Higher-order compatible discretization on hexahedrals

TL;DR

This paper introduces a geometric structure-based compatible discretization method for the Stokes problem, achieving divergence-free solutions and optimal convergence rates through a mixed variational formulation.

Contribution

It develops a novel compatible discretization approach that respects geometric and topological properties, ensuring divergence-free solutions for fluid flow problems.

Findings

Achieves pointwise divergence-free solutions for Stokes problem

Obtains optimal convergence rates in numerical tests

Validates theoretical properties through numerical experiments

Abstract

We derive a compatible discretization method that relies heavily on the underlying geometric structure, and obeys the topological sequences and commuting properties that are constructed. As a sample problem we consider the vorticity-velocity-pressure formulation of the Stokes problem. We motivate the choice for a mixed variational formulation based on both geometric as well as physical arguments. Numerical tests confirm the theoretical results that we obtain a pointwise divergence-free solution for the Stokes problem and that the method obtains optimal convergence rates.