An arbitrary-order fully discrete Stokes complex on general polyhedral meshes

TL;DR

This paper introduces a novel arbitrary-order fully discrete Stokes complex on polyhedral meshes, enhancing the de Rham complex with a gradient operator, and demonstrates its advantages for fluid and MHD problems.

Contribution

It develops a new fully discrete Stokes complex with added gradient operator on polyhedral meshes, providing exactness and consistency results for complex-valued PDE schemes.

Findings

The complex exhibits exactness properties and uniform Poincaré inequalities.

Schemes based on this complex are nonconforming but benefit from its exactness.

Numerical tests confirm the convergence rates of the proposed scheme.

Abstract

In this paper we present an arbitrary-order fully discrete Stokes complex on general polyhedral meshes. We enriche the fully discrete de Rham complex with the addition of a full gradient operator defined on vector fields and fitting into the complex. We show a complete set of results on the novelties of this complex: exactness properties, uniform Poincar\'e inequalities and primal and adjoint consistency. The Stokes complex is especially well suited for problem involving Jacobian, divergence and curl, like the Stokes problem or magnetohydrodynamic systems. The framework developed here eases the design and analysis of scheme for such problems. Schemes built that way are nonconforming and benefit from the exactness of the complex. We illustrate with the design and study of a scheme to solve the Stokes equations and validate the convergence rates with various numerical tests.