Quantum-assisted h{\lambda}-adaptive finite element method

TL;DR

This paper introduces a quantum-assisted adaptive finite element method for singularly perturbed problems, combining quantum stabilization with classical error control to improve approximation accuracy.

Contribution

It presents a novel hybrid quantum-classical finite element scheme with proven error estimates for singular perturbation problems.

Findings

The method effectively overcomes singular perturbations.

Numerical comparisons show improved accuracy over traditional adaptive schemes.

Error control and refinement are automated through the proposed approach.

Abstract

We propose a novel finite element method scheme for singularly perturbed advection-diffusion-reaction problems, which combines certain quantum-assisted stabilization scheme with a classical h-adaptive approach to provide automatic error control and corresponding approximation refinement. Appropriate finite element a posteriori error estimates are proved. Described approach demonstrates the possibility to overcome singular perturbations by applying error-controlled smoothening to finite element approximation. Possible benefits of the proposed finite element scheme are discussed and the numerical comparison with the typical adaptive scheme is provided.