On the $h$-adaptive PUM and the $hp$-adaptive FEM approaches applied to PDEs in quantum mechanics

摘要: In this paper the $h$-adaptive partition-of-unity method and $h$- $hp$-adaptive finite element are applied to partial differential equations arising in quantum mechanics, namely, Schrodinger equation with Coulomb harmonic potentials, Poisson problem. Implementational details of related enforcing continuity hanging nodes degeneracy basis discussed. The is equipped an a posteriori error estimator, thus enabling implementation error-controlled adaptive mesh refinement strategies. To that end, local interpolation estimates derived for enriched class exponential functions. results same as thereby admit usage standard residual indicators. efficiency compared method. latter implemented by adopting analyticity estimate from Legendre coefficients. An extension approach multiple solution vectors proposed. Numerical confirm remarkable accuracy approach. case Hydrogen atom, linear was found be comparable target eigenvalue $10^{-3}$.