On a three step two-grid finite element method for the Oldroyd model of order one,ZAMM - Journal of Applied Mathematics and Mechanics

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On a three step two-grid finite element method for the Oldroyd model of order one
ZAMM - Journal of Applied Mathematics and Mechanics ( IF 2.3 ) Pub Date : 2021-07-16 , DOI: 10.1002/zamm.202000373
Bikram Bir ₁ , Deepjyoti Goswami ₁

Affiliation

In this work, an optimal error analysis of a three step two-grid method for the equations of motion arising in the 2D Oldroyd model of order one is discussed. The model, which can be thought of as an integral perturbation of Navier-Stokes equations (NSE), represents linear viscoelastic fluid flows. This non-linear model is analyzed here using a three step numerical scheme. In the first step the problem is solved on a coarse grid, and we use this course grid solution to linearize the problem and solve it in second step, on a finer grid. Third step is a correcting step done on the finer grid. Optimal error estimates for the velocity in

L^{\infty} (L^{2})

and

L^{\infty} (H^{1})

-norms and for the pressure in

L^{\infty} (L^{2})

-norm in the semidiscrete case are established. Estimates are shown to be uniform under the uniqueness assumption. Then, based on backward Euler method, a completely discrete scheme is analyzed and optimal a priori error estimates are derived. All the analysis are carried out for the non-smooth initial data. Finally we present some numerical results to validate our theoretical results. These examples show that the three step two-grid method is efficient than solving a nonlinear problem directly, as is expected.

中文翻译：

一阶Oldroyd模型的三步二网格有限元法

在这项工作中，讨论了对一阶二维Oldroyd 模型中出现的运动方程的三步两网格方法的最佳误差分析。该模型可以被认为是纳维-斯托克斯方程 (NSE) 的积分扰动，代表线性粘弹性流体流动。此处使用三步数值方案分析该非线性模型。在第一步中，问题是在粗网格上解决的，我们使用这个课程网格解决方案来线性化问题并在第二步中在更细的网格上解决它。第三步是在更精细的网格上完成的校正步骤。速度的最佳误差估计