Recently, the Shifted Boundary Method (SBM) was proposed within the class of unfitted (or immersed, or embedded) finite element methods. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that {\it shifts} the location and values of the boundary conditions. In this article, we present enhanced variational SBM formulations for the Poisson and Stokes problems with improved flexibility and robustness. These simplified variational forms allow to relax some of the assumptions required by the mathematical proofs of stability and convergence of earlier implementations. First, we show that these new SBM implementations can be proved asymptotically stable and convergent even without the rather restrictive assumption that the inner product between the normals to the true and surrogate boundaries is positive. Second, we show that it is not necessary to introduce a stabilization term involving the tangential derivatives of the solution at Dirichlet boundaries, therefore avoiding the calibration of an additional stabilization parameter. Finally, we prove enhanced $L^{2}$-estimates without the cumbersome assumption - of earlier proofs - that the surrogate domain is convex. Instead we rely on a conventional assumption that the boundary of the true domain is smooth, which can also be replaced by requiring convexity of the true domain. The aforementioned improvements open the way to a more general and efficient implementation of the Shifted Boundary Method, particularly in complex three-dimensional geometries. We present numerical experiments in two and three dimensions.

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第二代移动边界法及其数值分析

最近，在未拟合（或浸入或嵌入）有限元方法类中提出了移动边界法 (SBM)。通过在代理（近似）计算域上重新构建原始边界值问题，SBM 避免了对切割单元的积分以及与数值稳定性和矩阵条件相关的相关问题。通过使用泰勒展开式修改原始边界条件来保持精度。因此，该方法的名称是 {\it shifts} 边界条件的位置和值。在本文中，我们针对泊松和斯托克斯问题提出了增强的变分 SBM 公式，具有更高的灵活性和鲁棒性。这些简化的变分形式允许放宽早期实现的稳定性和收敛性数学证明所需的一些假设。首先，我们表明，即使没有相当严格的假设，即真实边界和代理边界的法线之间的内积是正的，也可以证明这些新的 SBM 实现是渐近稳定和收敛的。其次，我们表明没有必要引入涉及狄利克雷边界处解的切向导数的稳定项，因此避免了额外稳定参数的校准。最后，我们证明了增强的 $L^{2}$-estimates 没有繁琐的假设 - 早期的证明 - 代理域是凸的。相反，我们依赖于真实域的边界是平滑的传统假设，这也可以通过要求真实域的凸性来代替。上述改进为移动边界法的更通用和更有效的实现开辟了道路，特别是在复杂的三维几何图形中。我们提供二维和三维的数值实验。