Localized Lagrange multipliers with weak enforcement of constraints and Mortar method equivalence

Domain decomposition methods based on Lagrange multipliers are widely used to couple non-conforming finite element discretizations. Among them, the Mortar Method (MM) and the Localized Lagrange Multiplier (LLM) method are often regarded as fundamentally different approaches. While MM enforces interface constraints variationally, LLM is commonly associated with collocation constraints based on Dirac delta multipliers. This work revisits the original concept of localized Lagrange multipliers and argues that the defining feature of LLM is the localization of multipliers to individual subdomains through a frame construction rather than the use of Dirac delta functions. Based on this interpretation, a weak localized Lagrange multiplier formulation is derived. It is shown that the derived method is mathematically equivalent to the classical mortar method, but only when appropriate interpolation spaces are employed. The solvability and stability of the resulting discrete system are investigated, and the frame-based formulation is shown to naturally accommodate crosspoints involving multiple subdomains. Numerical examples confirm the theoretical findings and demonstrate that weak LLM achieves the same accuracy as MM while preserving the implementation advantages associated with localized multipliers.