In this thesis, an adaptive numerical method is proposed to solve 2D Schrödinger equation with imaginary time propagation approach. The time-independent Schrödinger equation is transfered to the time-dependent one by Wick rotation. Then the temporal equation is discretized spatially with finite element methods, and temporally with Crank-Nicolson scheme. To handle the possible singular hebavior of the solution, the moving mesh method based on the harmonic maps is introduced. Several linear and nonlinear problems are examined by our method, which successfully show that our method is an effective alternative way to find the ground state of the system by solving the eigenvalue problems. v To My Parents