28 citations · 130 across the 20 of their papers we have counts for
21 papers
Exponential Convergence of -FEM for the Integral Fractional Laplacian on cuboids
Björn Bahr, Markus Faustmann, Carlo Marcati +2
For the Dirichlet integral fractional Laplacian, we prove root exponential convergence of tensor-product -finite element approximations on , for forcing that is an…
An implementation of hp-FEM for the fractional Laplacian
Björn Bahr, Markus Faustmann, Jens Markus Melenk
We consider the discretization of the -integral Dirichlet fractional Laplacian by -finite elements. We present quadrature schemes to set up the stiffness matrix and load ve…
Weighted analytic regularity for the integral fractional Laplacian in polyhedra
Markus Faustmann, Carlo Marcati, Jens Markus Melenk +1
On polytopal domains in , we prove weighted analytic regularity of solutions to the Dirichlet problem for the integral fractional Laplacian with analytic right-hand s…
FEM-BEM coupling in Fractional Diffusion
Markus Faustmann, Alexander Rieder
We derive and analyze a fully computable discrete scheme for fractional partial differential equations posed on the full space . Based on a reformulation using the w…
Fractional Diffusion in the full space: decay and regularity
Markus Faustmann, Alexander Rieder
We consider fractional partial differential equations posed on the full space . Using the well-known Caffarelli-Silvestre extension to as equivalent defini…
Two-level error estimation for the integral fractional Laplacian
Markus Faustmann, Ernst Peter Stephan, David Wörgötter
For the singular integral definition of the fractional Laplacian, we consider an adaptive finite element method steered by two-level error indicators. For this algorithm, we show l…
Exponential Convergence of hp FEM for the Integral Fractional Laplacian in Polygons
Markus Faustmann, Carlo Marcati, Jens Markus Melenk +1
We prove exponential convergence in the energy norm of finite element discretizations for the integral fractional diffusion operator of order subject to homogene…
Exponential convergence of hp-FEM for the integral fractional Laplacian in 1D
Markus Faustmann, Carlo Marcati, Jens Markus Melenk +1
We prove weighted analytic regularity for the solution of the integral fractional Poisson problem on bounded intervals with analytic right-hand side. Based on this regularity resul…
Exponential meshes and -matrices
Niklas Angleitner, Markus Faustmann, Jens Markus Melenk
In our previous works, we proved that the inverse of the stiffness matrix of an -version finite element method (FEM) applied to scalar second order elliptic boundary value probl…
Weighted analytic regularity for the integral fractional Laplacian in polygons
Markus Faustmann, Carlo Marcati, Jens Markus Melenk +1
We prove weighted analytic regularity of solutions to the Dirichlet problem for the integral fractional Laplacian in polygons with analytic right-hand side. We localize the problem…
-inverses for RBF interpolation
Niklas Angleitner, Markus Faustmann, Jens Markus Melenk
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 ma…
-matrix approximability of inverses of FEM matrices for the time-harmonic Maxwell equations
Markus Faustmann, Jens Markus Melenk, Maryam Parvizi
The inverse of the stiffness matrix of the time-harmonic Maxwell equation with perfectly conducting boundary conditions is approximated in the blockwise low-rank format of ${\mathc…
Caccioppoli-type estimates and -Matrix approximations to inverses for FEM-BEM couplings
Markus Faustmann, Jens Markus Melenk, Maryam Parvizi
We consider three different methods for the coupling of the finite element method and the boundary element method, the Bielak-MacCamy coupling, the symmetric coupling, and the John…
Approximating inverse FEM matrices on non-uniform meshes with -matrices
Niklas Angleitner, Markus Faustmann, Jens Markus Melenk
We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse -matrix format. For a large class of shape regular but possibly…
Local convergence of the FEM for the integral fractional Laplacian
Markus Faustmann, Michael Karkulik, Jens Markus Melenk
We provide for first order discretizations of the integral fractional Laplacian sharp local error estimates on proper subdomains in both the local -norm and the localized ener…
On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion
Markus Faustmann, Jens Markus Melenk, Maryam Parvizi
We provide an endpoint stability result for Scott-Zhang type operators in Besov spaces. For globally continuous piecewise polynomials these are bounded from into $B^{3/2}…
Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian
Markus Faustmann, Jens Markus Melenk, Dirk Praetorius
For the discretization of the integral fractional Laplacian , , based on piecewise linear functions, we present and analyze a reliable weighted residual a poster…
Local convergence of the boundary element method on polyhedral domains
Markus Faustmann, Jens Markus Melenk
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…
Existence of -matrix approximants to the inverse of BEM matrices: the hyper-singular integral operator
Markus Faustmann, Jens Markus Melenk, Dirk Praetorius
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 blockw…
Existence of -matrix approximants to the inverses of BEM matrices: the simple-layer operator
Markus Faustmann, Jens Markus Melenk, Dirk Praetorius
We consider the question of approximating the inverse of the Galerkin stiffness matrix obtained by discretizing the simple-layer operator $…
H-matrix approximability of the inverses of FEM matrices
Markus Faustmann, Jens Markus Melenk, Dirk Praetorius
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…