On weak Dirichlet boundary conditions for elliptic problems in the continuous Galerkin method

M. Vymazal, D. Moxey, S. Sherwin, C. D. Cantwell, R. M. Kirby

J. Comput. Phys., vol. 394, pp. 732-744 (2019)

@article{vymazal-2019,
  title = {On weak Dirichlet boundary conditions for elliptic problems in the continuous Galerkin method},
  author = {Vymazal, M. and Moxey, D. and Sherwin, S. and Cantwell, C. D. and Kirby, R. M.},
  year = {2019},
  journal = jcp,
  volume = {394},
  pages = {732-744},
  doi = {10.1016/j.jcp.2019.05.021},
  url = {https://davidmoxey.uk/assets/pubs/2019-weak-bcs.pdf},
  abstract = {We combine continuous and discontinuous Galerkin methods in the setting of a model diffusion problem.  Starting from a hybrid discontinuous formulation, we replace element interiors by more general subsets of the computational domain - groups of elements that support a piecewise-polynomial continuous expansion. This step allows us to identify a~new weak formulation of Dirichlet boundary condition in the continuous framework. We show that the boundary condition leads to a stable discretization with a single parameter insensitive to mesh size and polynomial order of the expansion. The robustness of the approach is demonstrated on several numerical examples.}
}

Dirichlet boundary conditions in the continuous Galerkin method are normally imposed strongly. Starting instead from a hybrid discontinuous formulation and replacing element interiors with groups of elements carrying a continuous expansion, this paper arrives at a new weak way of imposing them. The resulting discretisation is stable, with a single parameter that is insensitive to both mesh size and polynomial order.