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

Egor Kornev (), Sergey Dolgov, Michael Perelshtein and Artem Melnikov
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Egor Kornev: Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland
Sergey Dolgov: Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland
Michael Perelshtein: Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland
Artem Melnikov: Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland

Mathematics, 2024, vol. 12, issue 20, 1-19

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.

Keywords: tensor decompositions; tensor trains; finite element analysis; partial differential equations; incompressible Navier–Stokes equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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