We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse \({\mathcal{H}}\)-matrix format. For a large class of shape regular but possibly non-uniform meshes including algebraically graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the \({\mathcal{H}}\)-matrix format at an exponential rate in the block rank. Since the storage complexity of the hierarchical matrix is logarithmic-linear and only grows linearly in the block-rank, we obtain an efficient approximation that can be used, e.g., as an approximate direct solver or preconditioner for iterative solvers.

中文翻译：

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使用 $${\mathcal{H}}$$ H 矩阵逼近非均匀网格上的逆 FEM 矩阵

我们考虑数据稀疏\({\mathcal{H}}\)矩阵格式中有限元刚度矩阵的逆的近似。对于一大类形状规则但可能不均匀的网格，包括代数渐变网格，我们证明刚度矩阵的逆可以以指数速率在\({\mathcal{H}}\) -矩阵格式中近似在块等级。由于分层矩阵的存储复杂度是对数线性的并且仅在块秩中线性增长，因此我们获得了可以用作例如近似直接求解器或迭代求解器的预调节器的有效近似。