Direct Meshless Local Petrov–Galerkin Research Articles

Numerical investigation based on direct meshless local Petrov Galerkin (direct MLPG) method for solving generalized Zakharov system in one and two dimensions and generalized Gross–Pitaevskii equation

In the current investigation, we propose two efficient meshless numerical techniques for solving two models in optic engineering, i.e. the generalized Gross–Pitaevskii equation and the generalized Zakharov system. Two local meshless methods have been employed for solving these models: local radial basis functions collocation method and direct meshless local Petrov–Galerkin method. In this paper, we discrete the space direction using the local radial basis functions collocation and direct meshless local Petrov–Galerkin techniques and to obtain high-order numerical results, we use the fourth-order exponential time differencing Runge–Kutta method for discretizing the temporal direction. The obtained numerical results are compared with some well-known numerical techniques. Moreover, several examples are given that show the acceptable accuracy and efficiency of the proposed scheme.

Direct meshless local Petrov–Galerkin method for the two-dimensional Klein–Gordon equation

In this paper we apply the direct meshless local Petrov–Galerkin (DMLPG) method to solve the two dimensional Klein–Gordon equations in both strong and weak forms. Low computational cost is the main property of this method compared with the original MLPG technique. The reason lies behind the approach of generalized moving least squares approximation where the discretized functionals, obtained from the PDE problem, are directly approximated from nodal values. This shifts the integration over polynomials rather than the MLS shape functions, leading to an extremely faster scheme. We will see that this method can successfully solve the problem with a reasonable accuracy.

Remediation of contaminated groundwater by meshless local weak forms

In this paper, four meshless local weak form methods such as direct meshless local Petrov–Galerkin method (DMLPG), meshless local Petrov–Galerkin method (MLPG), local weak radial point interpolation method (LWRPIM) and local weak moving kriging method (LWMK) are applied to find the numerical solution of coupled non-linear advection–diffusion–reaction system arising in the prevention of groundwater contamination problem. A comparison between these methods is done from the perspective of accuracy and computational efficiency. An efficient fourth-order exponential time differencing Runge–Kutta method is utilized for the time discretization. The computational efficiency is the most significant advantage of the DMLPG method in comparison with the other meshless local weak form methods, because DMLPG reduces the computational costs, significantly. This is due to the fact that DMLPG shifts the numerical integrations over low-degree polynomials rather than over complicated shape functions. The main aim of this paper is to show that the meshless local weak form methods can be used for solving the system of non-linear partial differential equations especially coupled non-linear advection–diffusion–reaction system. The numerical results confirm the good efficiency of the proposed methods for solving our model.

Application of direct meshless local Petrov–Galerkin (DMLPG) method for some Turing-type models

Mathematical modeling of pattern formation in developmental biology leads to non-linear reaction---diffusion systems which are usually highly stiff in both diffusion and reaction terms. In this paper, the Direct Meshless Local Petrov---Galerkin (DMLPG) procedure is applied to find the numerical solution of some non-linear time-dependent reaction---diffusion systems such as Schnakenberg model, Gierer---Meinhardt model, FitzHugh---Nagumo model and Gray---Scott model. As far as we know, it is the first time that DMLPG method is applied for solving non-linear partial differential equations (PDEs) and systems of PDEs. Computational efficiency is the most significant advantage of the DMLPG method in comparison with the classic Meshless Local Petrov---Galerkin (MLPG) method. This is due to the fact that DMLPG shifts the numerical integrations over low-degree polynomials instead of over complicated moving least squares (MLS) shape functions and this reduces the computational costs, significantly. The main aim of this paper is to show that the DMLPG method is also suitable for solving the non-linear time-dependent systems, especially reaction---diffusion systems. Numerical results support the good efficiency of the proposed method for solving non-linear reaction---diffusion systems. Also it is shown that DMLPG provides considerable savings in computational time in comparison with the classical MLPG method.

Direct meshless local Petrov–Galerkin method for elastodynamic analysis

This article describes a new and fast meshfree method based on a generalized moving least squares (GMLS) approximation and the local weak forms for vibration analysis in solids. In contrast to the meshless local Petrov–Galerkin method, GMLS directly approximates the local weak forms from meshless nodal values, which shifts the local integrations over the low-degree polynomial basis functions rather than over the complicated MLS shape functions. Besides, if the method is set up properly, all local integrals have the same value if all local subdomains have the same shape. These features reduce the computational costs, remarkably. The new technique is called direct meshless local Petrov–Galerkin (DMLPG) method. In DMLPG, the stiff and mass matrices are constructed by integration against polynomials. This overcomes the main drawback of meshfree methods in comparison with the finite element methods (FEM). The Newmark scheme is adapted as a time integration method, and numerical results are presented for various dynamic problems. The results are compared with the exact solutions, if available, and the FEM solutions.

DMLPG solution of the fractional advection–diffusion problem

The aim of this work is application of the direct meshless local Petrov–Galerkin (DMLPG) method for solving a two-dimensional time fractional advection–diffusion equation. This method is based on the generalized moving least squares (GMLS) approximation, and makes a considerable reduction in the cost of numerical integrations in weak forms. In fact, DMLPG shifts the integrals over the close form polynomials rather than the complicated MLS shape functions. Moreover, the values of integrals on subdomains with the same shapes are equal. Thus DMLPG is a weak-based meshless technique in the cost-level of collocation or integration-free methods. In time domain, a simple and suitable finite difference approximation is employed. Some examples show the advantages of the new method in comparison with the traditional MLPG method.

A new low-cost meshfree method for two and three dimensional problems in elasticity

In this paper, we continue the development of the Direct Meshless Local Petrov–Galerkin (DMLPG) method for elasto-static problems. This method is based on the generalized moving least squares approximation. The computational efficiency is the most significant advantage of the new method in comparison with the original MLPG. Although, the “Petrov–Galerkin” strategy is used to build the primary local weak forms, the role of trial space is ignored and direct approximations for local weak forms and boundary conditions are performed to construct the final stiffness matrix. In this modification the numerical integrations are performed over polynomials instead of complicated MLS shape functions. In this paper, DMLPG is applied for two and three dimensional problems in elasticity. Some variations of the new method are developed and their efficiencies are reported. Finally, we will conclude that DMLPG can replace the original MLPG in many situations.

Direct meshless local Petrov–Galerkin method for elliptic interface problems with applications in electrostatic and elastostatic

In recent years, there have been extensive efforts to find the numerical methods for solving problems with interface. The main idea of this work is to introduce an efficient truly meshless method based on the weak form for interface problems. The proposed method combines the direct meshless local Petrov–Galerkin method with the visibility criterion technique to solve the interface problems. It is well-known in the classical meshless local Petrov–Galerkin method, the numerical integration of local weak form based on the moving least squares shape function is computationally expensive. The direct meshless local Petrov–Galerkin method is a newly developed modification of the meshless local Petrov–Galerkin method that any linear functional of moving least squares approximation will be only done on its basis functions. It is done by using a generalized moving least squares approximation, when approximating a test functional in terms of nodes without employing shape functions. The direct meshless local Petrov–Galerkin method can be a very attractive scheme for computer modeling and simulation of problems in engineering and sciences, as it significantly uses less computational time in comparison with the classical meshless local Petrov–Galerkin method. To create the appropriate generalized moving least squares approximation in the vicinity of an interface, we choose the visibility criterion technique that modifies the support of the weight (or kernel) function. This technique, by truncating the support of the weight function, ignores the nodes on the other side of the interface and leads to a simple and efficient computational procedure for the solution of closed interface problems. In the proposed method, the essential boundary conditions and the jump conditions are directly imposed by substituting the corresponding terms in the system of equations. Also, the Heaviside step function is applied as the test function in the weak form on the local subdomains. Some numerical tests are given including weak and strong discontinuities in the Poisson interface problem. To demonstrate the application of these problems, linearized Poisson–Boltzmann and linear elasticity problems with two phases are studied. The proposed method is compared with analytical solution and the meshless local Petrov–Galerkin method on several test problems taken from the literature. The numerical results confirm the effectiveness of the proposed method for the interface problems and also provide significant savings in computational time rather than the classical meshless local Petrov–Galerkin method.

Meshless techniques for anisotropic diffusion

Meshless Local Petrov Galerkin (MLPG) methods are pure meshless techniques for solving Partial Differential Equations. They have been successfully used for solving many real-life problems. Their efficiency is downgraded by the requirement of performing numerical multi-dimensional integration of tricky, non-polynomial factors. Recently, Direct MLPG (DMLPG) methods have been proposed. DMLPG techniques require lower computational costs with respect to their MLPG counterparts. The DMLPG accuracy has been initially analyzed in few papers, but its performance is quite unexplored. In this paper, we perform numerical comparisons between MLPG and DMLPG accuracy and efficiency in solving anisotropic diffusion problems. In particular, we set different boundary conditions, in order to check if and when MLPG and/or DMLPG suffer locking effects.

Solving heat conduction problems by the Direct Meshless Local Petrov-Galerkin (DMLPG) method

As an improvement of the Meshless Local Petrov---Galerkin (MLPG), the Direct Meshless Local Petrov---Galerkin (DMLPG) method is applied here to the numerical solution of transient heat conduction problem. The new technique is based on direct recoveries of test functionals (local weak forms) from values at nodes without any detour via classical moving least squares (MLS) shape functions. This leads to an absolutely cheaper scheme where the numerical integrations will be done over low---degree polynomials rather than complicated MLS shape functions. This eliminates the main disadvantage of MLS based methods in comparison with finite element methods (FEM), namely the costs of numerical integration.

Direct Meshless Local Petrov–Galerkin (DMLPG) method: A generalized MLS approximation

The Meshless Local Petrov–Galerkin (MLPG) method is one of the popular meshless methods that has been used very successfully to solve several types of boundary value problems since the late nineties. In this paper, using a generalized moving least squares (GMLS) approximation, a new direct MLPG technique, called DMLPG, is presented. Following the principle of meshless methods to express everything “entirely in terms of nodes”, the generalized MLS recovers test functionals directly from values at nodes, without any detour via shape functions. This leads to a cheaper and even more accurate scheme. In particular, the complete absence of shape functions allows numerical integrations in the weak forms of the problem to be done over low-degree polynomials instead of complicated shape functions. Hence, the standard MLS shape function subroutines are not called at all. Numerical examples illustrate the superiority of the new technique over the classical MLPG. On the theoretical side, this paper discusses stability and convergence for the new discretizations that replace those of the standard MLPG. However, it does not treat stability, convergence, or error estimation for the MLPG as a whole. This should be taken from the literature on MLPG.