Superconvergence analysis of bi-k-degree rectangular elements for two-dimensional time-dependent Schrödinger equation

Wang, Jianyun; Chen, Yanping

doi:10.1007/s10483-018-2369-9

Cited by 5 publications

(2 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Schrödinger equation is widely used in many various areas, for example, quantum mechanics, optics, seismology, and plasma physics. In the past several decades, numerical methods for the Schrödinger equation have been studied extensively, such as [14,15] for finite difference methods, [16,17] for spectral methods, [18][19][20][21][22][23][24] for finite element methods, [25,26] for mixed finite element methods, and [27,28] for local discontinuous Galerkin methods. Huang et al [17] applied the time-splitting Fourier pseudospectral method on generalized sparse grids to solve the space-fractional Schrödinger equation.…”

Section: Introductionmentioning

confidence: 99%

“…Akrivis et al [18] approximated the solutions of a nonlinear Schrödinger equations by two Crank-Nicolson fully discrete finite element schemes. Wang and Chen [22] analyzed the superconvergencent result for a two-dimensional time-dependent Schrödinger equation with the finite element method. Shi et al [26] presented a new mixed finite element scheme in space and a linearized backward Euler scheme in time for nonlinear Schrödinger equations.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Two‐grid mixed finite element method for two‐dimensional time‐dependent Schrödinger equation

Tian

Chen

Huang

et al. 2023

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

In this paper, we study the semidiscrete mixed finite element scheme and construct a two‐grid algorithm for the two‐dimensional time‐dependent Schrödinger equation. We analyze error results of the mixed finite element solution in L2$$ {L}^2 $$‐norm by some projection operators. Then, we propose a two‐grid method of the semidiscrete mixed finite element. With this method, the solution of the Schrödinger equation on a fine grid is reduced to the solution of original problem on a much coarser grid together with the solution of two elliptic equations on the fine grid. We also obtain the error estimate of two‐grid solution with exact solution in L2$$ {L}^2 $$‐norm. Finally, a numerical experiment indicates that our two‐grid algorithm is more efficient than the standard mixed finite element method.

show abstract