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Preconditioners for parallel adaptive hp FEM for incompressible flows

机译：用于不可压缩流动的并联自适应HP FEM的预处理器

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摘要

Adaptive hp finite element (FEM) approximations of incompressible flow make special demands on parallel solution algorithms. Load balancing and minimizing interprocessor communication play a large role in solution efficiency. Because of this, iterative methods with an efficient sparse storage scheme for storing the stiffness matrix are normally used in solving these problems in a parallel environment. For hp FEM discretizations, the condition number of the resulting system of linear equations can become quite high. Thus, preconditioning is very important in ensuring that the iterative method will converge in a small number of iterations. While FEM discretizations of the Poisson equation or linear elasticity can be preconditioned by standard preconditioners, such as a Block Jacobi or Incomplete LU factorization, FEM discretizations of the incompressible Navier-Stokes equations will have zeroes on the diagonal of the stiffness matrix and thus the standard preconditioners will not work. Through a Schur complement of the interior velocity degrees of freedom, the zero diagonals corresponding to the continuity equation become non-zero and then the standard preconditioners can again be used. Results will be shown for 2-dimensional problems for Jacobi, Block Jacobi and Additive Schwartz preconditioners.
机译：不可压缩流量的自适应HP有限元（FEM）近似值对并行解决方案算法进行了特殊要求。负载平衡和最小化进程间通信在解决方案效率中起着很大的作用。因此，具有用于存储刚度矩阵的有效稀疏存储方案的迭代方法通常用于解决并行环境中的这些问题。对于HP FEM离散化，线性方程的所得系统的条件数会变得非常高。因此，在确保迭代方法将收敛在少量迭代中，预处理非常重要。虽然Poisson方程或线性弹性的有效离散化可以通过标准的预处理器预处理，例如块jacobi或不完全的Lu分解，但不可压缩的Navier-Stokes方程的有限离子离散化将在刚度矩阵的对角线上具有零，因此是标准预处理者将无法工作。通过SCUR补充的内部速度自由度，对应于连续性方程的零对角线变为非零，然后可以再次使用标准的预处理器。将显示Jacobi，Block Jacobi和Cantive Schwartz Precketers的二维问题。

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