1. Comparison of CFEM and DG methods

2. Discontinuous Galerkin (DG) vs
Discontinuous Galerkin (DG) vs. Continuous Finite Element Methods (CFEMs)
DG methods use discontinuous basis functions

3. Formulation: Finite Element Methods (CFEMs)

4. Formulation: Discontinuous Galerkin (DG) methods

5. Target Values (fluxes) in DG methods
Target values (fluxes) are determined by the values on the two sides of the interface.
These fluxes can be designed to make the method more stable or achieve other objectives.
For hyperbolic problems target values are designed to preserve the characteristic structure of the waves eliminate many numerical artifacts observed in CFEMs.

6. Comparison of CFEM and DG methods:
Advantages of CFEMs
Preliminaries
Advantages of DG methods

7. Number of unknowns: Degrees of freedom (dof)

8. Number of unknowns: Average dof per element
Consider FEs for a scalar field and polynomial order p = 1

9. Number of unknowns: Average dof per element
Consider FEs for a scalar field and polynomial order p = 2
As p increases dofs of DG becomes closer to dofs of CFEM
For high polynomial order DG becomes more competitive p
dof ratio DG/CFEM 1 4 2
2.25 3
1.78
1.56 5
1.44

10. Comparison of CFEM and DG methods:
Advantages of CFEMs
Preliminaries
Advantages of DG methods

11. Preliminaries 1: types of PDEs

12. Preliminaries 1: types of PDEs
Burger’s equation (nonlinear)
t = 0, smooth solution
t > 0, shock has formed

13. Preliminaries 2: Semi-discrete formulation of dynamic problems
For a static problem (elliptic PDE) the FEM formulation results in:
where for a linear problem we have:
For a dynamic problem time derivatives in the equations result in equations in the form of:
We generally use a discrete method, e.g. Finite Difference (FD), for the solution of these systems of equations.
Implicit vs. Explicit refers to how the finite difference scheme is expressed:
Sample parabolic equation, using forward& backward Euler method we get,

14. Preliminaries 2: Semi-discrete formulation of dynamic problems
Implicit method:
Unconditionally stable
(can be) nonlinear
Explicit method:
Maximum allowable time step Dt based on element sizes
Linear
For a linear problem we have:
The form of C matrix dictates the solution complexity of explicit methods (similar to mass matrix for elastodynamics)

15. Preliminaries 3: Connectivity of elements / form of FEM matrices
9 connected elements
4 connected elements
This ratio can drastically increase depending on the element shape and dimension. For example for d = 3, tetrahedral elements the number are: CFEM: 24, DG: 4 ratio = 6.
The connectivity of elements greatly influences the form of FEM matrices and parallel efficiencies.

16. Comparison of CFEM and DG methods:
Advantages of CFEMs
Preliminaries
Advantages of DG methods

17. CFEMs: DGs: 1. FEM adaptivity h-adaptivity: element size changes
p-adaptivity: polynomial order changes
Transition elements
Because of strong continuity of elements transition elements are required
DGs:
h-adaptivity:
p-adaptivity:
Arbitrary change in size and polynomial order
as jump conditions are weakly enforced

18. 2. Efficiency for dynamic problems Form of the system of equations
Reminder: for explicit method (forward Euler) we had:
Based on the connectivity of elements and the use of explicit method Mass matrix in DG methods is block-diagonal
Example
CFEM DG

19. 2. Efficiency for dynamic problems
Block-diagonal “mass” matrix for DG methods with explicit integration:
Linear solution complexity in number of elements O(N)
Comments:
For a full C matrix linear solution schemes scale as O(N2.376) (e.g. Coppersmith–Winograd algorithm, link:
Example: By increasing the number of elements by 100:
DG solution cost scales by 100
CFEM solution cost scales by ~ 56000
For CFEMs often mass-lumping is used which results in diagonal mass matrix and yielding O(N) solution complexity. However, mass lumping severely affects the order of convergence of the method for high order elements.
Consistent mass matrix
Mass matrix with mass lumping

20. 3. Parallel computing
DG methods are much better for parallel computing for:
More local connectivity (e.g. factor of 6 & 30 fewer connected elements for d = 3,4. tetrahedral elements). Reduction in processor communications.
Higher ratio of FEM computation to communications because DG methods are often used for higher order polynomials. p
dof ratio DG/CFEM 1 4 2
2.25 3
1.78
1.56 5
1.44

21. 4. Resolving shocks and discontinuities for hyperbolic problems
Reminder: hyperbolic problems preserve discontinuities and generate shocks from smooth initial conditions for nonlinear problems.
How do CFEMs perform for problems with discontinuities and shocks?
Burger’s equation (nonlinear)
t = 0, smooth solution
t > 0, shock has formed
Global numerical oscillations
Results obtained by COMSOL

22. 4. Resolving shocks and discontinuities for hyperbolic problems
Due to the judicious use of target (numerical) fluxes ()* DG methods have a much better performance for these problems.
Borrowing ideas from Finite Volume methods (FV) DG methods use (approximate) Riemann-fluxes for target values.
FV methods:
Stable for hyperbolic problems
Not suitable for high orders
Not suitable for complicated geometries
CFEM methods:
High order & flexible
Sample DG solutions with no evident numerical
artifacts
Example: Laplacian in FV
DG methods:
Telescoping stencil

23. 5. Recovering balance laws at the element level
Since the weight functions can be set to unity at each individual element, balance properties can be recovered at element level.

24. Summary of CFEMs and DG methods
Advantages of DG methods:
1. FEM adaptivity
Resolving shocks and discontinuities for hyperbolic problems
Recovering balance laws at the element level
2. Efficiency /dynamic problems (block diagonal “mass” matrix)
3. Parallel computing (more local communication and use of higher order elements with DG methods)
4. Superior performance for resolving discontinuities (discrete solution space better resembles the continuum solution space)
5. Can recover balance properties at the element level (vs global domain)
Disadvantages:
Higher number of degrees of freedom:
Particularly important for elliptic problems (global system is solved).
Recently hybridizable DG methods (HDG), use Schur decomposition (static condensation) to eliminate elements internal dofs, making DG methods competitive or even better for elliptic problems as well.
no transition elements needed
Arbitrary change in size and polynomial order