Quantum computing holds the potential to revolutionize the computational framework for solving finite element problems, which are inherently resource-intensive on classical systems. By leveraging quantum speedups, these architectures offer a pathway to tackling large-scale, complex finite element analyses that remain challenging or impractical with traditional methods. This talk focuses on iterative solution algorithms tailored to two primary quantum computing architectures: quantum annealers and gate-based quantum computers. Quantum annealers specialize in optimization problems but are constrained by their applicability and the need for extensive qubits in complex cases. Gate-based quantum systems, on the other hand, exhibit versatility and promise exponential computational speedups for linear systems and differential equations, critical components of finite element methods. However, their scalability is hindered by current hardware limitations and the need for advanced error correction. We will discuss the development and implementation of iterative algorithms designed to overcome these challenges, highlighting their potential to enable efficient and precise solutions to finite element problems on quantum computing platforms.