Advanced finite-element discretizations and preconditioners for models of poroelasticity have attracted significant attention in recent years. The equations of poroelasticity offer significant challenges in both areas, due to the potentially strong coupling between unknowns in the system, saddle-point structure, and the need to account for wide ranges of parameter values, including limiting behavior such as incompressible elasticity. This paper was motivated by an attempt to develop monolithic multigrid preconditioners for the discretization developed in [48]; we show here why this is a difficult task and, as a result, we modify the discretization in [48] through the use of a reduced quadrature approximation, yielding a more "solver-friendly" discretization. Local Fourier analysis is used to optimize parameters in the resulting monolithic multigrid method, allowing a fair comparison between the performance and costs of methods based on Vanka and Braess-Sarazin relaxation. Numerical results are presented to validate the LFA predictions and demonstrate efficiency of the algorithms. Finally, a comparison to existing block-factorization preconditioners is also given.
翻译:近些年来,孔径度的方程式在这两个领域都提出了重大挑战,因为系统中的未知物、马鞍点结构以及需要考虑广泛的参数值,包括限制不压缩弹性等行为。本文的动机是试图为[48] 所开发的离散性模型开发单立性多格预设物;我们在这里说明为什么这是一个困难的任务,因此,我们通过使用减缩的四面形近近似来修改[48] 的离散性,从而产生一种更“离散性”的离散性。还进行了局部四面形分析,以优化由此产生的单面形多格方法中的参数,从而可以对基于Vanka和Braess-Sarazin的放松方法的性能和成本进行公平的比较。提出了数值结果,以验证LFA预测并展示算法的效率。最后,还进行了与现有块化前置物的比较。