Petrov-Galerkin Approximation Research Articles

Petrov–Galerkin approximation of time-fractional coupled Korteweg–de Vries equation for propagation of long wave in shallow water

The time-fractional coupled Korteweg–de Vries equations (TFCKdVEs) describe various interesting real-world phenomena including wave propagation and the description of shallow water waves on a viscous fluid. This paper presents an accurate and robust numerical technique to solve the TFCKdVE. The cubic B-spline is introduced as a basis function and a quadratic B-spline is used as a test function in a finite element method (FEM) is known as Petrov–Galerkin method. The temporal fractional part is simplified via L1 formula, while the B-spline is employed for the space approximation. The Lax–Richtmyer stability criterion is applied to analyze the stability of the proposed scheme. Four test problems are solved to check performance and validation of the scheme. The accuracy and efficiency of the proposed method are checked via various error norms. The obtained results show good agreement with the exact solutions and earlier work available in the literature.

Analysis and Petrov–Galerkin numerical approximation for variable coefficient two-sided fractional diffusion, advection, reaction equations

In this paper we investigate the variable coefficient two-sided fractional diffusion, advection, reaction equations on a bounded interval. It is known that the fractional diffusion operator may lose coercivity due to the variable coefficient, which makes both the mathematical and numerical analysis challenging. To resolve this issue, we design appropriate test and trial functions to prove the inf-sup condition of the variable coefficient fractional diffusion, advection, reaction operators in suitable function spaces. Based on this property, we prove the well-posedness and regularity of the solutions, as well as analyze the Petrov–Galerkin approximation scheme for the proposed model. Numerical experiments are presented to substantiate the theoretical findings and to compare the behaviors of different models.

Discontinuous Petrov–Galerkin Approximation of Eigenvalue Problems

Abstract In this paper, the discontinuous Petrov–Galerkin approximation of the Laplace eigenvalue problem is discussed. We consider in particular the primal and ultraweak formulations of the problem and prove the convergence together with a priori error estimates. Moreover, we propose two possible error estimators and perform the corresponding a posteriori error analysis. The theoretical results are confirmed numerically, and it is shown that the error estimators can be used to design an optimally convergent adaptive scheme.

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.

Optimized low-dispersion and low-dissipation two-derivative Runge–Kutta method for wave equations

The paper is devoted to the optimization of the explicit two-derivative sixth-order Runge–Kutta method in order to obtain low dissipation and dispersion errors. The method is dependent on two free parameters, used for the optimization. The optimized method demonstrates the lowest dispersion error in comparison with other widely-used high-order Runge–Kutta methods for hyperbolic problems. The efficiency of the method is demonstrated on the solutions of problems for five linear and nonlinear partial differential equations by the method-of-lines. Spatial derivatives are discretized by finite differences and Petrov–Galerkin approximations. The work-precision and error—CPU time plots, as the dependence of CPU time on space grid resolution, are considered. Also, the optimized method compared with other two-derivative methods. In most examples, the optimized method has better properties, especially for the cases of computations with adaptive stepsize control at high accuracy. The lowest CPU time takes place for the optimized method, especially in the cases of fine space grids.

A generalized-Jacobi-function spectral method for space-time fractional reaction-diffusion equations with viscosity terms

In this work, we study a new spectral Petrov-Galerkin approximation of space-time fractional reaction-diffusion equations with viscosity terms built by Riemann-Liouville fractional-order derivatives. The proposed method is reliant on generalized Jacobi functions (GJFs) for our problems. The contributions are threefold: First, thanks to the theoretical framework of variational problems, the well-posedness of the problem is proved. Second, new GJF-basis functions are established to fit weak solutions, which take full advantages of the global properties of fractional derivatives. Moreover, the basis functions conclude singular terms, in order to solve our problems with given smooth source term. Finally, we get a numerical analysis of error estimates to depend on GJF-basis functions. Numerical experiments confirm the expected convergence. In addition, they are given to show the effect of the viscosity terms in anomalous diffusion.

A novel model order reduction framework via staggered reduced basis space-time finite elements in linear first order transient systems

A novel model order reduction framework for space and time domain discretizations is proposed. Iterative convergence of a Galerkin approximation in space and a Least Squares Petrov Galerkin approximation in time is obtained through a staggered reduced basis method in space-time. In every iteration, one of the two domains (space or time) is refined; and the other is reduced and a posteriori error indicators in space and time are used to drive the convergence iterations. Numerical results for 2D heat transfer and convection-diffusion problems demonstrate the significant computational efficiency of the proposed methodology. Comparisons of wall-clock times and solution accuracy with traditional time integration algorithms has been presented to validate the efficacy of the proposed framework and demonstrate computational savings of an order of magnitude.

A moving Kriging‐based MLPG method for nonlinear Klein–Gordon equation

In this paper, the meshless local Petrov–Galerkin approximation is proposed to solve the 2‐D nonlinear Klein–Gordon equation. We used the moving Kriging interpolation instead of the MLS approximation to construct the meshless local Petrov–Galerkin shape functions. These shape functions possess the Kronecker delta function property. The Heaviside step function is used as a test function over the local sub‐domains. Here, no mesh is needed neither for integration of the local weak form nor for construction of the shape functions. So the present method is a truly meshless method. We employ a time‐stepping method to deal with the time derivative and a predictor–corrector scheme to eliminate the nonlinearity. Several examples are performed and compared with analytical solutions and with the results reported in the extant literature to illustrate the accuracy and efficiency of the presented method. Copyright © 2016 John Wiley & Sons, Ltd.

Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations

We analyze the convergence of compressive sensing based sampling techniques for the efficient evaluation of functionals of solutions for a class of high-dimensional, affine-parametric, linear operator equations which depend on possibly infinitely many parameters. The proposed algorithms are based on so-called non-intrusive sampling of the high-dimensional parameter space, reminiscent of Monte-Carlo sampling. In contrast to Monte-Carlo, however, a functional of the parametric solution is then computed via compressive sensing methods from samples of functionals of the solution. A key ingredient in our analysis of independent interest consists in a generalization of recent results on the approximate sparsity of generalized polynomial chaos representations (gpc) of the parametric solution families, in terms of the gpc series with respect to tensorized Chebyshev polynomials. In particular, we establish sufficient conditions on the parametric inputs to the parametric operator equation such that the Chebyshev coefficients of the gpc expansion are contained in certain weighted $\ell_p$-spaces for $0<p\leq 1$. Based on this we show that reconstructions of the parametric solutions computed from the sampled problems converge, with high probability, at the $L_2$, resp. $L_\infty$ convergence rates afforded by best $s$-term approximations of the parametric solution up to logarithmic factors.

The a posteriori error estimates of Chebyshev–Petrov–Galerkin methods for second-order equations

In this paper, the a posteriori error estimates of Chebyshev–Petrov–Galerkin approximations are investigated. For simplicity, we choose the Poisson equation with Dirichlet boundary conditions to discuss the a posteriori error estimators, and deduce their efficient and reliable properties. Some numerical experiments are performed to verify the theoretical analysis for the a posteriori error estimators.

Model order reduction with oblique projections for large scale wave propagation

There are many applications in which it is necessary to solve large scale parametrized wave propagation problems repeatedly. This is still quite a challenging task, even with the largest available computer clusters. In this paper we will discuss the application of Model Order Reduction (MOR) to problems in seismic petroleum exploration, with the aim of diminishing the necessary computing time by a significant factor. We consider POD and some variants. POD is a Model Order Reduction technique that uses snapshots of a few simulations in order to quickly compute related problems with similar accuracy. The method of lines via a Petrov–Galerkin approximation that uses the snapshots as basis functions is the considered approach. The order reduction comes from projecting the wave equation discretized in space to the subspace spanned by the snapshots. This has been shown earlier to work well in two dimensions. The challenge in three dimensions comes from the size of the spatial meshes required and the fact that the method usually requires a number of snapshots that do not fit in fast memory, even for current high end multicore machines. Parallelization is not an option since it is already used for other aspects of this massive problem.

Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations

We analyze the convergence of compressive sensing based sampling techniques for the ecient evaluation of functionals of solutions for a class of high-dimensional, ane-parametric, linear operator equations which depend on possibly innitely many parameters. The proposed algorithms are based on so-called \non-intrusive sampling of the high-dimensional parameter space, reminiscent of Monte-Carlo sampling. In contrast to Monte-Carlo, however, the parametric solution is then computed via compressive sensing methods from samples of (a functional of) the solution. A key ingredient in our analysis of independent interest consists in a generalization of recent results on the approximate sparsity of generalized polynomial chaos representations (gpc) of the parametric solution families, in terms of the gpc series with respect to tensorized Chebyshev polynomials. In particular, we establish sucient conditions on the parametric inputs to the parametric operator equation such that the Chebyshev coecients of gpc expansion are contained in certain weighted 'p-spaces for 0 < p 1. Based on this we show that reconstructions of the parametric solutions computed from the sampled problems converge, with high probability, at the L2, resp. L1 convergence rates aorded by best s-term approximations of the parametric solution up to logarithmic factors.

Approximation of solutions to operator equations in spaces of smooth functions and in spaces of distributions

The Galerkin method, in particular, the Galerkin method with finite elements (called finite element method) is widely used for numerical solution of differential equations. The Galerkin method allows us to obtain approximations of weak solutions only. However, there arises in applications a rich variety of problems where approximations of smooth solutions and solutions in the sense of distributions have to be found. This article is devoted to the employment of the Petrov–Galerkin method for solving such problems. The article contains general results on the Petrov–Galerkin approximations of solutions to linear and nonlinear operator equations. The problem on construction of the subspaces, which ensure the convergence of the approximations, is investigated. We apply the general results to two‐dimensional (2D) and 3D problems of the elasticity, to a parabolic problem, and to a nonlinear problem of the plasticity. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 406–450, 2014

Information-flux method: a meshfree maximum-entropy Petrov–Galerkin method including stabilised finite element methods

An information-flux method incorporating a novel approach to stable methods is proposed. The method may be considered as a meshfree Petrov–Galerkin approximation scheme with basis functions based on the principle of maximum entropy. The two goals of accuracy and stability are distinctly assigned to solution and weighting functions, respectively. It is emphasised that stability can be ensured if the weighting functions are chosen such that they resemble the information flux of the underlying physical problem. In this study, the proposed method is applied to convection-dominated convection–diffusion problems. A seamless transition of the proposed method to stabilised finite element methods is demonstrated for increasing locality of the basis functions in one dimension; for lower localities, a superior convergence behaviour can be shown compared to stabilised finite elements. The method is presented and discussed for the general multi-dimensional case, with numerical results shown for one- and two-dimensional problems.

Stability and passivity preserving Petrov–Galerkin approximation of linear infinite-dimensional systems

This contribution presents two approximation methods for linear infinite-dimensional systems that ensure the preservation of stability and passivity. The first approach allows one to approximate internal source free infinite-dimensional systems such that the resulting approximation is a port-controlled Hamiltonian system with dissipation. The second method deals with the class of systems that are not required to have conjugated outputs but only a dissipative system operator. It yields approximations with a dissipative system matrix for which bounds of their stability margin are provided. Both approaches are based on a state space formulation of the infinite-dimensional system. This makes it possible to use the Petrov–Galerkin approximation whose free parameters are partly used for achieving the structure preservation. Since still free parameters remain, further application specific objectives, such as, e.g., moment matching, can be achieved. Both approaches are applied to the approximation of an Euler–Bernoulli beam.

Comparison of the performance of SSPH and MLS basis functions for two-dimensional linear elastostatics problems including quasistatic crack propagation

We use symmetric smoothed particle hydrodynamics (SSPH) and moving least squares (MLS) basis functions to analyze six linear elastostatics problems by first deriving their Petrov-Galerkin approximations. With SSPH basis functions one can approximate the trial solution and its derivatives by using different basis functions whereas with MLS basis functions the derivatives of the trial solution involve derivatives of the basis functions used to approximate the trial solution. The class of allowable kernel functions for SSPH basis functions includes constant functions which are excluded in MLS basis functions if derivatives of the trial solution are also to be approximated. We compare results for different choices of weight functions, size of the compact support of the weight function, order of complete polynomials, and number of particles in the problem domain. The two basis functions are also used to analyze crack initiation and propagation in plane stress mode-I deformations of a plate made of a linear elastic isotropic and homogeneous material with particular emphasis on the computation of the T-stress. The crack trajectories predicted by using the two basis functions agree well with those found experimentally.

Global error bounds for the Petrov-Galerkin discretization of the neutron transport equation

In this paper, we prove that the numerical solution of the mono-directional neutron transport equation by the Petrov-Galerkin method converges to the true solution in the L{sup 2} norm at the rate of h{sup 2}. Since consistency has been shown elsewhere, the focus here is on stability. We prove that the system of Petrov-Galerkin equations is stable by showing that the 2-norm of the inverse of the matrix for the system of equations is bounded by a number that is independent of the order of the matrix. This bound is equal to the length of the longest path that it takes a neutron to cross the domain in a straight line. A consequence of this bound is that the global error of the Petrov-Galerkin approximation is of the same order of h as the local truncation error. We use this result to explain the widely held observation that the solution of the Petrov-Galerkin method is second accurate for one class of problems, but is only first order accurate for another class of problems.

Self-consistent boundary conditions with blurred derivatives

It is shown in this paper that self-consistent boundary conditions for numerical methods based on blurred derivatives can be derived from a suitable change of variables of the fundamental blurred approximation of the differential equation, followed by application of Leibnitz theorem for differentiation of an integral. The simplest scheme obtained in this way resembles the weak Local Petrov-Galerkin approximation, although interpretation of the operators appearing in the final equations is quite different—as is the derivation itself. Subsequent transformation leads to integral equations similar to the starting point for boundary integral methods of solution. In this way, a number of well-known computational methods are shown to be derivable from adequate manipulation of the blurred derivative technique. However, other approximations, which are not derivable with standard methods can also be obtained, hinting at a greater generality of blurred derivatives.

Meshless analysis of Timoshenko beams based on a locking-free formulation and variational approaches

A meshless method is developed, based on subdomain variational formulations and a local Petrov–Galerkin approximation, along with the locking-free formulation, for the analysis of bending of a thick beam. The local point interpolation method is employed to construct both trial and test functions. The present method is a truly meshless method based only on a number of randomly located nodes. No global background integration mesh is needed, no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. Effects of the sizes of local subdomain and interpolation domain on the performance of the present method are investigated. Problems of thick and thin beams under various loading, and boundary conditions are analysed by the proposed method and the numerical results are compared with analytical solutions. The contact problem for beam by means of subdomain variational inequalities and the meshless method has also been investigated.

Finite Difference Schemes and Block Rayleigh Quotient Iteration for Electronic Structure Calculations on Composite Grids

We present an original numerical method to discretize the Kohn–Sham equations by a finite difference scheme in real-space when computing the electronic structure of a molecule. The singular atomic potentials are replaced by pseudopotentials and the discretization of the 3D problem is done on a composite mesh refined in part of the domain. A “Mehrstellenverfahren” finite difference scheme is used to approximate the Laplacian on the regular parts of the grid. The nonlinearity of the potential operator in the Kohn–Sham equations is treated by a fixed point algorithm. At each step an iterative scheme is applied to determine the searched solutions of the eigenvalue problem for a given fixed potential. The eigensolver is a block generalization of the Rayleigh quotient iteration which uses Petrov–Galerkin approximations. The algorithm is adapted to a multigrid resolution of the linear systems obtained in the inverse iterations. Numerical tests of the different algorithms are presented on problems coming from the electronic structure calculation of some molecules.