Isogeometric multi-patch $C^1$-mortar coupling for the biharmonic equation

Abstract: We propose an isogeometric mortar method for solving fourth-order elliptic problems. Specifically, our approach focuses on discretizing the biharmonic equation on $C^0$-conforming multi-patch domains, employing the mortar technique to weakly enforce $C^1$-continuity across interfaces. To guarantee discrete inf-sup stability, we introduce a carefully designed Lagrange multiplier space. In this formulation, the multipliers are splines with a degree reduced by two relative to the primal space and feature enhanced smoothness (or merged elements for splines with maximum smoothness) near the vertices. Within this framework, we establish optimal a priori error estimates and validate the theoretical results through numerical tests.