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An adaptive FEM with ITP approach for steady Schrödinger equation. (English) Zbl 1390.65147

Summary: In this paper, an adaptive numerical method is proposed for solving a 2D Schrödinger equation with an imaginary time propagation approach. The differential equation is first transferred via a Wick rotation to a real time-dependent equation, whose solution corresponds to the ground state of a given system when time approaches infinity. The temporal equation is then discretized spatially via a finite element method, and temporally utilizing a Crank-Nicolson scheme. A moving mesh strategy based on harmonic maps is considered to eliminate possible singular behaviour of the solution. Several linear and nonlinear examples are tested by using our method. The experiments demonstrate clearly that our method provides an effective way to locate the ground state of the equations through underlying eigenvalue problems.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q40 PDEs in connection with quantum mechanics
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