This research investigates the possibility of solving one dimensional Poisson's equation on quantum computers using the Variational Quantum Linear Solver (VQLS) as a simplified test case for fluid dynamics applications. In this work, Poisson's equation is discretized with the finite element method and the resulting matrix is decomposed as a linear combination of unitaries to be cast in VQLS. When using Pauli Gates as a basis, the matrix is inefficiently decomposed with an exponential number of terms in the number of qubits. On the other hand, using gates with higher entanglement allowed for an efficient decomposition but requires high qubit interconnectivity. Numerical experiments were carried out on a quantum simulator: iterations to success were larger than the best classical counterpart and scaled exponentially for increasing qubits numbers. Across all the experiments performed, scalability was an issue mainly because of the vanishing of the gradient in the cost function