Alternating direction implicit (A. D. I. ) schemes have been proved valuable in the approximation of the solutions of parabolic partial differentialequations in multi-dimensional space. Consider equations in the formTwo A. D. I. schemes, Peaceman-Rachford scheme and Douglas scheme will be studied. In the literature, stability and convergence have been analysed with FourierMethod, which cannot be extended beyond the model problem with constantcoefficients. Additionally, L2 energy method has been introduced to analyse the caseof non-constant coefficients, however, the conclusions are too weak and incomplete because Of the so-called "equivalence between L2 norm and H1 semi-norm". In this paper, we try to improve these conclusions by H1 energy estimating method. The principal results ore that both of the two A. D. I. schemes are absolutely stable and converge to the exact solution with error estimations O(Δt2 + h2 ) in discrete H1norm. This implies essential improvement Of existing conclusions.
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