We develop two unfitted cut finite element methods for the Stokes equations based on $\mathbf{H}^{\text{div}}$-conforming finite elements which exhibit optimal convergence order for the velocity, pointwise divergence-free velocity fields, and well-posed linear systems, independently of the position of the boundary relative to the computational mesh. The first method is based on the Brezzi-Douglas-Marini (BDM) elements and involves interior penalty terms to enforce tangential continuity of the velocity at interior edges in the mesh. The second method is a 3-field formulation involving the vorticity, velocity, and pressure and uses the Raviart-Thomas (RT) space for the velocity. We present mixed ghost penalty stabilization terms for both methods so that the resulting discrete problems are stable and the divergence-free property of the $\mathbf{H}^{\text{div}}$-conforming elements is preserved also on unfitted meshes. In both methods boundary conditions are imposed weakly. We show that imposing Dirichlet boundary conditions weakly introduces additional challenges; 1) The divergence-free property of the RT and the BDM finite elemens may be lost depending on how the normal component of the velocity field at the boundary is imposed. 2) Pressure robustness is affected by how well the boundary conditions are satisfied and may not hold even if the incompressibility condition holds pointwise. We study two approaches of weakly imposing the normal component of the velocity at the boundary; we either use a penalty parameter and Nitsche's method or a Lagrange multiplier method. We show that appropriate conditions on the velocity space has to be imposed when Nitsche's method is used. Pressure robustness can hold with both approaches by reducing the error at the boundary but this impacts the condition numbers of linear systems, independent of if the mesh is fitted or unfitted to the boundary.

翻译：我们发展两种基于$\mathbf{H}^{\text{div}}$相容有限元的非拟合切割有限元方法求解Stokes方程，该方法在边界相对于计算网格任意位置下，均能实现速度的最优收敛阶、逐点无散度速度场及适定线性系统。第一种方法采用Brezzi-Douglas-Marini（BDM）单元，并通过引入内罚项在网格内部边上强制速度的切向连续性。第二种方法为包含涡量、速度和压力的三场公式，速度空间采用Raviart-Thomas（RT）单元。我们对两种方法提出混合鬼罚稳定化项，使得离散问题稳定，并在非拟合网格上保持$\mathbf{H}^{\text{div}}$相容单元的无散度性质。两种方法均采用弱形式施加边界条件。研究表明，弱施加Dirichlet边界条件会引入额外挑战：1）RT与BDM有限元的无散度性质可能因边界法向速度分量的施加方式而丧失；2）压力鲁棒性受边界条件满足程度影响，即使不可压缩条件逐点成立也可能无法保持。我们研究两种边界法向速度分量的弱施加途径：采用罚参数与Nitsche方法，或采用拉格朗日乘子方法。结果表明，采用Nitsche方法时需对速度空间施加适当约束。两种途径均可通过减小边界误差实现压力鲁棒性，但这将影响线性系统的条件数，且该影响与网格是否拟合边界无关。