Lagrangian finite elements in Sobolev-like spaces of order $3/2$

This paper introduces a Sobolev-like space of order $3/2$, denoted as $\widehat{H}^{3/2}$, for Lagrangian finite elements, especially for $C^0$ elements. It is motivated by the limitations of current stability analysis of the evolving surface finite element method (ESFEM), which relies exclusively on an energy estimate framework. To establish a PDE-based analysis framework for ESFEM, we encounter a fundamental regularity mismatch: the ESFEM adopts the $C^0$ elements, while the PDE regularity theory requires $H^{3/2}$ regularity for solutions. To overcome this difficulty, we first examine the properties of the continuous $H^{3/2}$ space, then introduce a Dirichlet lift and Scott-Zhang type interpolation operators to bridge to the discrete $\widehat{H}^{3/2}$ space. Our new $\widehat{H}^{3/2}$ space is shown to be compatible with the elliptic PDE regularity theory, the trace inequality, and the inverse inequality. Notably, we extend the critical domain deformation estimate in ESFEM to the $\widehat{H}^{3/2}$ setting. The $\widehat{H}^{3/2}$ theory provides a foundation for establishing a PDE-based convergence analysis framework of ESFEM.