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

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">H^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm. For the spatial discretization, we consider the finite element method with quadrature using <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Superscript k"> <mml:semantics> <mml:msup> <mml:mi>P</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">P^k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> basis on a simplicial mesh and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q Superscript k"> <mml:semantics> <mml:msup> <mml:mi>Q</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">Q^k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">P^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q Superscript k"> <mml:semantics> <mml:msup> <mml:mi>Q</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">Q^k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spectral element methods in two and three dimensions are provided to validate the efficiency of the proposed method.