High Order Methods with Exact Conservation Properties René Hiemstra and Marc Gerritsma Abstract Conservation laws, in for example, electromagnetism, solid and fluid mechanics, allow an exact discrete representation in terms of line, surface and volume integrals. In this paper, we develop high order interpolants, from any basis that constitutes a partition of unity, which satisfy these integral relations exactly, at cell level. The resulting gradient, curl and divergence conforming spaces have the property that the conservation laws become completely independent of the basis functions. Hence, they are exactly satisfied at the coarsest level of discretization and on arbitrarily curved meshes. As an illustration we apply our approach to B-splines and compute a 2D Stokes flow inside a lid driven cavity, which displays, amongst others, a point-wise divergence-free velocity field. 1 Introduction Conventional numerical methods, in particular finite difference and nodal finite element methods, expand their unknowns in terms of nodal interpolations only, and run into trouble when it comes to conservation. This can lead to instabilities, and perhaps more dangerously, to internal inconsistencies, such as the violation of fundamental conservation principles. Where instabilities lead to outright failure of R. Hiemstra () The institute for computational Engineering and Sciences, The University of Texas at Austin, 201 East 24th St, Stop C0200 Austin, TX 78712-1229, USA e-mail: rene@ices.utexas.edu M. Gerritsma Technical University Delft, Department of Aerospace engineering, Kluyverweg 2, 2629HT, Delft, The Netherlands e-mail: m.i.gerritsma@tudelft.nl M. Azaïez et al. (eds.), Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2012, Lecture Notes in Computational Science and Engineering 95, DOI 10.1007/978-3-319-01601-6__23, © Springer International Publishing Switzerland 2014 285