Abstract
SUMMARYNew bridging techniques are introduced to match high degree polynomials. This permits piecewise resolutions of elliptic partial differential equations (PDEs) in the framework of a boundary meshless method introduced recently. This new meshless method relies on the computation of Taylor series approximations deduced from the PDE, the shape functions being high degree polynomials. In this way, the PDE is solved quasi‐exactly inside the subdomains so that only discretization of the boundary and the interfaces are needed, which leads to small size matricial problems. The bridging techniques are based on the introduction of Lagrange multipliers and a set of collocation points on the boundary and the interfaces. Several numerical applications establish that the method is robust and permits an exponential convergence with the degree. Copyright © 2013 John Wiley & Sons, Ltd.
Highlights
In this paper, we improve a high-order boundary meshless method that has been recently introduced [1, 2]
The solution is sought in the form of a high-order polynomial, and the number of its coefficients is reduced by vanishing the Taylor series of the partial differential equations (PDEs) at a given expansion point
Two bridging techniques have been proposed to link high-order polynomial approximations, in the framework of a new discretization method named Taylor meshless method (TMM), in which the shape functions are deduced from the PDE via Taylor series expansions
Summary
We improve a high-order boundary meshless method that has been recently introduced [1, 2]. The basic idea of this method is to reduce the number of unknowns by analytically solving the partial differential equation (PDE) using the technique of a Taylor series. Contrary to MFS and because the PDE is solved quasi-exactly through a Taylor series, the present method remains truly a boundary technique even for non-homogeneous and nonlinear equations [13]. The continuity is enforced only in some nodes of the interface, and this continuity is understood in a mean sense in the Arlequin case Such weak continuity requirements seem more relevant to couple quite different functions. The coupling between several meshless approximations has not yet been studied, and the exponential convergence property with the degree (p-convergence) has not been discussed For this purpose, we propose some variants of mortar and Arlequin techniques that are consistent with the framework of collocation meshless methods.
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