A higher order nonconforming least-squares spectral element method for the stationary Stokes interface problem in three dimensions

We extend our earlier two-dimensional nonconforming least-squares spectral element method (NSEM) 22https://arxiv.org/abs/2502.20039. to the three-dimensional stationary Stokes interface problem with piecewise-constant viscosity. The computational mesh is fitted to the interface where the interface is represented by matching faces of hexahedral cells. We employ nonconforming trial and test spaces on the cells. The inter-element continuity conditions, interface conditions, and boundary conditions are directly included as the residuals in the least-squares functional in suitable Sobolev trace norms. The resulting normal equations are symmetric and positive definite and do not require an inf-sup condition because the formulation is based on the least squares formulation. We solve the system with a preconditioned conjugate gradient method equipped with a block-diagonal, separation of variables preconditioner whose condition number grows polylogarithmically with the polynomial degree W, i.e., O((lnW)2). Exponential convergence (velocity in H1, pressure in L2) is proven and verified through benchmarks under strong viscosity jumps, which shows exponential error decay and good mass conservation, highlighting the robustness and efficiency of high-order NSEM for 3D Stokes interface problems.