Abstract: Building precise simulations of the real world and using numerical methods to solve quantitative problems is an essential task in engineering and physics. We present FEABench, a benchmark to evaluate the ability of large language models (LLMs) and LLM agents to simulate and solve physics, mathematics and engineering problems using finite element analysis (FEA) software. We introduce a multipronged evaluation scheme to investigate the ability of LLMs to solve these problems using COMSOL Multiphysics$^\textregistered$. We further design an LLM agent equipped with the ability to interact with the software through its Application Programming Interface (API), examine its outputs and use tools to improve its solution over several iterations. Our best performing strategy generates executable API calls 88\% of the time. However, this benchmark still proves to be challenging enough that the LLMs and agents we tested were not able to completely and correctly solve any problem. LLMs that can successfully interact with and operate FEA software to solve problems such as those in our benchmark would push the frontiers of automation in engineering. Acquiring this capability would augment LLMs' reasoning skills with the precision of numerical solvers and advance the development of autonomous systems that can tackle complex problems in the real world.