A low-rank QTT-based finite element method for elasticity problems

We present an efficient and robust numerical algorithm for solving two-dimensional linear elasticity problems that combines the Quantized Tensor Train format and a domain partitioning strategy. This method substantially decreases memory usage and achieves a notable rank reduction compared to established Finite Element implementations like the FEniCS platform. This performance gain, however, requires a fundamental rethinking of how core finite element operations are implemented, which includes changes to mesh discretization, ordering of nodes and degrees of freedom, stiffness matrix, and internal nodal force assembly, and the execution of algebraic matrix-vector operations. In this work, we discuss all these aspects in detail and assess the method's performance in the numerical approximation of three representative test cases.