Fully discretized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem

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This paper studies the numerical approximation of the ground state of the Gross-Pitaevskii (GP) eigenvalue problem with a fully discretized Sobolev gradient flow induced by the H 1 H^1 norm. For the spatial discretization, we consider the finite element method with quadrature using P k P^k basis on a simplicial mesh and Q k Q^k basis on a rectangular mesh. We prove the global convergence to a critical point of the discrete GP energy, and establish a local exponential convergence to the ground state under the assumption that the linearized discrete Schrödinger operator has a positive spectral gap. We also show that for the P 1 P^1 finite element discretization with quadrature on an unstructured shape regular simplicial mesh, the eigengap satisfies a mesh-independent lower bound, which implies a mesh-independent local convergence rate for the proposed discrete gradient flow. Numerical experiments with discretization by high-order Q k Q^k spectral element methods in two and three dimensions are provided to validate the efficiency of the proposed method.