Stable full-field simulation of a multiscale elliptic equation by means of Quantized Tensor Trains
For computational scientists solving multiscale elliptic PDEs, this method enables unprecedented mesh sizes, though it is domain-specific to elliptic equations.
The paper presents a QTT-based solver for linear elliptic equations with heterogeneous coefficients that handles up to 10^37 degrees of freedom in 3D, recovering accurate solutions and gradients. It achieves full-field simulations with 20 orders of magnitude more DoFs than classical solvers.
In this article, we design an original solver based on Quantized Tensor Trains (QTT) for linear elliptic equations with heterogeneous coefficient field, that allows for extremely fine meshes. It can achieve full-field simulations in dimensions $d=2$ and $d=3$ with a number of Degrees of Freedom (DoFs) up to $20$ orders of magnitude beyond the classical solvers, recovering accurately the solution as well as its gradient in the $\LL^2$ norm. For treating such an enormous amount of data, the solver crucially relies on the exponential compression properties of QTTs. This significantly improves upon the existing literature. The main ingredient of the proposed solver consists in the introduction of a penalization term involving the Helmholtz--Leray projector in the equation governing the gradient unknown. For practical reasons related to the expression of the Helmholtz--Leray projector, the penalized equation is solved in Fourier space. The primal solution is then obtained from the gradient via the Green operator. A core property of the solver is that it is unconditionally stable with respect to the mesh size. Based on numerical evidence supported by mathematical analysis, we show that reliable gradients and solutions can be obtained, and guaranteed by the proposed a posteriori error estimator. As an illustration, we successfully solve an elliptic equation in a microstructured material with up to $10^{37}$ virtual degrees of freedom in dimension $d=3$.