Application of weak Galerkin finite element method for nonlinear chemotaxis and haptotaxis models
作者：
Highlights：
• We consider weak Galerkin finite element method for 2D Keller-Segel chemotaxis models including the blow-up problem in square domains, two-species chemotaxis blow-up problem, chemotactic bacteria pattern formation in a liquid medium, and closely related haptotaxis models to simulate tumor invasion into surrounding healthy tissue.
• The weak Galerkin finite element method (WGFEM) is applied to approximate the spatial variables in the chemotaxis system to capture the rapidly growing solutions in small neighborhoods of concentration points (blow-up).
• The proposed method produced numerical results that are nonnegative, oscillation-free, and suitable for solving models that have a blow-up of the cell density.
• Error bounds for cell density and chemoattractant equation are obtained.
• Numerical simulations for haptotaxis models are obtained for showing the process of tumor cell invasion.
• For examples 6.1 and 6.2 in the Keller-Segel model CPU times are computed. (Response to comment 5)
• We consider the linear instability criterion around a steady-state solution. (Response to comment 8)
• We construct weak Galerkin finite element space WG(P1, P0, [RT0,]2) for solving the Keller-Segel system. The errors measured in the L. norm converge with the rateO(h2). (Response to comment 1)
摘要
•We consider weak Galerkin finite element method for 2D Keller-Segel chemotaxis models including the blow-up problem in square domains, two-species chemotaxis blow-up problem, chemotactic bacteria pattern formation in a liquid medium, and closely related haptotaxis models to simulate tumor invasion into surrounding healthy tissue.•The weak Galerkin finite element method (WGFEM) is applied to approximate the spatial variables in the chemotaxis system to capture the rapidly growing solutions in small neighborhoods of concentration points (blow-up).•The proposed method produced numerical results that are nonnegative, oscillation-free, and suitable for solving models that have a blow-up of the cell density.•Error bounds for cell density and chemoattractant equation are obtained.•Numerical simulations for haptotaxis models are obtained for showing the process of tumor cell invasion.•For examples 6.1 and 6.2 in the Keller-Segel model CPU times are computed. (Response to comment 5)•We consider the linear instability criterion around a steady-state solution. (Response to comment 8)•We construct weak Galerkin finite element space WG(P1, P0, [RT0,]2) for solving the Keller-Segel system. The errors measured in the L. norm converge with the rateO(h2). (Response to comment 1)
论文关键词：Time-dependent nonlinear partial differential equations,Weak Galerkin finite element method,Weak gradient,Error estimate,Chemotaxis model,Haptotaxis model,Mathematical biology,Cancer invasion model,Blow-up,Nonlinear parabolic equations
论文评审过程：Received 15 June 2020, Revised 15 April 2021, Accepted 2 June 2021, Available online 18 June 2021, Version of Record 18 June 2021.
论文官网地址：https://doi.org/10.1016/j.amc.2021.126436
