ISOGEOMETRIC DIVERGENCE-CONFORMING B-SPLINES FOR THE DARCY–STOKES–BRINKMAN EQUATIONS

Abstract

We develop divergence-conforming B-spline discretizations for the numerical solution of the Darcy–Stokes–Brinkman equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart–Thomas elements. The new discretizations are (at least) patchwise C0 and can be directly utilized in the Galerkin solution of Darcy–Stokes–Brinkman flow for single-patch configurations. When applied to incompressible flows, these discretizations produce pointwise divergence-free velocity fields and hence exactly satisfy mass conservation. In the presence of no-slip boundary conditions and multi-patch geometries, the discontinuous Galerkin framework is invoked to enforce tangential continuity without upsetting the conservation or stability properties of the method across patch boundaries. Furthermore, as no-slip boundary conditions are enforced weakly, the method automatically defaults to a compatible discretization of Darcy flow in the limit of vanishing viscosity. The proposed discretizations are extended to general mapped geometries using divergence-preserving transformations. For sufficiently regular single-patch solutions, we prove a priori error estimates which are optimal for the discrete velocity field and suboptimal, by one order, for the discrete pressure field. Our estimates are in addition robust with respect to the parameters of the Darcy–Stokes–Brinkman problem. We present a comprehensive suite of numerical experiments which indicate optimal convergence rates for both the discrete velocity and pressure fields for general configurations, suggesting that our a priori estimates may be conservative. The focus of this paper is strictly on incompressible flows, but our theoretical results naturally extend to flows characterized by mass sources and sinks.