arXiv subjects

Markus Faustmann

Publications and source records attributed to Markus Faustmann.

22 records · Page 2Linked to original sources

arXiv2017

Local convergence of the boundary element method on polyhedral domains

The local behavior of the lowest order boundary element method on quasi-uniform meshes for Symm's integral equation and the stabilized hyper-singular integral equation on polygonal/polyhedral Lipschitz domains is analyzed. We prove local a priori estimates in $L^2$ for Symm's integral equation and in $H^1$ for the hypersingular equation. The local rate of convergence is limited by the local regularity of the sought solution and the sum of the global regularity and additional regularity provided by the shift theorem for a dual problem.

math.NA↗

arXiv2015

Existence of $\mathcal{H}$-matrix approximants to the inverse of BEM matrices: the hyper-singular integral operator

We consider discretizations of the hyper-singular integral operator on closed surfaces and show that the inverses of the corresponding system matrices can be approximated by blockwise low-rank matrices at an exponential rate in the block rank. We cover in particular the data-sparse format of $\mathcal{H}$-matrices. We show the approximability result for two types of discretizations. The first one is a saddle point formulation, which incorporates the constraint of vanishing mean of the solution. The second discretization is based on a stabilized hyper-singular operator, which leads to symmetric positive definite matrices. In this latter setting, we also show that the hierarchical Cholesky factorization can be approximated at an exponential rate in the block rank.

math.NA↗

arXiv2014

Existence of $\mathcal{H}$-matrix approximants to the inverses of BEM matrices: the simple-layer operator

We consider the question of approximating the inverse $\mathbf W = \mathbf V^{-1}$ of the Galerkin stiffness matrix $\mathbf V$ obtained by discretizing the simple-layer operator $V$ with piecewise constant functions. The block partitioning of $\mathbf W$ is assumed to satisfy any of the standard admissibility criteria that are employed in connection with clustering algorithms to approximate the discrete BEM operator $\mathbf V$. We show that $\mathbf W$ can be approximated by blockwise low-rank matrices such that the error decays exponentially in the block rank employed. Similar exponential approximability results are shown for the Cholesky factorization of $\mathbf V$.

math.NA↗

arXiv2013

H-matrix approximability of the inverses of FEM matrices

We study the question of approximability for the inverse of the FEM stiffness matrix for (scalar) second order elliptic boundary value problems by blockwise low rank matrices such as those given by the H-matrix format. We show that exponential convergence in the local block rank can be achieved. We also show that exponentially accurate LU-decompositions in the H-matrix format are possible for the stiffness matrices arising in the FEM. Unlike prior works, our analysis avoids any coupling of the block rank r and the mesh width h and also covers mixed Dirichlet-Neumann-Robin boundary conditions.

math.NA↗