The authors introduce a new global spectral Galerkin framework, the QTT Chebyshev–Galerkin method, that represents both field variables and operators as quantics tensor trains (QTT). This quantum‑inspired compression allows the method to assemble and solve large linear systems directly in compressed form, without ever forming full coefficient arrays.

Starting from the weak form of elliptic problems, the algorithm builds matrix‑product operators (MPOs) for differentiation, multiplication and integration. Mixed Dirichlet–Robin boundary conditions are enforced by compressed algebraic operations, preserving the symmetric positive‑definite structure that tensor‑network solvers such as AMEn exploit. Parameters and random inputs are treated as extra coordinates, so a single tensor‑network construction delivers the solution for an entire parametric space.

On a three‑dimensional linear‑elastic benchmark, the method reaches spectral convergence with L² errors below 10⁻⁸, outperforming both QTT finite‑difference discretizations and quadratic and quartic finite‑element solvers over the tested resolution range. A six‑parameter cantilever sweep that covers 1.7 × 10⁷ distinct designs is solved in a single 92‑second run on a CPU, instead of the 2.5 × 10⁷ s that would be required for a conventional sweep. In a stochastic elasticity problem with ten random variables, the QTT representation consumes 2.4 × 10⁶ times less memory than the full explicit coefficient array.