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Numerical and computational aspects of some block-preconditioners for saddle point systems

机译:鞍点系统的某些块预处理器的数值和计算方面

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Linear systems with two-by-two block matrices are usually preconditioned by block lower-or upper-triangular systems that require an approximation of the related Schur complement. In this work, in the finite element framework, we consider one special such approximation, namely, the element-wise Schur complement. It is sparse and its construction is perfectly parallelizable, making it an appropriate ingredient when building preconditioners for iterative solvers executed on both distributed and shared memory computer architectures. For saddle point matrices with symmetric positive (semi-)definite blocks we show that the Schur complement is spectrally equivalent to the so-constructed approximation and derive spectral equivalence bounds. We also illustrate the quality of the approximation for nonsymmetric problems, where we observe the same good numerical efficiency.
机译:具有两乘二块矩阵的线性系统通常由块下三角系统或上三角系统进行预处理,这些系统需要近似相关舒尔补码。在这项工作中,在有限元框架中,我们考虑一种特殊的近似方法,即逐元素的Schur补码。它是稀疏的,其构造是完全可并行化的,因此,当为在分布式和共享内存计算机体系结构上执行的迭代求解器构建预处理器时,使其成为适当的组成部分。对于具有对称正(半)定块的鞍点矩阵,我们证明了Schur补谱在频谱上等价于如此构造的近似值,并推导了频谱等效边界。我们还说明了非对称问题的近似质量,其中我们观察到了相同的良好数值效率。

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