Brief comparison of SDG method with other discontinuous Galerkin methods

Discontinuous Galerkin (DG) methods have several advantages over continuous finite element methods. The recovery of balance laws at the element level and their inherent strength in dealing with nonsmooth features greatly improve the accuracy. The support for nonconforming meshes and arbitrary element polynomial orders enhance the efficiency of h- and hp-adaptive schemes. Spacetime Discontinuous Galerkin (SDG) replaces separate time integration by direct discretization of spacetime. SDG has the following advantages over other DG and FEM methods:

  1. Moving interfaces can be directly tracked by element boundaries in spacetime.
  2. Local solution feature: For hyperbolic problems, the solution of an otherwise globally coupled system reduces to solving many small local systems, yielding linear computational complexity.
  3. Adaptive and multiscale benefits:
    1. No global time step: Particularly a limiting factor in multiscale simulations using other methods.
    2. Local-effect adaptivity: From A2, rejection of elements with large error do not require reanalysis of the entire domain.
    3. Arbitrary order in time: Element-wise high order interpolation in time in contrast to typically uniform and low order accuracy of time-marching schemes.
  4. Asynchronous structure because of 3.1; ideal for parallel computing.
  5. Riemann-free scheme eliminates an expensive and complex feature of other DGs.

For more detail about SDG method including its special spacetime meshing refer to “Spacetime Discontinuous Galerkin (SDG) Finite Element method (FEM)” at the active research page.

For the description of advantages of SDG method for adaptive and multiscale grids refer to “Adaptive mesh operations for SDG method” at the active research page.