  This paper presents a quantum algorithm for the solution of prototypical
second-order linear elliptic partial differential equations discretized by
$d$-linear finite elements on Cartesian grids of a bounded $d$-dimensional
domain. An essential step in the construction is a BPX preconditioner, which
transforms the linear system into a sufficiently well-conditioned one, making
it amenable to quantum computation. We provide a constructive proof
demonstrating that our quantum algorithm can compute suitable functionals of
the solution to a given tolerance $\texttt{tol}$ with a complexity linear in
$\texttt{tol}^{-1}$ for a fixed dimension $d$, neglecting logarithmic terms.
This complexity is proportional to that of its one-dimensional counterpart and
improves previous quantum algorithms by a factor of order $\texttt{tol}^{-2}$.
We also detail the design and implementation of a quantum circuit capable of
executing our algorithm, and present simulator results that support the quantum
feasibility of the finite element method in the near future, paving the way for
quantum computing approaches to a wide range of PDE-related challenges.
