Is a Lovely Boundary Element Research Tool

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 P⁰ for fluxes and piecewise affine and globally continuous S¹ for traces of concentrations.

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.

• Paper on HILBERT [Applied Numerical Mathematics, 67 (2014), open access]
• Documentation of HILBERT (PDF)
• Download: HILBERT, Release 3 (ZIP)

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 P⁰ ansatz and test functions,
• simple-layer potential V for P⁰ ansatz and test functions,
• double-layer potential K for S¹ ansatz and P⁰ test functions,
• hypersingular integral operator W for S¹ ansatz and test functions.

Moreover, HILBERT provides functions, also implemented in C through the Matlab MEX interface, for the point evaluations of these operators as well as of the adjoint double-layer potential. The remaining Matlab codes are fully vectorized. By others, HILBERT includes

• 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.

• 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)