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The accuracy of numerical solutions near singular points is crucial for numerical methods. In this paper we develop an efficient mechanical quadrature method (MQM) with high accuracy. The following advantages of MQM show that it is very promising and beneficial for practical applications: (1) the $ O(h_{\rm {max}}^{3})$ convergence rate; (2) the $O(h_{\rm {max}}^{5})$ convergence rate after...
Series expansions of fundamental solutions are essential to algorithms and analysis of the null field method (NFM) and to analysis of the method of fundamental solutions (MFS). For linear elastostatics, new Fourier series expansions of FS are derived, directly from integration. The new expansions of the FS are simpler than those in Chen et al. (J Mech 26(3):393–401, 2010), thus facile to application...
Consider the over-determined system Fx = b where $${{\bf F}\in\mathcal{R}^{m \times n}, m \geq n}$$ and rank (F) = r ≤ n, the effective condition number is defined by $${{\rm Cond_{-}eff }= \frac {\|{\bf b}\|}{\sigma_r\|{\bf x}\|}}$$ , where the singular values of F are given as σmax = σ1 ≥ σ2 ≥ . . . ≥ σr > 0 and σr+1 = . . . = σn = 0. For the general perturbed system (A+ΔA...
Since the stability of the method of fundamental solutions (MFS) is a severe issue, the estimation on the bounds of condition number Cond is important to real application. In this paper, we propose the new approaches for deriving the asymptotes of Cond, and apply them for the Dirichlet problem of Laplace’s equation, to provide the sharp bound of Cond for disk domains. Then the new bound of Cond is...
In this short article, we recalculate the numerical example in Křížek and Neittaanmäki (1987) for the Poisson solution u=xσ(1−x)sinπy in the unit square S as . By the finite difference method, an error analysis for such a problem is given from our previous study by where h is the meshspacing of the uniform square grids used, and C1 and C2 are two positive constants. Let ε=u−uh, where uh...
Abstract To solve the elliptic boundary value problems with singularities, the simplified hybrid combinations of the Ritz-Galerkin and finite element methods (RGM-FEM) are explored to lead to the global superconvergence rates on the entire solution domain, based on an a posteriori interpolation techniques of Lin and Yan [12] that only cost a little more computation. Let the solution domain S=S1S20...
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