Quantum dots can grow spontaneously during molecular beam epitaxy of two materials with different lattice parameters, Stranski-Krastanow growth mode. We study a mathematical model based on the continuum approximation of the growing layer in two dimensions. Nonlinear evolution equation is solved using finite element method and spectral methods together with stress calculation based on the Goursat analytic functions. Effect of the film/substrate interface is included in the dependence of surface tension on height of the film. The effect of directional anisotropy of surface tension on the evolution is established together with stability of faceted island arrays and their shape transitions. Corner between facets and strong anisotropy lead to difficulties in the numerical analysis, which are prevented by including the regularization terms. Simulation results qualitatively agree with experiments.
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