A priori and a posteriori error analyses of a high order unfitted mixed-FEM for Stokes flow
A priori and a posteriori error analyses of a high order unfitted mixed-FEM for Stokes flow
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斯托克斯流高阶未拟合混合有限元的先验和后验误差分析
DOI：
10.1016/j.cma.2019.112780
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发表时间：
2020
影响因子：
7.2
通讯作者：
Paulo Zúñiga
中科院分区：
文献类型：
--
作者：
Ricardo Oyarzúa;Manuel E. Solano;Paulo Zúñiga
本文提出并分析了一种高精度不适应混合有限元方法，用于求解具有Dirichlet边界条件的流体域Ω上的拟应力-速度方程，其边界为弯曲的Γ。该方法包括用多面体子域Dh来逼近Ω，边界Γh，其中用Galerkin方法逼近解，并基于积分速度的离散外推梯度的变换技术来逼近计算边界Γh上的Dirichlet边界数据。相应的Galerkin格式是由k次≥0阶RaviartThomas元定义的伪应力和k次间断速度多项式。在边界Γ附近的网格上作了适当的假设，利用Babu-ška-Brezzi理论证明了Galerkin格式的适定性，并建立了相应的O(hk+1)的最优收敛。此外，对于Γh是通过Γ的分段线性内插构造的情形，我们提出了一种可靠且准有效的基于残差的后验误差估计。它的定义是在解足够光滑的情况下，利用具有更高精度的后处理速度来获得与该方法相同的收敛速度。数值实验验证了该方案的性能，展示了相关自适应算法的行为，验证了理论。
We propose and analyze a high order unfitted mixed finite element method for the pseudostress-velocity formulation of the Stokes problem with Dirichlet boundary condition on a fluid domain Ω with curved boundary Γ. The method consists of approximating Ω by a polyhedral subdomain D h, with boundary Γ h, where a Galerkin method is applied to approximate the solution, and on a transferring technique, based on integrating the extrapolated discrete gradient of the velocity, to approximate the Dirichlet boundary data on the computational boundary Γ h. The associated Galerkin scheme is defined by Raviart–Thomas elements of order k≥ 0 for the pseudostress and discontinuous polynomials of degree k for the velocity. Provided suitable hypotheses on the mesh near the boundary Γ, we prove well-posedness of the Galerkin scheme by means of the Babuška–Brezzi theory and establish the corresponding optimal convergence of O (h k+ 1). Moreover, for the case when Γ h is constructed through a piecewise linear interpolation of Γ, we propose a reliable and quasi-efficient residual-based a posteriori error estimator. Its definition makes use of a postprocessed velocity with enhanced accuracy to achieve the same rate of convergence of the method when the solution is smooth enough. Numerical experiments illustrate the performance of the scheme, show the behavior of the associated adaptive algorithm and validate the theory.
具有浸没边界的可压缩流的高阶间断伽辽金法
影响因子：
2.9
作者：
Müller;Krämer-Eis;Kummer;Oberlack
通讯作者：
Oberlack