Optimized Schwarz and finite element cell-centered method for heterogeneous anisotropic diffusion problems
Optimized Schwarz and finite element cell-centered method for heterogeneous anisotropic diffusion problems
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DOI：
10.1016/j.apnum.2020.01.009
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发表时间：
2020-05
影响因子：
2.8
通讯作者：
Thanh Hai Ong;Thi-Thao-Phuong Hoang
中科院分区：
文献类型：
--
作者：
Thanh Hai Ong;Thi-Thao-Phuong Hoang
本文讨论了用有限元单元中心（FECC）格式离散非均匀各向异性扩散问题的优化施瓦茨型方法的推导和分析。与标准有限元法（FEM）不同，FECC方法仅涉及单元未知数，并通过采用对偶网格和多点通量近似技术构造离散梯度算子，从而满足通量局部守恒。因此，如果将域分解为不重叠的子域，则与FECC方案相关的传输条件（子域之间的接口上）与标准FEM的传输条件不同。在FECC离散化的框架下，我们推导出离散的罗宾型传输条件，其中包括由于FECC的离散梯度算子的构造而产生的罗宾项的弱形式和强形式。严格证明了相关的迭代算法分解成带状子域的收敛性。二维数值结果的各向同性和各向异性的扩散张量与大跳跃的系数，以说明所提出的方法与优化罗宾参数的性能。
The paper is concerned with the derivation and analysis of the optimized Schwarz type method for heterogeneous, anisotropic diffusion problems discretized by the finite element cell-centered (FECC) scheme. Differently from the standard finite element method (FEM), the FECC method involves only cell unknowns and satisfies local conservation of fluxes by using a technique of dual mesh and multipoint flux approximations to construct the discrete gradient operator. Consequently, if the domain is decomposed into nonoverlapping subdomains, the transmission conditions (on the interfaces between subdomains) associated with the FECC scheme are different from those of the standard FEM. We derive discrete Robin-type transmission conditions in the framework of FECC discretization, which include both weak and strong forms of the Robin terms due to the construction of the FECC's discrete gradient operator. Convergence of the associated iterative algorithm for a decomposition into strip-shaped subdomains is rigorously proved. Two dimensional numerical results for both isotropic and anisotropic diffusion tensors with large jumps in the coefficients are presented to illustrate the performance of the proposed methods with optimized Robin parameters.