论文标题
通过有限元法解决了预处理扩散和弹性问题的所有特征值的两侧边界
Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method
论文作者
论文摘要
最近在K.A. T. Gergelits中引入了一种估计具有DIRICHLET边界条件的预处理标量扩散算子的所有特征值的方法。 Mardal，B.F。Nielsen，Z。Strakoš：椭圆形PDE的Laplacian预处理：离散操作员特征值的定位，Siam Journal在数值分析上57（3）（2019），1369-1394。在本文的激励下，我们提供了一种略有不同的方法，可以将先前的结果扩展到某些方向。也就是说，我们在所有越来越有序的特征值的范围内，具有张量数据的一般扩散或弹性运算符，并通过符合有限元法离散，并由同一操作员的矩阵的倒数和不同的数据进行预处理。我们的结果适用于将原始和预处理问题应用于原始问题和周期性边界条件。边界是双面的，可保证的，易于访问，并且仅取决于材料数据。
A method of estimating all eigenvalues of a preconditioned discretized scalar diffusion operator with Dirichlet boundary conditions has been recently introduced in T. Gergelits, K.A. Mardal, B.F. Nielsen, Z. Strakoš: Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator, SIAM Journal on Numerical Analysis 57(3) (2019), 1369-1394. Motivated by this paper, we offer a slightly different approach that extends the previous results in some directions. Namely, we provide bounds on all increasingly ordered eigenvalues of a general diffusion or elasticity operator with tensor data, discretized with the conforming finite element method, preconditioned by the inverse of a matrix of the same operator with different data. Our results hold for mixed Dirichlet and Robin or periodic boundary conditions applied to the original and preconditioning problems. The bounds are two-sided, guaranteed, easily accessible, and depend solely on the material data.
