What are the main advantages and disadvantages of DG methods compared to other FEA methods?

The discontinuous Galerkin (DG) method is a finite element method formulated relative to a weak formulation of a particular model system. Unlike traditional Continuous Galerkin (CG) methods that are conforming, the DG method works over a trial space of functions that are only piecewise continuous and thus often comprise more inclusive function spaces. The DG method has also been used to discretize the time domain rather than classical finite difference integration in time. In this context, it is called the TDG Time-Discontinuous Galerkin (TDG) method. My PhD dissertation, written 30 years ago, was mainly focused on the TDG method for coupled structural acoustics. There are many advantages of TDG over traditional time-stepping methods.

The Discontinuous Galerkin (DG) method is a powerful numerical technique for solving differential equations. Combining finite element and finite volume methods, it uses discontinuous basis functions for flexible solution representation, upholds local conservation laws, ensures high-order accuracy, and adapts to complex geometries. DG is widely used in fields like computational fluid dynamics, electromagnetics, plasma physics, structural mechanics, and environmental modeling. Its versatility and precision make it an essential tool in scientific and engineering disciplines.

The Discontinuous Galerkin (DG) method is a robust numerical simulation tool offering high-order accuracy, local conservation, geometric flexibility, and adaptability. It excels in handling discontinuities, is highly parallelizable, and allows customizable basis functions. With a compact stencil and wide applicability in fields like fluid dynamics and structural mechanics, DG is ideal for complex geometries and high-accuracy requirements.

The Discontinuous Galerkin (DG) method offers several advantages but also comes with significant drawbacks. These include higher computational costs, complex implementation, stability concerns, challenging boundary conditions, limited software maturity, and increased memory demands. Despite these issues, the DG method is valuable for specific applications and ongoing research aims to mitigate its challenges.

Choosing the best Finite Element Analysis (FEA) method depends on the type of partial differential equation (PDE), domain geometry, desired accuracy, efficiency, and available computational resources. Hyperbolic PDEs benefit from Discontinuous Galerkin (DG) methods, parabolic PDEs from Finite Difference (FD) or Finite Volume (FV) methods, and elliptic PDEs from Continuous Galerkin (CG) or FV methods. For simple geometries, FD methods are suitable, while complex geometries may require FV or DG methods. DG methods offer high accuracy, whereas FD and FV methods are more efficient and resource friendly. Consider time-dependent vs. steady-state problems, conservation laws, and discontinuities when making your choice.