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In this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results. [ABSTRACT FROM AUTHOR]

This study explores the Petrov–Galerkin method’s application in solving a linear fourth-order ordinary beam equation of the form u″″+qu=fu^{\prime\prime} ^{\prime\prime} +qu=f. The equation entails two distinct boundary conditions: pinned–pinned conditions on uu and u′u^{\prime} , and clamped–clamped conditions on uu and u″{u}^{^{\prime\prime} }. To satisfy these boundary conditions, we have built two sets of basis functions. The explicit forms of all spectral matrices were reported. The nonhomogeneous boundary conditions were easily handled using perfect transformations, ensuring the numerical solution’s accuracy. Detailed analysis of the method’s convergence was studied. Some numerical examples were presented, accompanied by comparisons with other existing methods in the literature.

In this paper, the error analysis of the Petrov–Galerkin finite volume element method (FVEM) is investigated for a nonlinear parabolic integro-differential equation that arises in the mathematical modeling of the penetration of a magnetic field into a substance, accounting for temperature-dependent changes in electrical conductivity. Starting from Maxwell’s equations, we derive a one-dimensional model problem, which forms the basis of our analysis. Our main goal is to develop a general framework for obtaining finite volume element approximations and to study the error analysis. For simplicity, we consider only the lowest-order (linear and L-splines) finite volume elements. The novel contribution lies in the application of FVEM to this problem, leading to the establishment of an unconditionally stable numerical scheme and the derivation of optimal error estimates in the L∞(L2(Ω)) and L2(H01(Ω)) norms for both semi-discrete and linearized backward Euler fully-discrete schemes, using a generalized projection method that carefully manages the nonlinear terms. Lastly, numerical experiments are provided to support the theoretical conclusions.

This article presents the implementation of a randomized neural network (RNN) approach in approximating the solution of fractional order boundary value problems using a Petrov–Galerkin framework with Lagrange basis test functions. Traditional methods, like Physics Informed Neural Networks (PINNs), use standard deep learning techniques, which suffer from a computational bottleneck. In contrast, RNNs offer an alternative by employing a random structure with random coefficients, only solving for the output layer. We allow for the application of numerical analysis principles by using RNNs as trial functions and piecewise Lagrange polynomials as test functions. The article covers the construction and properties of the RNN basis, the definition and solution of fractional boundary value problems, and the implementation of the RNN Petrov–Galerkin method. We derive the stiffness matrix and solve it using least squares. Error analysis shows that the method meets the requirements of the Lax–Milgram lemma along with a Ceá inequality, ensuring optimal error estimates, depending on the regularity of the exact solution. Computational experiments demonstrate the method’s efficacy, including multiples cases with both regular and irregular solutions. The results highlight the utility of RNN-based Petrov–Galerkin methods in solving fractional differential equations with experimental convergence.

Herein, we construct an explicit modal numerical solver based on the spec-tral Petrov–Galerkin method via a specific combination of shifted Cheby-shev polynomial basis for handling the nonlinear time-fractional Burger-type partial differential equation in the Caputo sense. The process reduces the problem to a nonlinear system of algebraic equations. Solving this alge-braic equation system will yield the approximate solution’s unknown coef-ficients. Many relevant properties of Chebyshev polynomials are reported, some connection and linearization formulas are reported and proved, and all elements of the obtained matrices are evaluated neatly. Also, conver-gence and error analyses are established. Various illustrative examples demonstrate the applicability and accuracy of the proposed method and depict the absolute and estimated error figures. Besides, the current ap-proach’s high efficiency is proved by comparing it with other techniques in the literature.

This article proposes a family of the Petrov–Galerkin–FEM methods that can be used to solve the nonlinear Klein–Gordon equation. The discrete schemes were formulated based on the solution of the problem and its time derivative. They ensure that the total energy is conserved at a discrete level. The simplest two-layer scheme was studied numerically. Based on the solution of the test problems with smooth solutions, it was shown that the scheme can determine the solution of the problem, as well as its time derivative with an error of the order of O(h2 + τ 2) in the continuous L2 norm, where τ and h characterize the grid steps in time and space, respectively.

Herein, we construct an explicit modal numerical solver based on the spectral Petrov-Galerkin method via a specific combination of shifted Chebyshev polynomial basis for handling the nonlinear time-fractional Burger-type partial differential equation in the Caputo sense. The process reduces the problem to a nonlinear system of algebraic equations. Solving this algebraic equation system will yield the approximate solution's unknown coefficients. Many relevant properties of Chebyshev polynomials are reported, some connection and linearization formulas are reported and proved, and all elements of the obtained matrices are evaluated neatly. Also, convergence and error analyses are established. Various illustrative examples demonstrate the applicability and accuracy of the proposed method and depict the absolute and estimated error figures. Besides, the current approach's high efficiency is proved by comparing it with other techniques in the literature. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Basrah Researches (Sciences) is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this research, a compact combination of Chebyshev polynomials is created and used as a spatial basis for the time fractional fourth-order Euler–Bernoulli pinned–pinned beam. The method is based on applying the Petrov–Galerkin procedure to discretize the differential problem into a system of linear algebraic equations with unknown expansion coefficients. Using the efficient Gaussian elimination procedure, we solve the obtained system of equations with matrices of a particular pattern. The L∞{L}_{\infty } and L2{L}_{2} norms estimate the error bound. Three numerical examples were exhibited to verify the theoretical analysis and efficiency of the newly developed algorithm.

This article presents an implicit two-layer finite element scheme for solving the Kirchhoff equation, a nonlinear nonlocal equation of hyperbolic type with the Dirichlet integral. The discrete scheme was designed considering the solution of the problem and its derivative for the time variable. It ensures total energy conservation at a discrete level. The use of the Newton method was proven to be effective for solving the scheme on the time layer despite the nonlocality of the equation. The test problems with smooth solutions showed that the scheme can define both the solution of the problem and its time derivative with an error of O(h2+τ2) in the root-mean-square norm, where τ and h are the grid steps in time and space, respectively.

Copyright of Arab Journal of Science & Research Publishing is the property of Mussasat Al-Majallah Al-Arabiyah lil-Ulum Wa-Nashr Al-Abhath and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this contribution, it is demonstrated that the mesh sensitivity of linear elastic Reissner–Mindlin finite‐element plate formulations can be significantly reduced by using a Petrov–Galerkin‐based approach. In contrast to the usual Bubnov–Galerkin method, Petrov–Galerkin methods are generally characterized by the fact that the test function and the trial function are approximated using different shape functions. To provide an overview, established Petrov–Galerkin methods for 2D solid elements, which have been shown to reduce mesh sensitivity, are reviewed first. It is then investigated whether a suitable Petrov– Galerkin plate formulation can be developed. In this context, it is demonstrated that a full Petrov–Galerkin method leads to problems in the treatment of transverse shear locking. However, the proposed partial Petrov–Galerkin method shows the desired mesh‐insensitive behavior. [ABSTRACT FROM AUTHOR]

The primary focus of this article is on applying specific generalized Jacobi polynomials (GJPs) as basis functions to obtain the solution of linear and non-linear even-order two-point BVPs. These GJPs are orthogonal polynomials that are expressed as Legendre polynomial combinations. The linear even-order BVPs are treated using the Petrov–Galerkin method. In addition, a formula for the first-order derivative of these polynomials is expressed in terms of their original ones. This relation is the key to constructing an operational matrix of the GJPs that can be used to treat the non-linear two-point BVPs. In fact, a numerical approach is proposed using this operational matrix of derivatives to convert the non-linear differential equations into effectively solvable non-linear systems of equations. The convergence of the proposed generalized Jacobi expansion is investigated. To show the precision and viability of our suggested algorithms, some examples are given. [ABSTRACT FROM AUTHOR]

We extend the fictitious domain spectral method presented in Gu and Shen (SIAM J Sci Comput 43:A309–A329, 2021) for elliptic PDEs in bounded domains to the Helmhotlz equation in exterior domains. We first reduce the problem in an exterior domain to a bounded domain using the exact Dirichlet-to-Neumann operator. Next, we formulate the reduced problem into an equivalent problem in an annulus by using a fictitious domain approach. Then, we apply the Fourier-spectral method in the radial direction to reduce the problem in an annulus to a sequence of 1-D Bessel-type equations, each with a one-sided open boundary condition that are to be determined by the boundary condition of the original Helmholtz equation. We solve these 1-D Bessel-type equations by the Legendre-spectral method, and determine the open boundary conditions with a least square approach. We derive a wave number explicit error estimate for the special case of a circular obstacle, and provide ample numerical results to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper studies the transition to turbulence of pipe flow for a system rotating about the streamwise direction. The linearized Navier–Stokes equations are numerically solved using the Petrov–Galerkin discretization. Both destabilizing and stabilizing effects of the rotation are highlighted. We show that in comparison to the linear stability situation at the Reynolds number going up to 10 7 , for the non-rotating case, pipe flow is linearly unstable to infinitesimal perturbations at low rotation numbers. By fixing the Reynolds numbers, small rotation numbers increase the amplitude of perturbations, which attain the maximum values in the region of the unstable modes. As they exceed their maximum values, these perturbations are attenuated. When the rotation number is increased further, perturbations are progressively damped until a critical value above which the flow becomes linearly stable. We determine the region in perturbation amplitude which separates perturbations that trigger instabilities and perturbations that decay toward the laminar profile. This study also reveals that the large values of rotation numbers cause a symmetric structure onto the flow, in a domain that is periodically continued in the axial direction. [ABSTRACT FROM AUTHOR]

The primary focus of this article is on applying specific generalized Jacobi polynomials (GJPs) as basis functions to obtain the solution of linear and non-linear even-order two-point BVPs. These GJPs are orthogonal polynomials that are expressed as Legendre polynomial combinations. The linear even-order BVPs are treated using the Petrov–Galerkin method. In addition, a formula for the first-order derivative of these polynomials is expressed in terms of their original ones. This relation is the key to constructing an operational matrix of the GJPs that can be used to treat the non-linear two-point BVPs. In fact, a numerical approach is proposed using this operational matrix of derivatives to convert the non-linear differential equations into effectively solvable non-linear systems of equations. The convergence of the proposed generalized Jacobi expansion is investigated. To show the precision and viability of our suggested algorithms, some examples are given.

This paper presents a novel Petrov-Galerkin method for a class of linear systems of fractional differential equations. New fractional-order generalized Jacobi functions are introduced, and their approximation properties are investigated. We show that these functions satisfy the given supplementary conditions and have the same asymptotic behavior as the exact solution, which are essential properties to design a high-order spectral Petrov-Galerkin method. For implementing our scheme, we represent the approximate solution by a linear combination of fractional-order generalized Jacobi functions and minimize the residual using shifted fractional Jacobi functions. The numerical solvability of the resultant algebraic system is justified as well as convergence and stability properties of the proposed scheme are explored. Finally, the reliability of the method is evaluated using various analytical and realistic problems. [ABSTRACT FROM AUTHOR]

This paper presents new efficient numerical methods for solving Volterra integro-differential equations and a system of nonlinear delay integro-differential equations which arises in biology. The principal idea of these approaches is based on a careful blend of the Petrov-Galerkin technique and the Sumudu transform method. In the proposed methods, using Lagrange polynomials and zeros of Jacobi polynomials, the considered system of linear and nonlinear integro-differential equations, with their associated initial conditions are reduced to linear and nonlinear systems of algebraic equations in the unknown expansion coefficients. Solving the resulting algebraic systems by Gaussian elimination and Newton's methods respectively, approximate solutions of the integro-differential problems are constructed. Detailed error analysis of the proposed methods is carried out to establish and ascertain the reliability and effectiveness of the methods. The methods are then tested on several examples, and the results are compared with those obtained via existing methods in the literature. The numerical results showed that the proposed methods are accurate, efficient, and reliable for solving all kinds of integro-differential equations. [ABSTRACT FROM AUTHOR]

• A theorem on existence and uniqueness for SFDEs of Riemann-Liouville type is proved. • A PG finite element method based on fractional Jacobi polyfractonomials is proposed. • Error estimates are derived and numerical experiments are provided. • Paths of the RL fBm widely used in flow in telecommunication networks is generated. In this paper, we are concerned with existence, uniqueness and numerical approximation of the solution process to an initial value problem for stochastic fractional differential equation of Riemann-Liouville type. We propose and analyze a Petrov-Galerkin finite element method based on fractional (non-polynomial) Jacobi polyfractonomials as basis and test functions. Error estimates in L 2 norm are derived and numerical experiments are provided to validate the theoretical results. As an illustrative application, we generate sample paths of the Riemann-Liouville fractional Brownian motion which is of importance in many applications ranging from geophysics to traffic flow in telecommunication networks. [ABSTRACT FROM AUTHOR]

• A Petrov-Galerkin method is developed by two-sided Jacobi polyfractonomials. • We derive a priori error estimates in both L ω (− k , − s) 2 and fractional norms for the Petrov-Galerkin method. • We develop a novel postprocessing technique to enhance the Petrov-Galerkin method. • We define a-posteriori error estimators and prove they are asymptotically accuracy. • A posteriori error analysis is established rigorously. The analysis of diffusion-reaction equations with general two-sided fractional derivative characterized by a parameter p ∈ [ 0 , 1 ] is investigated in this paper. First, we present a Petrov-Galerkin method, derive a proper weak formulation and show the well-posedness of its weak solution. Moreover, on the basis of the two-sided Jacobi polyfractonomials, a priori error analysis of Petrov-Galerkin method is derived. Further, a posteriori error analysis is established rigorously. More precisely, we develop a novel postprocessing technique to enhance the Petrov-Galerkin method by adding a small amount of computation, and analyze asymptotically exact a-posteriori error estimators. Finally, we demonstrate the theoretical results with numerical examples. [ABSTRACT FROM AUTHOR]

We introduce the concept of machine-learning minimal-residual (ML-MRes) finite element discretizations of partial differential equations (PDEs), which resolve quantities of interest with striking accuracy, regardless of the underlying mesh size. The methods are obtained within a machine-learning framework during which the parameters defining the method are tuned against available training data. In particular, we use a provably stable parametric Petrov–Galerkin method that is equivalent to a minimal-residual formulation using a weighted norm. While the trial space is a standard finite element space, the test space has parameters that are tuned in an off-line stage. Finding the optimal test space therefore amounts to obtaining a goal-oriented discretization that is completely tailored towards the quantity of interest. We use an artificial neural network to define the parametric family of test spaces. Using numerical examples for the Laplacian and advection equation in one and two dimensions, we demonstrate that the ML-MRes finite element method has superior approximation of quantities of interest even on very coarse meshes. [ABSTRACT FROM AUTHOR]

The article focuses on the use of the meshfree numerical method in the field of slope stability computations. There are many meshfree implementations of numerical methods. The article shows the results obtained using the meshfree localized Petrov-Galerkin method (MLPG) – localized weak-form of the equilibrium equations with an often used elastoplastic material model based on Mohr-Coulomb (MC) yield criterion. The most important aspect of MLPG is that the discretization process uses a set of nodes instead of elements. Node position within the computational domain is not restricted by any prescribed relationship. The shape functions are constructed using just the set of nodes present in the simple shaped domain of influence. The benchmark slope stability numerical model was performed using the developed meshfree computer code and compared with conventional finite element (FEM) and limit equilibrium (LEM) codes. The results showed the ability of the implemented theoretical preliminaries to solve the geotechnical stability problems.

We propose a reconstruction-based a posteriori error estimate for linear advection problems in one space dimension. In our framework, a stable variational ultra-weak formulation is adopted, and the equivalence of the L2-norm of the error with the dual graph norm of the residual is established. This dual norm is showed to be localizable over vertex-based patch subdomains of the computational domain under the condition of the orthogonality of the residual to the piecewise affine hat functions. We show that this condition is valid for some well-known numerical methods including continuous/discontinuous Petrov–Galerkin and discontinuous Galerkin methods. Consequently, a well-posed local problem on each patch is identified, which leads to a global conforming reconstruction of the discrete solution. We prove that this reconstruction provides a guaranteed upper bound on the L2 error. Moreover, up to a generic constant, it also gives local lower bounds on the L2 error, where the constant only depends on the mesh shape-regularity. This, in particular, leads to robustness of our estimates with respect to the polynomial degree. All the above properties are verified in a series of numerical experiments, additionally leading to asymptotic exactness. Motivated by these results, we finally propose a heuristic extension of our methodology to any space dimension, achieved by solving local least-squares problems on vertex-based patches. Though not anymore guaranteed, the resulting error indicator is still numerically robust with respect to both advection velocity and polynomial degree in our collection of two-dimensional test cases including discontinuous solutions aligned and not aligned with the computational mesh. [ABSTRACT FROM AUTHOR]

Numerical methods for stochastic partial differential equations typically estimate moments of the solution from sampled paths. Instead, we shall directly target the deterministic equations satisfied by the mean and the spatio-temporal covariance structure of the solution process. In the first part, we focus on stochastic ordinary differential equations. For the canonical examples with additive noise (Ornstein-Uhlenbeck process) or multiplicative noise (geometric Brownian motion) we derive these deterministic equations in variational form and discuss their well-posedness in detail. Notably, the second moment equation in the multiplicative case is naturally posed on projective-injective tensor product spaces as trial-test spaces. We then propose numerical approximations based on Petrov-Galerkin discretizations with tensor product piecewise polynomials and analyze their stability and convergence in the natural tensor norms. In the second part, we proceed with parabolic stochastic partial differential equations with affine multiplicative noise. We prove well-posedness of the deterministic variational problem for the second moment, improving an earlier result. We then propose conforming space-time Petrov-Galerkin discretizations, which we show to be stable and quasi-optimal. In both parts, the outcomes are validated by numerical examples. [ABSTRACT FROM AUTHOR]

This paper concerns a numerical solution for the diffusion equation on the unit sphere. The given method is based on the spherical basis function approximation and the Petrov–Galerkin test discretization. The method is meshless because spherical triangulation is not required neither for approximation nor for numerical integration. This feature is achieved through the spherical basis function approximation and the use of local weak forms instead of a global variational formulation. The local Petrov–Galerkin formulation allows to compute the integrals on small independent spherical caps without any dependence on a connected background mesh. Experimental results show the accuracy and the efficiency of the new method. [ABSTRACT FROM AUTHOR]

This manuscript is devoted to present an efficient numerical method for finding numerical solution of Volterra-Fredholm integro-differential equations of fractionalorder. This technique is based on applying Müntz-Legendre polynomials and Petrov- Galerkin method. A new Riemann-Liouville operational matrix for Müntz-Legendre polynomials is proposed using Laplace transform. Employing this operational matrix and Petrov-Galerkin method, transforms the problem into a system of algebraic equations. Next, we solve this system by applying any iterative method. An estimation of the error is proposed. Moreover, some numerical examples are implemented in order to show the validity and accuracy of the suggested method. [ABSTRACT FROM AUTHOR]

In this paper, we present a Petrov–Galerkin finite element method for a class of chemotaxis models defined on surfaces, which describe the movement by one community in reaction to one chemical or biological signal on manifolds. It is desired for numerical methods to satisfy discrete maximum principle and discrete mass conservation property, which is a challenge due to the singular behavior of numerical solution. Thus a Petrov–Galerkin method is combined with an effective mass conservation factor to overcome the challenge. Furthermore, we prove two facts, this method maintains positivity and discrete mass conservation property. In addition, decoupled approach is applied based on the gradient and Laplacian recoveries to solve the coupling system. The relevant stability analyses is provided. Finally, numerical simulations of blowing-up problems and pattern formulations demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, the idea of direct discretization via radial basis functions (RBFs) is applied on a local Petrov–Galerkin test space of a partial differential equation (PDE). This results to a weak-based RBF-generated finite difference (RBF–FD) scheme that possesses some useful properties. The error and stability issues are considered. When the PDE solution or the basis function has low smoothness, the new method gives more accurate results than the already well-established strong-based collocation methods. Although the method uses a Galerkin formulation, it still remains meshless because not only the approximation process relies on scattered point layouts but also integrations are done over non-connected, independent and well-shaped subdomains. Some applications to potential and elasticity problems on scattered data points support the theoretical analysis and show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Many problems in electrical engineering or fluid mechanics can be modeled by parabolic-elliptic interface problems, where the domain for the exterior elliptic problem might be unbounded. A possibility to solve this class of problems numerically is the non-symmetric coupling of finite elements (FEM) and boundary elements (BEM) analyzed in [H. Egger, C. Erath and R. Schorr, On the nonsymmetric coupling method for parabolic-elliptic interface problems, SIAM J. Numer. Anal. 56 2018, 6, 3510–3533]. If, for example, the interior problem represents a fluid, this method is not appropriate since FEM in general lacks conservation of numerical fluxes and in case of convection dominance also stability. A possible remedy to guarantee both is the use of the vertex-centered finite volume method (FVM) with an upwind stabilization option. Thus, we propose a (non-symmetric) coupling of FVM and BEM for a semi-discretization of the underlying problem. For the subsequent time discretization we introduce two options: a variant of the backward Euler method which allows us to develop an analysis under minimal regularity assumptions and the classical backward Euler method. We analyze both, the semi-discrete and the fully-discrete system, in terms of convergence and error estimates. Some numerical examples illustrate the theoretical findings and give some ideas for practical applications. [ABSTRACT FROM AUTHOR]

We propose and analyze a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak formulation allows for the direct approximation of solutions in the Lebesgue L p {L^{p}} -space, 1 < p < ∞ {1

In this paper, we develop an h-p version of finite element method for one-dimensional fractional differential equation − 0 D x α u + Au = f (x) with Dirichlet boundary condition. First, we introduce the existence and uniqueness of the considered problem and show the wellposedness of the corresponding weak form. To solve the mentioned equation, the classical hierarchical polynomials are employed as the trial basis functions. Then, we develop a kind of novel test basis functions for the Petrov-Galerkin finite element method such that the stiffness matrix becomes an identity matrix and the coefficient matrix often has a small condition number. Moreover, we give some properties of the developed test basis functions, and discuss the implementation of the developed finite element method in detail. It is shown that the implementation of our method is easier than that of other finite (and spectral) element methods. Finally, we give a numerical example, and the numerical results show the effectiveness of the develped method. [ABSTRACT FROM AUTHOR]

In this paper, time-fractional Sharma–Tasso–Olver (STO) equation has been solved numerically through the Petrov–Galerkin approach utilizing a quintic B-spline function as the test function and a linear hat function as the trial function. The Petrov–Galerkin technique is effectively implemented to the fractional STO equation for acquiring the approximate solution numerically. The numerical outcomes are observed in adequate compatibility with those obtained from variational iteration method (VIM) and exact solutions. For fractional order, the numerical outcomes of fractional Sharma–Tasso–Olver equations are also compared with those obtained by variational iteration method (VIM) in Song et al. [Song L., Wang Q., Zhang H., Rational approximation solution of the fractional Sharma–Tasso–Olver equation, J. Comput. Appl. Math. 224:210–218, 2009]. Numerical experiments exhibit the accuracy and efficiency of the approach in order to solve nonlinear fractional STO equation. [ABSTRACT FROM AUTHOR]

We present a mass lumping approach based on an isogeometric Petrov–Galerkin method that preserves higher-order spatial accuracy in explicit dynamics calculations irrespective of the polynomial degree of the spline approximation. To discretize the test function space, our method uses an approximate dual basis, whose functions are smooth, have local support and satisfy approximate bi-orthogonality with respect to a trial space of B-splines. The resulting mass matrix is "close" to the identity matrix. Specifically, a lumped version of this mass matrix preserves all relevant polynomials when utilized in a Galerkin projection. Consequently, the mass matrix can be lumped (via row-sum lumping) without compromising spatial accuracy in explicit dynamics calculations. We also address the imposition of Dirichlet boundary conditions and the preservation of approximate bi-orthogonality under geometric mappings. In addition, we establish a link between the exact dual and approximate dual basis functions via an iterative algorithm that improves the approximate dual basis towards exact bi-orthogonality. We demonstrate the performance of our higher-order accurate mass lumping approach via convergence studies and spectral analyses of discretized beam, plate and shell models. • We present a mass lumping approach based on an isogeometric Petrov–Galerkin method. • It is based on smooth, locally supported, approximate dual functions to B-splines. • The resulting mass matrix can be lumped without compromising the spatial accuracy in explicit dynamics calculations. • We demonstrate its performance via beam, plate and shell benchmarks. [ABSTRACT FROM AUTHOR]

The finite element method (FEM) is applied to obtain numerical solutions to a recently derived nonlinear equation for the shallow water wave problem. A weak formulation and the Petrov–Galerkin method are used. It is shown that the FEM gives a reasonable description of the wave dynamics of soliton waves governed by extended KdV equations. Some new results for several cases of bottom shapes are presented. The numerical scheme presented here is suitable for taking into account stochastic effects, which will be discussed in a subsequent paper.

Copyright of Journal of College of Education is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The ion‐flow environment is an important corona design factor of HVDC transmission lines. The space charges generated by corona form a typical convection‐dominated ion‐flow field. For this special transport phenomenon, however, standard Galerkin FEM always presents serious deficiencies. In this study, a new approach is used to solve ion‐flow field based on Petrov–Galerkin method derived from the theory of fluid computation. A high‐order stabilisation technique is used to overcome the non‐physical oscillations in the presence of highly convective effects. This algorithm is equivalent to the proper modification of weight function according to the direction and magnitude of local drift velocity. It reflects the characteristic of information propagation in flow problem. The establishment procedure of weak form and matrix gives a further explanation on the deficiency of Galerkin method. Moreover, to guarantee convergence of the iteration for this fully‐coupled non‐linear problem, a relaxation method is introduced in the iterative loop. Calculations with the Petrov–Galerkin least square method are in good agreement with analytical solution and experiment values of ground‐level values and corona losses. Results show that the proposed stabilisation technique is able to preclude the numerical spurious oscillation and has a higher‐order accuracy than the fully‐upwind treatment. [ABSTRACT FROM AUTHOR]

In this paper, a numerical solution of partial differential equations on the unit sphere is given by using a kernel trial approximation in combination with a special Petrov-Galerkin test discretization. The solvability of the scheme is proved, and the error bounds are obtained for functions in appropriate Sobolev spaces. The condition number of the final system is estimated in terms of discretization parameters. The method is meshless because in the trial side the numerical solution parameterizes entirely in terms of scattered points and in the test side everything breaks down to simple numerical integrations over independent spherical caps. This means that no connected background mesh is required for either approximation or integration. [ABSTRACT FROM AUTHOR]