In this study, a least-squares (LS) finite element method with an adaptive mesh approach is investigated for Giesekus viscoelastic flow problems. We consider the weighted LS method on uniform and adaptive meshes for the Newton linearized viscoelastic problem, where adaptive grids are automatically generated by the least-squares solutions. We use a residual-type a-posteriori error estimator to adjust weights in the LS functional and compare the convergence behaviour of adaptive meshes generated using different grading functions. Numerical results demonstrate that the adaptive LS method shows at least the first-order convergence rate when equal-order linear interpolation functions are used for all variables, which agrees with the theoretical estimate. In addition, adaptive grids generated using the velocity outperform those based on the a-posteriori error estimator, yielding better numerical results.

在这项研究中，针对 Giesekus 粘弹性流动问题研究了具有自适应网格方法的最小二乘 (LS) 有限元方法。对于牛顿线性粘弹性问题，我们考虑均匀和自适应网格上的加权 LS 方法，其中自适应网格由最小二乘解自动生成。我们使用残差型后验误差估计器来调整 LS 函数中的权重，并比较使用不同分级函数生成的自适应网格的收敛行为。数值结果表明，当对所有变量使用等阶线性插值函数时，自适应LS方法至少显示出一阶收敛速度，这与理论估计是一致的。此外，