| Abstract: | In this paper, we propose a uniformly convergent adaptive finite element
method with hybrid basis (AFEM-HB) for the discretization of singularly perturbed
nonlinear eigenvalue problems under constraints with applications in Bose-Einstein
condensation (BEC) and quantum chemistry. We begin with the time-independent
Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed
nonlinear eigenvalue problem under a constraint. Matched asymptotic approximations
for the problem are reviewed to confirm the asymptotic behaviors of the solutions
in the boundary/interior layer regions. By using the normalized gradient flow, we
propose an adaptive finite element with hybrid basis to solve the singularly perturbed
nonlinear eigenvalue problem. Our basis functions and the mesh are chosen adaptively
to the small parameter ε. Extensive numerical results are reported to show the
uniform convergence property of our method. We also apply the AFEM-HB to compute
the ground and excited states of BEC with box/harmonic/optical lattice potential
in the semiclassical regime (0<ε?1). In addition, we give a detailed error analysis of
our 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/$(NlnN)^\frac{2}{3}$). |
