A Petrov–Galerkin finite element method for simulating chemotaxis models on stationary surfaces. (1st June 2020)

• Record Type: Journal Article
• Authors: Zhao, Shubo; Xiao, Xufeng; Zhao, Jianping; Feng, Xinlong
• Is Part Of: Computers & mathematics with applications. Volume 79:issue 11(2020)
• Page Start: 3189
• Page End: 3205
• Publication Date: 2020-06-01
• Subjects: Surface chemotaxis models -- Surface finite element method -- Petrov–Galerkin method -- Positivity preservation -- Mass conservation; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2020.01.019
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: In this paper, we present a Petrov–Galerkin finite element method for a class of chemotaxis models defined on surfaces, which describe the movement by one community in reaction to one chemical or biological signal on manifolds. It is desired for numerical methods to satisfy discrete maximum principle and discrete mass conservation property, which is a challenge due to the singular behavior of numerical solution. Thus a Petrov–Galerkin method is combined with an effective mass conservation factor to overcome the challenge. Furthermore, we prove two facts, this method maintains positivity and discrete mass conservation property. In addition, decoupled approach is applied based on the gradient and Laplacian recoveries to solve the coupling system. The relevant stability analyses is provided. Finally, numerical simulations of blowing-up problems and pattern formulations demonstrate the effectiveness of the proposed method.