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

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