We present and analyse an implicit-explicit timestepping procedure with finite element spatial approximation for a semilinear reaction-diffusion systems on evolving domains arising from biological models, such as Schnakenberg's (1979). We employ a Lagrangian formulation of the model equations which permits the error analysis for parabolic equations on a fixed domain but introduces technical difficulties, foremost the space-time dependent conductivity and diffusion. We prove optimal-order error estimates in the $\Lp{\infty}(0,T;\Lp{2}(\W))$ and $\Lp{2}(0,T;\Hil{1}(\W))$ norms, and a pointwise stability result. We remark that these apply to Eulerian solutions. Details on the implementation of the Lagrangian and the Eulerian scheme are provided. We also report on a numerical experiment for an application to pattern formation on an evolving domain.
Optimal strong convergence rates of some Euler-type timestepping schemes for the finite element discretization SPDEs driven by additive fractional Brownian motion and Poisson random measure
Mean-square stability and error analysis of implicit time-stepping schemes for linear parabolic SPDEs with multiplicative Wiener noise in the first derivative
A stabilized trace finite element method for partial differential equations on evolving surfaces
University of Göttingen（哥廷根大学）；University of Houston（休斯顿大学）；Sechenov University（谢切诺夫大学）；Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, AMSS, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所，国家数学与交叉科学中心）
A Stream-Function Formulation for the Surface Stokes-Helfrich Model - Surface Finite Element Discretization and Validation
TUD Dresden University of Technology（德累斯顿工业大学）；Institute of Scientific Computing, TUD Dresden University of Technology（德累斯顿工业大学科学计算研究所）；Center for Systems Biology Dresden (CSBD)（德累斯顿系统生物学中心）；Cluster of Excellence Physics of Life (PoL), TUD Dresden University of Technology（德累斯顿大学“生命物理”卓越集群）
Randomized Neural Networks for Partial Differential Equation on Static and Evolving Surfaces
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, P.R. China（西安交通大学数学与统计学院）；School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China（西安交通大学数学与统计学院）
