Optimal Time-Adaptivity for Parabolic Problems
This work provides a foundational theoretical advancement for numerical analysts and computational scientists working with time-dependent PDEs, enabling the development of more efficient and reliable adaptive solvers.
This paper addresses the long-standing problem of proving optimality for adaptive time-stepping methods in non-stationary partial differential equations (PDEs). By leveraging a new equivalence between quasi-orthogonality and inf-sup stability, and applying it to Radau IIA methods, the authors achieve the first provably rate-optimal adaptive time-stepping method for parabolic problems.
Since the first optimality proofs for adaptive mesh refinement algorithms in the early 2000s, the theory of optimal mesh refinement for PDEs was inherently limited to stationary problems. The reason for this is that time-dependent problems usually do not exhibit the necessary coercive structure that is used in optimality proofs to show a certain quasi-orthogonality, which is crucial for the theory. Recently, by using a new equivalence between quasi-orthogonality and inf-sup stability of the underlying problem, it was shown that an adaptive Crank-Nicolson scheme for the heat equation is optimal under a severe step size restriction. In this work, we use this new approach towards quasi-orthogonality together with Radau IIA methods of any order larger than one to obtain the first adaptive time stepping method for non-stationary PDEs that is provably rate optimal with respect to number of time steps vs. approximation error.