Higher-order time discretizations with ALE finite elements for parabolic problems on evolving surfaces
For researchers solving parabolic problems on moving domains, this work provides a stable and accurate numerical framework that overcomes mesh regularity issues of prior methods.
This paper develops and analyzes higher-order time discretizations for parabolic PDEs on evolving surfaces using ALE finite elements, proving unconditional stability and optimal convergence for Runge-Kutta and BDF methods up to order 5. Numerical experiments confirm the theoretical findings.
A linear evolving surface partial differential equation is first discretized in space by an arbitrary Lagrangian Eulerian (ALE) evolving surface finite element method, and then in time either by a Runge-Kutta method, or by a backward difference formula. The ALE technique allows to maintain the mesh regularity during the time integration, which is not possible in the original evolving surface finite element method. Unconditional stability and optimal order convergence of the full discretizations is shown, for algebraically stable and stiffly accurate Runge-Kutta methods, and for backward differentiation formulae of order less than 6. Numerical experiments are included, supporting the theoretical results.