# HILBERT

Is a
Lovely
Boundary
Element
Research
Tool

## Description

HILBERT is a Matlab library for *h*-adaptive Galerkin BEM. Currently,
only lowest-order elements for the 2D Laplacian are implemented, i.e.,
piecewise constants *P ^{0}* for fluxes and piecewise affine
and globally continuous

*S*for traces of concentrations.

^{1}
HILBERT was developed as the research code for the FWF project P21732
*Adaptive Boundary Element Method* (2009-2014). HILBERT is free for academic use and might provide
a good basis for academic education on BEM as well.

## Downloads

- Paper on HILBERT [Applied Numerical Mathematics, 67 (2014), open access]
- HILBERT, Release 1 can be downloaded from http://www.netlib.org/numeralgo/na38.zip.
- Documentation of HILBERT [pdf]
- HILBERT, Release 3 can be downloaded here

## Features of HILBERT (Release 3, June 2012)

All Galerkin matrices are implemented in C through the Matlab MEX interface. They can thus be easily linked to any other programming language like Fortran, C, or C++. So far, HILBERT provides the following three discrete integral operators

- Newton potential
**N**for*P*ansatz and test functions,^{0} - simple-layer potential
**V**for*P*ansatz and test functions,^{0} - double-layer potential
**K**for*S*ansatz and^{1}*P*test functions,^{0} - hypersingular integral operator
**W**for*S*ansatz and test functions.^{1}

- different error estimators,
- h-h/2 error estimators (proposed by Ferraz-Leite, Praetorius & co-workers),
- two-level estimator (introduced by Maischak, Stephan & co-workers),
- weigthed-residual error estimators (introduced by Carstensen, Stephan & co-workers),
- different marking strategies,
- Dörfler' bulk criterion,
- maximum criterion,
- optimal local mesh-refinement for boundary meshes,
- newest vertex bisection to refine volume meshes,
- several visualization tools.

For the ease of introduction to adaptive BEM, HILBERT provides example files and adaptive algorithms for the integral formulations for

- the Dirichlet problem (so-called
*weakly-singular integral equation*), - the Neumann problem (so-called
*hypersingular integral equation*), - the mixed boundary value problem with Dirichlet/Neumann boundary conditions,
- with/without volume data,
- for different adaptive strategies from the literature,
- also for indirect BEM formulations.

## HILBERT Team

- Markus Aurada (PhD student and postdoc)
- Michael Feischl (PhD student)
- Samuel Ferraz-Leite (PhD student)
- Thomas Führer (PhD student and postdoc)
- Petra Goldenits (PhD student)
- Michael Karkulik (PhD student)
- Markus Mayr (MSc student)
- Gregor Mitscha-Eibl (MSc student)
- Dirk Praetorius (Principal investigator)