ETD

• beam models
• finite element method
• geometric dimensionality reduction
• model order reduction
• partial differential equations
• slender domains

In physics and engineering, differential models for slender domains often rely on ad hoc simplifications, which fail to provide either a priori control over the introduced error or the flexibility to systematically enrich the simplified model.
To address these issues, we develop a projection-based framework for the geometric dimensionality reduction of elliptic partial differential equations on slender domains. The construction relies on a geometric projection from the full domain onto a representative lower-dimensional set, defining weighted average and extension operators along the corresponding fibers.
Under a reference-product assumption, these operators are bounded on Sobolev spaces and generate finite-dimensional modal subspaces in the fiber direction while retaining infinite-dimensional approximation spaces on the representative domain. We prove that the modal extension is bounded below, thus providing a well-posed reduced problem, and we derive a Céa-type quasi-optimality estimate that separates the analytic discretization error from the geometric reduction error.
Multiple examples are developed for the Poisson equation, leading to explicit mass and stiffness matrices. For general slender domains, the framework recovers the classical lower-dimensional Laplace or Laplace-Beltrami problem in the constant mode and produces coupled second-order systems when higher cross-sectional modes are retained. Moreover, under an observability condition, these systems can be rewritten as higher-order scalar equations related to beam models on elastic foundations.
The same modal restriction-extension mechanism also applies component-wise to vector-valued problems such as linear elasticity.
To validate the theoretical framework, numerical simulations are performed using the deal.II finite element library, assessing both the accuracy and the computational speed-up of the proposed approach. Ultimately, this work provides a stable, systematically enrichable bridge between asymptotic dimension reduction and projection-based model reduction.