation theory, we prove strong convergence rates for spatial spectral Galerkin approximations of solutions of semilinear SPDEs with nonglobally monotone nonlinearities including Cahn–Hilliard–Cook-type equations and stochastic Burgers equations. Further applications of the developed perturbation theory include regularity analyses of solutions of SDEs with respect to their initial values as well as small-noise analyses for ordinary and partial differential equations.

Galerkin spectral approximation of optimal control problems with L2-norm control constraint

In this paper, an optimal control problem governed by elliptic partial differential equations, whose objective functionals do not depend on the controls, with L2-norm constraint on control variable is considered, and Galerkin spectral approximation of the optimal control problem without penalty term is established. To analyze the optimal control problem, optimality conditions of the control problem are first derived in detail. By virtue of some lemmas and auxiliary equations, a priori error estimates of spectral approximation for optimal control problems are analyzed rigorously. Moreover, a posteriori error estimates of the optimal control problem are also established carefully. In the end, some numerical examples are carried out to confirm the analytical results that the errors decay exponentially fast as long as the data is smooth.

Spectral-Galerkin approximation and optimal error estimate for biharmonic eigenvalue problems in circular/spherical/elliptical domains

In this paper, we propose and analyze spectral-Galerkin methods for the biharmonic eigenvalue problem in circular/spherical/elliptical domains. We first analyze the eigenfunction formulated fourth-order equation under the polar coordinates, then we derive the pole condition and reduce the problem on a circular disk/sphere to a sequence of equivalent one-dimensional eigenvalue problems that can be solved in parallel. The novelty of our approach lies in the construction of suitably weighted Sobolev spaces according to the pole conditions, based on which, the optimal error estimate for approximated eigenvalue of each one-dimensional problem can be obtained. Further, we extend our method to the non-separable biharmonic eigenvalue problem in an elliptic domain and establish the optimal error bounds. Finally, we provide some numerical experiments to validate our theoretical results and algorithms.

Adjusted Sparse Tensor Product Spectral Galerkin Method for Solving Pseudodifferential Equations on the Sphere with Random Input Data

An adjusted sparse tensor product spectral Galerkin approximation method based on spherical harmonics is introduced and analyzed for solving pseudodifferential equations on the sphere with random input data. These equations arise from geodesy where the sphere is taken as a model of the earth. Numerical solutions to the corresponding $k$-th order statistical moment equations are found in adjusted sparse tensor approximation spaces which are accordingly designed to the regularity of the data and the equation. Established convergence theorem shows that the adjusted sparse tensor Galerkin discretization is superior not only to the full tensor product but also to the standard sparse tensor counterpart when the statistical moments of the data are of mixed unequal regularity. Numerical experiments illustrate our theoretical results.

A saturation property for the spectral-Galerkin approximation of a Dirichlet problem in a square

Both practice and analysis of p-FEMs and adaptive hp-FEMs raise the question what increment in the current polynomial degree p guarantees a p-independent reduction of the Galerkin error. We answer this question for the p-FEM in the simplified context of homogeneous Dirichlet problems for the Poisson equation in the two dimensional unit square with polynomial data of degree p. We show that an increment proportional to p yields a p-robust error reduction and provide computational evidence that a constant increment does not.

Spectral Galerkin approximation of optimal control problem governed by Riesz fractional differential equation

In this paper we investigate spectral Galerkin approximation of optimal control problem governed by Riesz fractional differential equation with control integral constraint. First order optimality condition is derived and the regularity of control problem is discussed. Based on first discretize then optimize approach a spectral Galerkin approximation of the control problem is developed, where two-sided Jacobi polyfractonomials are used to approximate the state variable and variational discretization is used to discretize the control variable. A priori error analysis for state variable, adjoint state variable and control variable is presented. Numerical experiments are carried out to illustrate the theoretical findings.

Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients

Strong convergence rates for (temporal, spatial, and noise) numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the scientific literature. Weak convergence rates for numerical approximations of such SEEs have been investigated since about 11 years and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117] for details. In this article we solve the weak convergence problem emerged from Debussche's article in the case of spectral Galerkin approximations and establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Our solution to the weak convergence problem does not use Malliavin calculus. Rather, key ingredients in our solution to the weak convergence problem emerged from Debussche's article are the use of appropriately modified versions of the spatial Galerkin approximation processes and applications of a mild It\^{o} type formula for solutions and numerical approximations of semilinear SEEs. This article solves the weak convergence problem emerged from Debussche's article merely in the case of spatial spectral Galerkin approximations instead of other more complicated numerical approximations. Our method of proof extends, however, to a number of other kind of spatial, temporal, and noise numerical approximations for semilinear SEEs.

An efficient spectral Galerkin approximation to a Helmholtz transmission eigenvalue problem in spherical geometries

In this paper, we present an efficient spectral method based on Legendre-Galerkin approximation for the Helmholtz transmission eigenvalue problem in spherical geometries. By means of the spherical coordinate transformation and spherical harmonic expansion, we decompose the original problem into a sequence of equivalent one-dimensional generalized eigenvalue problems. Then we establish corresponding weak forms and discrete schemes for each one-dimensional generalized eigenvalue problem. Especially for the case of solid spherical domain, we need to deal with singularities introduced by spherical coordinate transformation. In order to overcome the difficulty, we derive the pole conditions and introduce the weighted Sobolev space according to pole conditions. The corresponding weak forms and discrete schemes are also established according to the weighted Sobolev space. In addition, we prove the error estimates of eigenvalues and eigenfunctions by using the spectral theory of compact operators for each one-dimensional generalized eigenvalue problems.We also provide some numerical experiments to demonstrate the validity of the algorithm and the correctness of the theoretical results.

Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space

<p style='text-indent:20px;'>In this article, we propose and analyze some novel spectral methods for the Schödinger equation (including the associated eigenvalue problem) with an inverse square potential on an arbitrary whole space <inline-formula><tex-math id="M1">\begin{document}$\mathbb{R}^d$\end{document}</tex-math></inline-formula> for any dimension <inline-formula><tex-math id="M2">\begin{document}$d$\end{document}</tex-math></inline-formula>. We start from the investigation that the radial component of the eigenfunctions, corresponding to spherical harmonics of degree <inline-formula><tex-math id="M3">\begin{document}$n$\end{document}</tex-math></inline-formula>, of the Schrödinger operator <inline-formula><tex-math id="M4">\begin{document}$\displaystyle -Δ u + \frac{c^2}{r^2}u$\end{document}</tex-math></inline-formula> can be expressed by Bessel functions of fractional orders <inline-formula><tex-math id="M5">\begin{document}$α_n = \sqrt{c^2+(n+d/2-1)^2}$\end{document}</tex-math></inline-formula> together with the multiplier <inline-formula><tex-math id="M6">\begin{document}$r^{1-\frac{d}{2}}$\end{document}</tex-math></inline-formula>. This knowledge helps us to construct the Müntz-Hermite functions as the basis functions to fit the singularities of the eigenfunctions. In return, a novel spectral method is then proposed for solving the Schrödinger eigenvalue problem efficiently. Further, a Galerkin spectral approximation using genuine Hermite functions with a distinct Müntz sequence <inline-formula><tex-math id="M7">\begin{document}$\{α_n = α+n+d/2-1\}$\end{document}</tex-math></inline-formula> is also proposed for the Schrödinger source problem with a singular solution of type <inline-formula><tex-math id="M8">\begin{document}$r^{α}$\end{document}</tex-math></inline-formula>. Optimal error estimates are then established rigorously for both the source and eigenvalue problems. In contrast to classic Hermite spectral methods using tensorial basis functions, our new methods possess an exponential order of convergence for such singular problems while offer a banded structure of the stiffness and mass matrices. Finally, numerical experiments illustrate the efficiency and spectral accuracy of our new methods.

Error Analysis and Simulation of Galerkin Spectral Approximation for Flow Optimal Control with State Constraint

ABSTRACTWe investigate the spectral approximation of optimal control governed by Stokes equations with integral state constraint. A good choice for basis functions leads the discrete system with sparse matrices. The optimality conditions are derived, a priori and a posteriori error estimates are presented in both H1 and L2 norms. Numerical experiment indicates the high precision can be achieved with the proposed method.