A Closed-Form Master Element for the Fractal P-Wave Equation: Galerkin Formulation and Euclidean Limit

This paper studies a two-dimensional P-wave model in a fractal-continuum setting, where the classical Laplacian is replaced by the Hausdorff Laplacian. Using a Galerkin finite-element formulation on the reference triangle, we derive exact element-level expressions for the master element. The derivation makes explicit the role of the translated integration domain induced by the lower cutoffs ℓx and ℓy, which leads naturally to incomplete Beta/Gamma representations for the closed forms. In this way, the element integrations can be written analytically and the Euclidean limit can be checked in a consistent manner. A high-order quadrature verification confirms the shifted-domain closed-form expressions for the mass and stiffness matrices M1 and K1 while the corresponding Euclidean limit is recovered consistently. The resulting formulas provide reusable ingredients for implementations of related linear PDEs with the same operator structure, while avoiding element-level numerical quadrature.