Discontinuous Galerkin Methods
Discontinuous Galerkin methods use high-order polynomials per element with no continuity across faces, coupling neighbors through numerical fluxes.
Elements that do not share values
Discontinuous Galerkin (DG) methods represent the solution by an independent polynomial inside each element, with no requirement that values match across element boundaries. Neighboring elements communicate only through numerical fluxes evaluated on shared faces, exactly as in finite volume methods. DG thus fuses the high-order accuracy of finite elements with the local conservation and upwinding of finite volumes.
Because each element carries its own degrees of freedom, the mass matrix is block-diagonal and trivially invertible, which makes explicit time stepping efficient and the method highly parallel: each element's update depends only on its immediate neighbors.
Numerical fluxes and stability
The choice of numerical flux controls stability and accuracy. Upwind or Riemann-solver-based fluxes add controlled dissipation for hyperbolic problems, while central fluxes conserve energy for wave equations. For nonlinear conservation laws, slope or moment limiters suppress oscillations near discontinuities without destroying high-order accuracy in smooth regions.
Strengths and costs
DG handles complex geometry, supports hp-adaptivity naturally because there is no inter-element continuity to enforce, and captures shocks and steep gradients robustly. Its cost is more degrees of freedom than a continuous method of the same order, since face values are duplicated on both sides.
- Block-diagonal mass matrix for cheap explicit updates
- Local conservation through face fluxes
- Natural hp-adaptivity without continuity constraints
- Limiters preserve high order away from discontinuities
DG is increasingly used for kinetic and fluid plasma models, where its combination of conservation, high order, and shock robustness suits the steep gradients found in edge and scrape-off-layer transport.