AI 中文总结
本文针对三维隐式定义光滑闭曲面上的椭圆方程，提出一种结合曲面有限元与非拟合/迹方法特征的高阶曲面有限元法，通过背景四面体与曲面相交构造离散空间并经水平集参数化提升，推导了误差界并证明了稳定性。
AI 中文摘要
本文针对三维空间中作为某函数零水平集隐式定义的光滑闭曲面上的椭圆方程，提出了一种高阶曲面有限元方法。与从曲面的给定三角剖分出发的经典曲面有限元方法不同，该方法通过精确曲面与环境四面体网格的相交来构造离散曲面空间。更准确地说，会在隐式曲面周围生成一个活动四面体壳，每个被切割的四面体根据其与零水平集的相交模式，贡献一个三角形或四边形曲面片。有限元空间首先在得到的分段平面切割曲面上定义，随后通过基于水平集的局部参数化提升到精确曲面。该方法结合了曲面有限元法与非拟合/迹方法的特征：与曲面有限元法类似，其在提升后于精确曲面上生成符合要求的有限元空间；但与非拟合/迹方法类似，其几何结构和有限元空间由与背景四面体的相交诱导。本文证明，局部提升映射在公共面处逐点一致，并可组装为全局同胚；还推导了切向量、Gram矩阵、质量积分与刚度积分的显式公式，并证明了提升映射的稳定性和高阶导数估计，后者以破碎 Sobolev 范数表述，为所提出的高阶提升曲面有限元方法得出了 Cea 型能量范数误差界。
英文摘要
This paper develops a high-order surface finite element method for elliptic equations posed on a smooth closed surface implicitly defined as the zero level set of a function in three dimensions. In contrast to classical surface finite element methods that start from a prescribed triangulation of the surface, the proposed method constructs the discrete surface space from the intersections between the exact surface and an ambient tetrahedral mesh. More precisely, an active tetrahedral shell is generated around the implicit surface, and each cut tetrahedron contributes either a triangular or a quadrilateral surface patch according to its intersection pattern with the zero level set. The finite element space is first defined on the resulting piecewise planar cut surface and is then lifted to the exact surface by local level-set-based parameterizations. The resulting method combines features of surface FEM and unfitted/trace methods. Like surface FEM, it produces a conforming finite element space on the exact surface after lifting; however, the geometry and finite element space are induced by intersections with background tetrahedra like unfitted/trace methods. We prove that the local lifting maps agree pointwise across common faces and assemble into a global homeomorphism. We also derive explicit formulas for tangent vectors, Gram matrices, mass and stiffness integrals, and prove stability and high-order derivative estimates for the lifting maps. The latter estimates are formulated in broken Sobolev norms and lead to a Cea-type energy-norm error bound for the proposed high-order lifted surface finite element method.
