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We present an efficient algorithm for the construction of a basis of H2(Ω¯,∂Ω;ℤ) via the Poincaré-Lefschetz duality theorem. Denoting by g the first Betti number of Ω¯ the idea is to find, first g different 1-boundaries of Ω¯ with supports contained in ∂Ω whose homology classes in ℝ3∖Ω form a basis of H1(ℝ3∖Ω;ℤ), and then to construct a set of 2-chains in Ω¯ having these 1-boundaries as their boundaries. The Poincaré-Lefschetz duality theorem ensures that the relative homology classes of these 2-chains in Ω¯ modulo ∂Ω form a basis of H2(Ω¯,∂Ω;ℤ). We devise a simple procedure for the construction of the required set of 1-boundaries of Ω¯ that, combined with a fast algorithm for the construction of 2-chains with prescribed boundary, allows the efficient computation of a basis of H2(Ω¯,∂Ω;ℤ) via this very natural approach. Some numerical experiments show the efficiency of the method and its performance comparing with other algorithms. [ABSTRACT FROM AUTHOR]

This paper presents the solution of the Schrödinger-Poisson coupled problem for nanoscale electron devices obtained by means of the Discrete Geometric Approach (DGA). The paper illustrates a self-contained description of the DGA method for a Schrödinger-Poisson problem, discusses its implementation and compares the results of the DGA with respect to the ones obtained by the well established Pseudo-spectral (PS) method for two technologically relevant benchmark devices (i.e. a nanowire and a FinFET). Finally, the paper examines the merits of the DGA approach with respect to the Finite Differences (FD) and Finite Elements (FE), that are the most frequently used methods in the electron device community. [ABSTRACT FROM AUTHOR]