论文标题
Kelvin-Voigt粘弹性流体运动方程的不连续的Galerkin有限元法的先验错误估计
A Priori Error Estimates of a Discontinuous Galerkin Finite Element Method for the Kelvin-Voigt Viscoelastic Fluid Motion Equations
论文作者
论文摘要
本文在强迫函数在$ l^\ indty中（{\ bf l}^2）$ - 空间时，将不连续的Galerkin有限元方法应用于开尔文 - voigt粘弹性流体运动方程。最佳的先验错误估计值$ l^\ infty（{\ bf l}^2）$ - 速度和$ l^\ infty（l^2）$中的标准 - 对于半discrete不连续的galerkin方法的压力近似值的标准。建立误差估计的主要成分是标准椭圆双重参数和在适当破损坏的Sobolev空间上定义的Sobolev-Stokes运算符的修改版本。此外，在数据对数据的较小假设下，已经证明这些估计值在及时均匀地有效。然后，采用一阶准确的向后欧拉方法完全离散了半discrete的不连续的盖尔金开尔金·沃伊格特公式。建立了速度和压力的完全离散的最佳误差估计。最后，使用数值实验，验证了理论结果。在这里值得强调的是，在本文中，使用有限元分析应用于开尔文 - voigt模型的不连续的盖尔金方法是该方向的第一次尝试。
This paper applies a discontinuous Galerkin finite element method to the Kelvin-Voigt viscoelastic fluid motion equations when the forcing function is in $L^\infty({\bf L}^2)$-space. Optimal a priori error estimates in $L^\infty({\bf L}^2)$-norm for the velocity and in $L^\infty(L^2)$-norm for the pressure approximations for the semi-discrete discontinuous Galerkin method are derived here. The main ingredients for establishing the error estimates are the standard elliptic duality argument and a modified version of the Sobolev-Stokes operator defined on appropriate broken Sobolev spaces. Further, under the smallness assumption on the data, it has been proved that these estimates are valid uniformly in time. Then, a first-order accurate backward Euler method is employed to discretize the semi-discrete discontinuous Galerkin Kelvin-Voigt formulation completely. The fully discrete optimal error estimates for the velocity and pressure are established. Finally, using the numerical experiments, theoretical results are verified. It is worth highlighting here that the error results in this article for the discontinuous Galerkin method applied to the Kelvin-Voigt model using finite element analysis are the first attempt in this direction.
