Higher-order time discretizations with ALE finite elements for parabolic problems on evolving surfaces

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.