Why $\mathcal{H}$-Matrix Based Galerkin BEM?

  • Compared with the finite element method (FEM), the Galerkin BEM offers several advantages:

    • Open-domain problems

      BEM naturally handles unbounded open domains, which are common in electromagnetics and acoustics.

    • Surface-only discretization

      Only the boundary surface requires meshing, avoiding the complexity of generating full 3D meshes. This is particularly attractive for models with fine structures or extreme size ratios, where 3D meshing may become the bottleneck — or even fail entirely.

    • Surface quantities of interest

      Physical quantities such as electric field strength or mechanical stresses often concentrate on surfaces or interfaces, which are critical in engineering design. Galerkin BEM directly computes these boundary quantities from integral equations, whereas FEM typically approximates them from cell-based values.

  • Compared with collocation and Nyström BEM, the Galerkin formulation avoids the difficulties in handling singularities at corners and edges.

  • Compared with fast multiple method (FMM) and panel clustering, $\mathcal{H}$-matrix based BEM is a purely algebraic approach: it requires no problem-specific kernel expansions, making it more general and widely applicable across different physical models.

Features

  • Numerical quadrature
    • Accurate evaluation of singular boundary integrals using Sauter quadrature.
    • CUDA parallelization to accelerate computationally intensive quadrature.
    • Support for curved surfaces via high-order mappings from deal.II.
  • $\mathcal{H}$-matrix algebra
    • Formatted addition, matrix/matrix multiplication, and matrix/vector multiplication.
    • TBB task parallelization for matrix/vector multiplication.
    • TBB flow graph parallelization for LU and Cholesky factorization.
    • Load balancing for parallel $\mathcal{H}$-matrix assembly and matrix/vector multiplication using sequence partitioning.
  • Linear solvers and preconditioners
    • Iterative solvers for real and complex-valued systems, including preconditioned CG and GMRES.
    • $\mathcal{H}$-matrix LU and Cholesky factorization as preconditioners.
    • Operator preconditioning based on the pseudo-differential operator theory:
      • Ensures spectral equivalence between the system and preconditioning matrices.
      • Condition number of the preconditioned system becomes independent of discretization.
      • Enables efficient solution of large-scale BEM problems.
  • BEM solvers
    • Laplace solvers (real and complex) with Dirichlet, Neumann, and mixed boundary conditions.

Next Steps

  • Helmholtz solver — Develop a complex valued solver for the Helmholtz equation to address acoustic problems.
  • Multi-domain BEM — Extend the existing single domain BEM solver to handle multiple homogeneous subdomains using domain decomposition methods (DDM).
  • BEM/FEM coupling — Integrate HierBEM with the FEM framework in deal.II, combining the strengths of BEM for open and interface problems with FEM’s ability to model inhomogeneous and nonlinear domains.

Citation

If you use HierBEM in your research, please cite it as follows:

@misc{hierbem2025,
  author       = {Jihuan Tian, Xiaozhe Wang},
  title        = {HierBEM: A Hierarchical Matrix Based Galerkin Boundary Element Method Library},
  year         = {2025},
  publisher    = {GitHub},
  journal      = {GitHub repository},
  howpublished = {\url{https://github.com/jihuan-tian/hierbem}},
}

License

HierBEM library is licensed under the LGPL v3.

Links