Weak formulations are mathematical approaches used to solve partial differential equations (PDEs) by relaxing the requirements for solutions. Instead of requiring a function to satisfy the equation everywhere, weak formulations allow for solutions that meet the equation in an average sense, often using test functions. This is particularly useful in cases where traditional solutions may not exist or are difficult to find. In the context of finite element methods, weak formulations enable the approximation of solutions over complex domains. By converting strong formulations into weak ones, engineers and scientists can apply numerical techniques to analyze problems in fields like mechanics and fluid dynamics, making it easier to handle irregular geometries and boundary conditions.