Comparison of CFEM and DG methods

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

Formulation: Finite Element Methods (CFEMs)

Formulation: Discontinuous Galerkin (DG) methods

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.

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

Number of unknowns: Degrees of freedom (dof)

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

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

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

Preliminaries 1: types of PDEs

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

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,

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)

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.

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

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

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

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: http://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations#Matrix_algebra 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

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

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

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

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.

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