Adaptive Spectral Galerkin Methods with Dynamic Marking

TL;DR

This paper introduces a dynamic marking strategy for adaptive spectral Galerkin methods, enabling exponential convergence and improved performance over traditional fixed-parameter approaches, especially for solutions in Gevrey classes.

Contribution

The paper proposes a novel dynamic marking strategy that achieves super-linear error reduction and exponential convergence in spectral Galerkin methods, surpassing traditional fixed marking techniques.

Findings

Achieves exponential convergence with linear complexity.

Supports solutions in Gevrey approximation classes.

Overcomes limitations of fixed parameter Dörfler marking.

Abstract

The convergence and optimality theory of adaptive Galerkin methods is almost exclusively based on the D\"orfler 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.