A Contraction Property of an Adaptive Divergence-Conforming Discontinuous Galerkin Method for the Stokes Problem

TL;DR

This paper proves a contraction property for an adaptive divergence-conforming discontinuous Galerkin method applied to the Stokes problem, demonstrating error reduction and optimal complexity in degrees of freedom.

Contribution

It establishes the contraction property and quasi-orthogonality for the adaptive DG method, ensuring error reduction and optimal complexity for the Stokes problem.

Findings

Proves contraction property for the adaptive algorithm

Establishes quasi-orthogonality due to divergence-conformity

Shows quasi-optimal complexity in degrees of freedom

Abstract

We prove the contraction property for two successive loops of the adaptive algorithm for the Stokes problem reducing the error of the velocity. The problem is discretized by a divergence-conforming discontinuous Galerkin method which separates pressure and velocity approximation due to its cochain property. This allows us to establish the quasi-orthogonality property which is crucial for the proof of the contraction. We also establish the quasi-optimal complexity of the adaptive algorithm in terms of the degrees of freedom.