arXiv subjects

Niklas Angleitner

Publications and source records attributed to Niklas Angleitner.

3 recordsLinked to original sources

arXiv2022

Exponential meshes and $\mathcal{H}$-matrices

In our previous works, we proved that the inverse of the stiffness matrix of an $h$-version finite element method (FEM) applied to scalar second order elliptic boundary value problems can be approximated at an exponential rate in the block rank by $\mathcal{H}$-matrices. Here, we improve on this result in multiple ways: (1) The class of meshes is significantly enlarged and includes certain exponentially graded meshes. (2) The dependence on the polynomial degree $p$ of the discrete ansatz space is made explicit in our analysis. (3) The bound for the approximation error is sharpened, and (4) the proof is simplified.

math.NA↗

arXiv2021

$\mathcal{H}$-inverses for RBF interpolation

We consider the interpolation problem for a class of radial basis functions (RBFs) that includes the classical polyharmonic splines (PHS). We show that the inverse of the system matrix for this interpolation problem can be approximated at an exponential rate in the block rank in the $\mathcal{H}$-matrix format if the block structure of the $\mathcal{H}$-matrix arises from a standard clustering algorithm.

math.NA↗

arXiv2020

Approximating inverse FEM matrices on non-uniform meshes with $\mathcal{H}$-matrices

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 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.

math.NA↗