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TetraFEM: Numerical Solution of Partial Differential Equations Using Tensor Train Finite Element Method

Author

Listed:
  • Egor Kornev

    (Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland)

  • Sergey Dolgov

    (Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland
    Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK)

  • Michael Perelshtein

    (Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland)

  • Artem Melnikov

    (Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland)

Abstract

In this paper, we present a methodology for the numerical solving of partial differential equations in 2D geometries with piecewise smooth boundaries via finite element method (FEM) using a Quantized Tensor Train (QTT) format. During the calculations, all the operators and data are assembled and represented in a compressed tensor format. We introduce an efficient assembly procedure of FEM matrices in the QTT format for curvilinear domains. The features of our approach include efficiency in terms of memory consumption and potential expansion to quantum computers. We demonstrate the correctness and advantages of the method by solving a number of problems, including nonlinear incompressible Navier–Stokes flow, in differently shaped domains.

Suggested Citation

  • Egor Kornev & Sergey Dolgov & Michael Perelshtein & Artem Melnikov, 2024. "TetraFEM: Numerical Solution of Partial Differential Equations Using Tensor Train Finite Element Method," Mathematics, MDPI, vol. 12(20), pages 1-19, October.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:20:p:3277-:d:1501911
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