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arXiv · 2506.22782

Long-time error estimate and decay of finite element method to a generalized viscoelastic flow

Yingwen Guo·Yinnian He·Wenlin Qiu·Xiangcheng Zheng
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Abstract

This work analyzes the finite element approximation to a viscoelastic flow model, which generalizes the Navier-Stokes equation and Oldroyd's model by introducing the tempered power-law memory kernel. We prove regularity and long-time exponential decay of the solutions, as well as a long-time convolution-type Gr\"onwall inequality to support numerical analysis. A Volterra-Stokes projection is developed and analyzed to facilitate the parabolic-type duality argument, leading to the long-time error estimates and exponential decay of velocity and pressure. A benchmark problem of planar four-to-one contraction flow is simulated to substantiate the generality of the proposed model in comparison with the Navier-Stokes equation and Oldroyd's model.

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Yingwen Guo, Yinnian He, Wenlin Qiu, Xiangcheng Zheng. 2025-06-28. Long-time error estimate and decay of finite element method to a generalized viscoelastic flow. https://arxiv.org/abs/2506.22782

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