Divergence-free vector fields play an important role in many types of problems, including the incompressible Navier-Stokes equations and the equations for magnetohydrodynamics. In the discrete setting, these fields are often obtained by projection, resulting in a discrete approximation of the continuous field that is discretely divergence-free. For many applications, such as tracing particles, this discrete field must then be extended to the entire region using interpolation. This interpolated field is continuous and differentiable (almost everywhere), but in general it will not be divergence-free. In this paper, we construct approximation schemes with the property that discretely divergence-free data interpolates to an analytically divergence-free vector field. Our focus is on data stored in a MAC grid layout that is divergence free under the second order central difference stencil, a case that is common in projection methods for the Navier-Stokes equations. While existing schemes with this property are known, they tend to be global (the interpolated value at a point depends on data stored on the grid far from that point) or discontinuous. We construct $C^0$ and $C^1$ continuous approximation schemes for 2D and 3D that are local and satisfy the divergence-free property. We also construct interpolating versions of the schemes that reproduce the MAC data at face centers. All eight schemes are explicit piecewise polynomials over small stencils.

无散矢量场在许多类型的问题中都发挥着重要作用，包括不可压缩的 Navier-Stokes 方程和磁流体动力学方程。在离散设置中，这些场通常是通过投影获得的，从而导致连续场的离散近似，即离散无散度。对于许多应用，例如跟踪粒子，必须使用插值将这个离散场扩展到整个区域。这个插值场是连续且可微的（几乎在所有地方），但通常它不会是无散度的。在本文中，我们构建了具有离散无散度数据内插到解析无散度矢量场的近似方案。我们的重点是存储在 MAC 网格布局中的数据，该布局在二阶中心差分模板下无散度，这种情况在 Navier-Stokes 方程的投影方法中很常见。虽然具有此属性的现有方案是已知的，但它们往往是全局的（某个点的插值取决于存储在远离该点的网格上的数据）或不连续的。我们构建$C^0$和$C^1$2D 和 3D 的连续逼近方案是局部的并且满足无散特性。我们还构建了在人脸中心再现 MAC 数据的方案的插值版本。所有八种方案都是小模板上的显式分段多项式。