# Adaptive Maximum Continuity Spline Approximations for the Navier-Stokes-Cahn-Hilliard Equations

Prof. Harald van Brummelen

TU Eindhoven

**Abstract**

IGA offers many advantages relative to conventional Finite-Element techniques, including the possibility to construct approximation spaces with customized continuity/regularity. For example, C1-continuous splines are H2 conforming, which implies that such C1 splines are suitable for shell theories. Another example is provided by divergence-conforming vector-scalar pairs, where the order and continuity of the vector components is selected such that the image of the divergence operator coincides with the scalar-variable space. High-continuity spline spaces carry the added, significant benefit that the dimension of the space (number of degrees of freedom) is significantly lower than that of their conventional C0 counterpart. This implies that high-order approximations become available at a fraction of the computational cost. In fact, maximum continuity splines, viz. splines of which the continuity is one order lower than their polynomial degree, carry as few degrees of freedom as the lowest order conventional FEM space.

High order approximations are efficient for problems of which the solution is suitably smooth. In practice, the theoretical high-order asymptotic convergence behavior is only observed if the mesh width is smaller than the length scale of characteristic features of the solution. This implies that high-order approximations are in principle ideally suited for diffuse-interface models such as the Navier-Stokes-Cahn-Hilliard equations, because solutions to such models are typically smooth, but that the effectiveness of high-order methods only manifests if the mesh width is (locally) smaller than the diffuse-interface thickness. Because the diffuse interface is confined to the vicinity of a manifold, the high resolution requirements imposed by the diffuse interface are only local, and efficient high-order approximations can be obtained by means of local adaptive refinement.

n this presentation, we focus on an adaptive maximum continuity spline approximation method for diffuse-interface models of binary fluid flows, as described by the Navier-Stokes-Cahn-Hilliard equations. For the Navier-Stokes subsystem we consider, in particular, maximum-continuity equal-order velocity/pressure pairs in conjunction with skeleton stabilization. We highlight specific parts of the adaptive algorithm pertaining to high-continuity splines, and show some applications in elasto-capillary fluid-solid interaction.