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Accuracy of a coupled mixed and Galerkin finite element approximation for poroelasticity
Portugaliae Mathematica ( IF 0.5 ) Pub Date : 2019-06-06 , DOI: 10.4171/pm/2018
Silvia Barbeiro 1
Affiliation  

In this paper, we consider a coupling mixed finite element and continuous Galerkin finite element formulation for a coupled flow and geomechanics model. We use the lowest order Raviart-Thomas space for the spatial approximation of the flow variables and continuous piecewise linear finite elements for the deformation variable while we consider the backward Euler method for the time discretization. This numerical scheme appears to be one common approach applied to existing reservoir engineering simulators. Theoretical convergence error estimates are derived in a discrete-in-time setting. Previous a priori error estimates described in the literature e.g. [2][19], which are optimal, show first order convergency with respect to the L-norm for the pressure and for the average fluid velocity approximation errors and with respect to the H-norm for the displacement approximation error. Here we prove one extra order of convergence for the displacement approximation with respect to the L-norm. We also demonstrate that, by including a post-processing step in the scheme, the order of convergence for the approximation of pressure can be improved. Even though this result is critical for deriving the Lnorm error estimates for the approximation of the deformation variable, surprisingly the corresponding gain of one convergence order holds independently of including or not the post-processing step in the method.

中文翻译:

多孔弹性耦合混合和 Galerkin 有限元近似的精度

在本文中,我们考虑了耦合流动和地质力学模型的耦合混合有限元和连续 Galerkin 有限元公式。我们使用最低阶的 Raviart-Thomas 空间来逼近流变量的空间近似值,使用连续分段线性有限元来逼近变形变量,同时我们考虑使用后向欧拉方法来进行时间离散化。这种数值方案似乎是应用于现有油藏工程模拟器的一种常用方法。理论收敛误差估计是在离散时间设置中导出的。先前文献中描述的先验误差估计,例如[2][19],这是最优的,显示关于压力和平均流体速度近似误差的 L 范数以及位移近似误差的 H 范数的一阶收敛。在这里,我们证明了相对于 L 范数的位移近似的一个额外收敛阶。我们还证明,通过在方案中包含一个后处理步骤,可以改进压力近似的收敛顺序。尽管这一结果对于推导变形变量近似的 Lnorm 误差估计至关重要,但令人惊讶的是,一个收敛阶的相应增益独立于该方法中是否包含后处理步骤。在这里,我们证明了相对于 L 范数的位移近似的一个额外收敛阶。我们还证明,通过在方案中包含一个后处理步骤,可以改进压力近似的收敛顺序。尽管这一结果对于推导变形变量近似的 Lnorm 误差估计至关重要,但令人惊讶的是,一个收敛阶的相应增益独立于该方法中是否包含后处理步骤。在这里,我们证明了相对于 L 范数的位移近似的一个额外收敛阶。我们还证明,通过在方案中包含一个后处理步骤,可以改进压力近似的收敛顺序。尽管这一结果对于推导变形变量近似的 Lnorm 误差估计至关重要,但令人惊讶的是,一个收敛阶的相应增益独立于该方法中是否包含后处理步骤。
更新日期:2019-06-06
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