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In this paper, based on the moving Kriging interpolation (MKI), the meshless interpolating local Petrov–Galerkin (ILPG) method is presented to solve two- and three-dimensional potential problems. In the present method, the shape function constructed by the MKI has the property of the Kronecker δ function. Then in the ILPG method the essential boundary conditions can be implemented directly. The discrete equations are obtained using the local symmetric weak form. The Heaviside step function is used as the test function in each sub-domain to avoid some domain integral in the symmetric weak form, which will greatly improve the effectiveness of the present method. The ILPG method in this paper is a truly meshless method, which does not require a mesh either for obtaining shape function or for numerical integration in the local weak form. Several numerical examples of potential problems show that the ILPG method has higher computational efficiency and convergence rate than the MLPG method.

The flower-like ZnO with micro-nano hierarchical structure is successfully obtained by a simple hydrothermal synthesis, using sodium dodecyl benzene sulfonate (SDBS) as a structure direct agent. The resulted ZnO microflowers are very uniform in morphology with particle sizes around 1 μm. A number of techniques, including X-ray diffraction (XRD), field emission scan electron microscopy (FESEM), energy-dispersive spectroscopy (EDS), fourier transform infrared (FTIR) spectra and thermogravimetry analysis (TGA), are used to characterize the obtained ZnO. The self-assemble of ZnO nano-sheets under the direction of SDBS leads to the formation of ZnO micro-flowers. The room temperature photoluminescence property of the obtained flower-like ZnO exhibits a broad visible light emission. The surface of as-made ZnO shows a very hydrophilic property, while the special micro-nano hierarchical structure enables the ZnO micro-flower a superhydrophobic surface after modification of fluoroalkylsilane.

The main objective of this work is to demonstrate that sharp a posteriori error estimators can be employed as appropriate monitor functions for moving mesh methods. We illustrate the main ideas by considering elliptic obstacle problems. Some important issues such as how to derive the sharp estimators and how to smooth the monitor functions are addressed. The numerical schemes are applied to a number of test problems in two dimensions. It is shown that the moving mesh methods with the proposed monitor functions can effectively capture the free boundaries of the elliptic obstacle problems and reduce the numerical errors arising from the free boundaries. KEY WORDS:Finite element method; moving mesh method; a posteriori error estimator; obstacle problem.