High-order evolving surface finite element method for parabolic problems on evolving surfaces
This work provides rigorous convergence guarantees for high-order numerical methods on evolving surfaces, addressing a theoretical gap for computational scientists and engineers simulating surface PDEs.
The authors prove convergence of high-order evolving surface finite element methods for parabolic PDEs on evolving surfaces, including full discretizations with backward difference formulae and implicit Runge-Kutta methods, establishing high-order geometric approximation and perturbation error estimates.
High-order spatial discretisations and full discretisations of parabolic partial differential equations on evolving surfaces are studied. We prove convergence of the high-order evolving surface finite element method, by showing high-order versions of geometric approximation errors and perturbation error estimates and by the careful error analysis of a modified Ritz map. Furthermore, convergence of full discretisations using backward difference formulae and implicit Runge-Kutta methods are also shown.