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

• Published in: Applied mathematics and mechanics Vol. 39; no. 9; pp. 1353 - 1372
• Main Authors: Wang, Jianyun, Chen, Yanping
• Format: Journal Article
• Language: English
• Published: Shanghai Shanghai University 01.09.2018; Springer Nature B.V
• Edition: English ed.
• Subjects: Applications of Mathematics; Classical Mechanics; Finite element method; Fluid- and Aerodynamics; Interpolation; Mathematical analysis; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Numerical methods; Partial Differential Equations; Post-production processing; Schrodinger equation; Time dependence; interpolation post-processing; 65M15; elliptic projection; O241.82; 65M60; 65M12; superconvergence; Schrödinger equation
• ISSN: 0253-4827; 1573-2754
• DOI: 10.1007/s10483-018-2369-9

Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schrödinger equation with the finite element method. The error estimate and superconvergence property with order O ( h k +1 ) in the H 1 norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O ( h k +1 + τ 2 ) in the H 1 norm can be obtained in the Crank-Nicolson fully discrete scheme.