The Poisson equation has many applications across the broad areas of science and engineering. Most quantum algorithms for the Poisson solver presented so far either suffer from lack of accuracy and/or are limited to very small sizes of the problem, and thus have no practical usage. Here we present an advanced quantum algorithm for solving the Poisson equation with high accuracy and dynamically tunable problem size. After converting the Poisson equation to a linear system through the finite difference method, we adopt the HHL algorithm as the basic framework. Particularly, in this work we present an advanced circuit that ensures the accuracy of the solution by implementing non-truncated eigenvalues through eigenvalue amplification, as well as by increasing the accuracy of the controlled rotation angular coefficients, which are the critical factors in the HHL algorithm. Consequently, we are able to drastically reduce the relative error in the solution while achieving higher success probability as the amplification level is increased. We show that our algorithm not only increases the accuracy of the solutions but also composes more practical and scalable circuits by dynamically controlling problem size in NISQ devices. We present both simulated and experimental results and discuss the sources of errors. Finally, we conclude that though overall results on the existing NISQ hardware are dominated by the error in the CNOT gates, this work opens a path to realizing a multidimensional Poisson solver on near-term quantum hardware.