ALBERT

Description: 〈p〉Publication date: 15 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 379〈/p〉 〈p〉Author(s): Baoli Yin, Yang Liu, Hong Li, Siriguleng He〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this article, a fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme, which aims at solving nonlinear problems quickly, is considered to numerically solve the nonlinear space fractional Allen–Cahn equations with smooth and non-smooth solutions. The implicit second-order 〈em〉θ〈/em〉 scheme containing both implicit Crank–Nicolson scheme and second-order backward difference method is applied to time direction, a fast TT-M method is used to increase the speed of calculation, and the FE method is developed to approximate the spacial direction. The TT-M FE algorithm includes the following main computing steps: firstly, a nonlinear implicit second-order 〈em〉θ〈/em〉 FE scheme on the time coarse mesh 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉τ〈/mi〉〈/mrow〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈/msub〉〈/math〉 is solved by a nonlinear iterative method; secondly, based on the chosen initial iterative value, a linearized FE system on time fine mesh 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉τ〈/mi〉〈mo〉〈〈/mo〉〈msub〉〈mrow〉〈mi〉τ〈/mi〉〈/mrow〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈/msub〉〈/math〉 is solved, where some useful coarse numerical solutions are found by the Lagrange's interpolation formula. The analysis for both stability and a priori error estimates are made in detail. Finally, three numerical examples with smooth and non-smooth solutions are provided to illustrate the computational efficiency in solving nonlinear partial differential equations, from which it is easy to find that the computing time can be saved.〈/p〉〈/div〉