A unified theory for continuous-in-time evolving finite element space approximations to partial differential equations in evolving domains. (26th November 2020)

• Record Type: Journal Article
• Authors: Elliott, C M; Ranner, T
• Is Part Of: IMA journal of numerical analysis. Volume 41:Number 3(2021)
• Page Start: 1696
• Page End: 1845
• Publication Date: 2020-11-26
• Subjects: evolving finite element spaces -- abstract error analysis -- parabolic equations on moving domains -- advection–diffusion on evolving surfaces -- bulk-surface parabolic equations -- evolving surface finite element methods -- evolving bulk finite element methods; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/draa062
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: We develop a unified theory for continuous-in-time finite element discretizations of partial differential equations posed in evolving domains, including the consideration of equations posed on evolving surfaces and bulk domains, as well as coupled surface bulk systems. We use an abstract variational setting with time-dependent function spaces and abstract time-dependent finite element spaces. Optimal a priori bounds are shown under usual assumptions on perturbations of bilinear forms and approximation properties of the abstract finite element spaces. The abstract theory is applied to evolving finite elements in both flat and curved spaces. Evolving bulk and surface isoparametric finite element spaces defined on evolving triangulations are defined and developed. These spaces are used to define approximations to parabolic equations in general domains for which the abstract theory is shown to apply. Numerical experiments are described, which confirm the rates of convergence.