一种新的无参数部分惩罚浸入有限元及其最优收敛性分析
A new parameter free partially penalized immersed finite element and the optimal convergence analysis
摘要：
本文提出了一种新的无参数部分惩罚浸入有限元法，并分析了其收敛性。在界面边缘引入提升算子以保证方法的收敛性，而不需要一个自组织的稳定参数。通过一种新的方法证明了浸入有限元空间的最佳逼近能力，该方法比文献中的方法简单得多。在界面元上建立了一个新的迹不等式，它是证明浸入有限元法最优收敛性所必需的。最佳的误差估计推导严格与常数无关的接口相对于网格的位置。新的方法和分析也被扩展到变系数和三维问题。数值算例验证了理论分析和方法的有效性。
英文摘要：
This paper presents a new parameter free partially penalized immersed finite element method and convergence analysis for solving second order elliptic interface problems. A lifting operator is introduced on interface edges to ensure the coercivity of the method without requiring an ad-hoc stabilization parameter. The optimal approximation capabilities of the immersed finite element space is proved via a novel new approach that is much simpler than that in the literature. A new trace inequality which is necessary to prove the optimal convergence of immersed finite element methods is established on interface elements. Optimal error estimates are derived rigorously with the constant independent of the interface location relative to the mesh. The new method and analysis have also been extended to variable coefficients and three-dimensional problems. Numerical examples are also provided to confirm the theoretical analysis and efficiency of the new method.
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