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.

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 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.