Localized meshless methods based on polynomial basis functions for solving axisymmetric equations

Abstract

In this paper, two localized meshless methods based on polynomial basis functions are utilized to solve axisymmetric problems. In the first approach, we applied the localized method of particular solutions (LMPS) and the closed-form particular solution to simplify the two-stage approach using Chebyshev polynomial as the basis functions for solving axisymmetric problems. We also propose the modified local Pascal polynomial method (MLPM) to compare the results with LMPS. Since only the low order polynomial basis functions are used, no preconditioning treatment is required and the solution is quite stable. Four numerical examples are given to demonstrate the effectiveness of the proposed methods.

Introduction

Finite element method and finite difference method are traditionally the dominated numerical methods for solving science and engineering problems [1], [8], [15], [22], [23]. For three dimensional problems, these mesh-based methods have experienced tedious procedure to mesh both the boundary and domain of the given 3D problem. In many realistic real world problems, the geometry of the given 3D domain is axisymmetric. In the case when the forcing term of the given differential equation and its boundary condition are also axisymmetric, the associate partial differential equation can be reduced to 2D case due to the symmetry of the 3D domain. As a result, the complication of meshing the 3D domain can be greatly alleviated by solving the reduced axisymmetric problem in 2D. On the other hand, a singularity has been introduced for the 2D Laplace operator which causes another difficulty in the solution process. Various numerical techniques have been proposed for solving 2D axisymmetric problems [5], [10], [12], [13], [17], [20], [21].

In recent years, meshless methods have gained popularity for solving various kinds of partial differential equations. Among them, the method of particular solutions (MPS) and the method of fundamental solutions (MFS) have been applied for solving axisymmetric problems [17], [20] in which two-stage approach has been adopted. Such two-stage approach is restricted to certain types of differential equations when the fundamental solution of the associate differential operator is available. The recent development of the one-stage MPS [2], [3], [6] has further simplified the two-stage approach without the need of solving homogeneous equation. In the past, due to the use of the two-stage approach, Chebyshev polynomial [14], [19] was being used as a basis function for obtaining the particular solution. As such, it is required to assume that the forcing term of the differential equation can be smoothly extended to outside the domain. Furthermore, in the solution process, we need to expand the Chebyshev polynomial into a series of monomial since the closed-form particular solution is only available for monomial. As a result, the solution procedure is quite tedious and more restricted to certain types of differential equations. In contrast, the one-stage approach has become populous due to its simplicity and applicable for solving a larger class of differential equations. A localized method is crucial for solving large-scale problems [24], [26]. A localized version of the MPS (LMPS) has also been developed for solving large-scale problems. In this paper, we proposed to adopt the closed-form particular solution derived in [5] and the LMPS to solve various types of axi-symmetric equations. In contrast to the two-stage MPS, our proposed approach is simple, faster, and capable for solving a large-scale problems due to the usage of the localized method. It is well-known that the polynomial is notorious for severe ill-conditioning and special treatment for preconditioning is normally required. Since we focus on the localized method, the order of polynomial is basically pretty low and the conditioning number is not an issue. In this paper, we also consider a direct approach using polynomial which is similar to multiple-scale Pascal polynomial method [16], [18]. Note that we modify the global version of multiple-scale Pascal polynomial method to the modified local Pascal polynomial method (MLPM) so that it will work for the axisymmetric problems without using multiple-scale technique. Basically, MLPM is a direct approach and the LMPS is a reverse approach using particular solution. In general, in terms of theoretical aspect, the polynomial basis functions for the LMPS is more completed than the MLPM. However, as far as the numerical accuracy is concerned, we find there are little difference.

The paper is organized as follows. In Section 2, we reduce the three-dimensional axisymmetric equation to an equivalent 2D equation. In Section 3, we apply both the localized method of particular solution(LMPS) and modified local Pascal polynomial method (MLPM) to solve the axisymmetric problems. In Section 4, several numerical examples are given to demonstrate the effectiveness of the proposed method. Finally, some conclusions and ideas for future work are outlined in Section 5.