Robust and efficient preconditioners for the discontinuous Galerkin time-stepping method
Robust and efficient preconditioners for the discontinuous Galerkin time-stepping method
复制标题
用于不连续伽辽金时间步进方法的稳健且高效的预处理器
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Iain Smears
中科院分区：
文献类型：
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作者：
Iain Smears
间断Galerkin时间步进方法在求解抛物型方程时具有许多优越的性质。然而，它需要在每个时间步的大型非对称系统的解决方案。这项工作开发了一个完全强大的和有效的预处理策略来解决这些系统。利用抛物型inf-sup理论，我们首先构造一个左预条件子，将线性系统转化为一个对称正定问题，用预条件共轭梯度算法求解。然后，我们证明了变换后的系统可以进一步预条件的理想块对角预条件子，导致一个条件数有界的4为任何时间步长，任何逼近阶和任何正定自伴空间运营商。数值实验表明，该算法对理想预条件子和近似预条件子都具有较低的条件数和较快的收敛速度，并证明了该算法求解大问题高阶解的可行性.
The discontinuous Galerkin time-stepping method has many advantageous properties for solving parabolic equations. However, it requires the solution of a large nonsymmetric system at each time-step. This work develops a fully robust and efficient preconditioning strategy for solving these systems. Drawing on parabolic inf-sup theory, we first construct a left preconditioner that transforms the linear system to a symmetric positive definite problem to be solved by the preconditioned conjugate gradient algorithm. We then prove that the transformed system can be further preconditioned by an ideal block diagonal preconditioner, leading to a condition number bounded by 4 for any time-step size, any approximation order and any positive-definite self-adjoint spatial operators. Numerical experiments demonstrate the low condition numbers and fast convergence of the algorithm for both ideal and approximate preconditioners, and show the feasibility of the high-order solution of large problems.