Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems
Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems
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DOI：
10.1002/nla.1816
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发表时间：
2012-03
影响因子：
4.3
通讯作者：
Peter Bastian;Markus Blatt;Robert Scheichl
中科院分区：
文献类型：
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作者：
Peter Bastian;Markus Blatt;Robert Scheichl
本文提出了一种新的代数多重网格（AMG）算法，用于求解非均匀椭圆型问题的间断Galerkin（DG）离散所产生的线性方程组。该算法基于子空间修正的思想，第一层粗空间是由连续线性基函数张成的子空间。与此空间相关联的线性系统的代数构造使用Galerkin方法与自然嵌入作为延长算子。需要提供这个嵌入运算符，这意味着该方法不是完全代数的。为了在随后的较粗水平上构造线性系统，使用非平滑聚合AMG技术。在一系列数值实验中，我们首次建立了AMG方法对于各种对称和非对称内部惩罚DG方法（包括高阶情况）的效率和鲁棒性，这些方法用于系数中具有复杂，高对比度跳跃的问题。该求解器对于DG逼近空间的多项式次数的增加（至少高达6次）具有鲁棒性，计算效率高，并且仅受到系数跳跃和网格大小h的轻微影响（即，迭代次数= O（log h−1））。版权所有© 2012约翰威利父子有限公司.
We present a new algebraic multigrid (AMG) algorithm for the solution of linear systems arising from discontinuous Galerkin (DG) discretizations of heterogeneous elliptic problems. The algorithm is based on the idea of subspace corrections, and the first coarse level space is the subspace spanned by continuous linear basis functions. The linear system associated with this space is constructed algebraically using a Galerkin approach with the natural embedding as the prolongation operator. This embedding operator needs to be provided, which means that the approach is not fully algebraic. For the construction of the linear systems on the subsequent coarser levels, non‐smoothed aggregation AMG techniques are used. In a series of numerical experiments, we establish for the first time the efficiency and robustness of an AMG method for various symmetric and non‐symmetric interior penalty DG methods (including the higher‐order cases) on problems with complicated, high contrast jumps in the coefficients. The solver is robust with respect to an increase in the polynomial degree of the DG approximation space (at least up to degree 6), computationally efficient, and affected only mildly by the coefficient jumps and by the mesh size h (i.e., number of iterations = O(log h−1)). Copyright © 2012 John Wiley & Sons, Ltd.