| Abstract: | In this paper, we propose a uniformly convergent adaptive finite elementmethod with hybrid basis (AFEM-HB) for the discretization of singularly perturbednonlinear eigenvalue problems under constraints with applications in Bose-Einsteincondensation (BEC) and quantum chemistry. We begin with the time-independentGross-Pitaevskii equation and show how to reformulate it into a singularly perturbednonlinear eigenvalue problem under a constraint. Matched asymptotic approximationsfor the problem are reviewed to confirm the asymptotic behaviors of the solutionsin the boundary/interior layer regions. By using the normalized gradient flow, wepropose an adaptive finite element with hybrid basis to solve the singularly perturbednonlinear eigenvalue problem. Our basis functions and the mesh are chosen adaptivelyto the small parameter ε. Extensive numerical results are reported to show theuniform convergence property of our method. We also apply the AFEM-HB to computethe ground and excited states of BEC with box/harmonic/optical lattice potentialin the semiclassical regime (0<ε≪1). In addition, we give a detailed error analysis ofour AFEM-HB to a simpler singularly perturbed two point boundary value problem,show that our method has a minimum uniform convergence order $mathcal{O}$(1/$(NlnN)^frac{2}{3}$). |
