Finite element method (FEM) solvers are foundational to scientific and engineering simulation, yet their susceptibility to \emph{efficiency attacks} remains largely unexplored. An adversary can craft syntactically valid and runnable FEM programs that complete successfully while forcing excessive computational work, enabling computational denial-of-service (DoS) attacks. Finding such inputs is challenging because FEM programs occupy a vast, structured space constrained by numerical, geometric, and constitutive requirements. We introduce CloakFEM, a domain-guided LLM-agent framework for automatically generating runnable adversarial FEM programs that amplify solver computation. CloakFEM combines LLM-guided search with an FEM-specific intermediate representation and a deterministic validity gate that enforces runnable constraints. Across three FEM stacks, 17 problem classes, and 1,068 seeds, CloakFEM discovers adversarial inputs that substantially amplify solver work while remaining valid and solvable. In the most extreme case, solver iterations increase from 43 to 247,367, a \(5{,}752.72\times\) amplification. CloakFEM outperforms five generic black-box baselines and retains an advantage over evolutionary search under a controlled bounded action space. Cross-solver analysis further reveals both problem-intrinsic and solver-specific sources of vulnerability. Our results uncover an underexplored efficiency-attack surface in FEM solvers and show that runnable FEM programs can serve as adversarial inputs.

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