Adaptive Spectral Galerkin Methods with Dynamic Marking
This work provides a new theoretical framework for adaptive spectral methods, improving convergence rates for problems with Gevrey-regular solutions.
The paper addresses the limitation of Dörfler marking in adaptive spectral Galerkin methods, which causes a fixed contraction constant and poor performance. They propose a dynamic marking strategy that achieves exponential convergence with linear computational complexity for solutions in a Gevrey class.
The convergence and optimality theory of adaptive Galerkin methods is almost exclusively based on the Dörfler marking. This entails a fixed parameter and leads to a contraction constant bounded below away from zero. For spectral Galerkin methods this is a severe limitation which affects performance. We present a dynamic marking strategy that allows for a super-linear relation between consecutive discretization errors, and show exponential convergence with linear computational complexity whenever the solution belongs to a Gevrey approximation class.