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The exact spectra of two electrons confined by two-dimensional and three-dimensional quantum dots ͑2D and 3D QD's͒ with parabolic potentials are obtained. Using the present results, the size and shape effects of QD's on the spectra are revealed. It is found that the spectra are dramatically changed with the variation of the dot size, and then the crossover of two levels can appear. The variation of spectra with size and the positions of crossover points are quite different between 2D and 3D QD's. It is well explained based on the study of electron-electron interaction energies in 2D and 3D QD's. The size and shape effects predict a possibility to observe phenomena related to electron-electron interactions in QD's. ͓S0163-1829͑97͒06120-1͔

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We propose a projection iterative algorithm based on a fixed point equation for solving a certain class of Signorini problem. The satisfaction of the Signorini boundary conditions is verified in a projection iterative manner, and at each iterative step, an elliptic mixed boundary value problem is solved by a boundary element method which is suitable for any domain. We prove the convergence of the algorithm by the property of projection. The advantage of this algorithm is that it is easy to be implemented and converge quickly. Some numerical results show the accuracy and effectiveness of the algorithm.