Isogeometric Divergence-conforming B-splines for the Darcy-Stokes-Brinkman Equations

Abstract

Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 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 di erential forms and may be interpreted as smooth generalizations of Raviart-Thomas elements. The new discretizations are (at least) patch-wise C0 and can be directly utilized in the Galerkin solution of Darcy-Stokes-Brinkman ow for single-patch con gurations. When applied to incompressible ows, these discretizations produce pointwise divergence-free velocity elds 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 ow in the limit of vanishing viscosity. The proposed discretizations are extended to general mapped geometries using divergence-preserving transformations. For su ciently regular single-patch solutions, we prove a priori error estimates which are optimal for the discrete velocity eld and suboptimal, by one order, for the discrete pressure eld. Our estimates are additionally 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 elds for general con gurations, suggesting that our a priori estimates may be conservative. The focus of the current paper is strictly on incompressible ows, but our theoretical results naturally extend to ows characterized by mass sources and sinks. AbstractWe 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) patch-wise C 0 and can be directl...

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