11
votes
Accepted
Stability of hyperbolic PDE and DG-FEM
Stability does indeed mean that small changes in the data lead to small changes in the solution. This can be shown for the linear advection equation through the energy method; in proving the stability ...
- 182
10
votes
Accepted
Intro to DG Finite Element methods
For parabolic/elliptic PDE's, I highly recommend Beatrice Riviere's book: Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation.
For hyperbolic PDE'...
- 11.9k
9
votes
Accepted
Library for generating Discontinuous Galerkin FEM mesh
You are confused about different concepts. A mesh is really just a collection of cells defined by the vertices of the mesh and which vertices together form each cell. Consequently, a mesh is an ...
- 52.4k
8
votes
Discontinuous Galerkin method VS Continuous Galerkin method Degrees of freedom
The argument is misleading. The different spaces that are used are $P_1=\text{span}\{1,x,y\}$ and $Q_1=\text{span}\{1,x,y,xy\}$. For higher orders, they are
$P_2=\text{span}\{1,x,y,x^2,y^2\}$ and $Q_2=...
- 52.4k
6
votes
Choosing the penalty for Discontinuous Galerkin
The penalty usually scales as $O(p^2/h)$ for polynomial discontinuous Galerkin methods regardless of dimension, but for a more precise constant, it's usually enough to take the penalty large enough to ...
- 3,052
6
votes
Absorbing boundary conditions for acoustics in Discontinuous Galerkin
The problem with ABC in the form of derivative operators at the boundary is that the exact ABCs are non-local in space and often in time, which makes them hard to use in numerical simulations. You can ...
- 1,238
6
votes
Accepted
CFL condition in Discontinuous Galerkin schemes
The restrictive CFL of DG schemes typically comes from the combination of high order accuracy and a compact stencil (see this reference for example). The CFL depends on bounding the variational form ...
- 3,052
6
votes
Accepted
Gauss-Lobatto quadrature and nodal points for FEM
Ok, here comes the answer promised in the comment section. Let's start the other way round, going from a general grid to Gaussian grids and further constructions such as spectral elements.
In grid ...
- 2,937
5
votes
Role of the numerical flux in DG-FEM
The numerical flux is chosen to ensure that information in the problem travels in the direction of the characteristic curves of the equation (upwinding). As mentioned in the comments, the numerical ...
- 791
5
votes
Accepted
Spectral methods, Spectral Volume methods, Spectral Difference methods
"Spectral methods" usually means methods which make use of global basis functions. Fourier spectral methods use sine and cosine, and are used when you have periodic boundary conditions. Chebyshev ...
- 2,993
5
votes
Matlab implementation of 2D Interior penalty discontinuous Galerkin poisson problem
Since you read Riviere's book and know how to assemble element integrals I assume you are familiar with concepts such as
transformation to reference elements,
numerical quadrature.
These techniques ...
- 51
5
votes
what is the difference between non-conformal and conformal?
The concept of $p$-nonconforming meshes can also apply to continuous FEM, not just DG. For continuous FEM, continuity is enforced by enforcing a single set of degrees of freedom on a face shared ...
- 3,052
5
votes
Has a uniform estimate in k of the inf-sup constant for hp-DG methods for the Stokes problem been established?
The problem was solved recently: Lederer/Schöberl: Polynomial robust stability analysis for H(div)-conforming finite elements for the Stokes equations
- 1,114
5
votes
Accepted
Size of jump for piecewise discontinuous approximations
I believe that for odd polynomial degrees you can get superconvergence by one order at the jump. For even degrees I don't think that's the case.
For the odd case, we consider a smooth function $u:\...
- 791
5
votes
How to compute turbulent energy cascade
Of course, the Fourier transform is a linear operator. So, you have the kinetic energy defined as: $E(\mathbf{r}) = \frac{1}{2} \mathbf{u}(\mathbf{r}) \cdot \mathbf{u}(\mathbf{r})$. The Fourier ...
- 2,322
4
votes
The meaning of conservative discretization in Galerkin FEM and Discontinuous Galerkin
No. The statement means that the finite element method, without modifications, will lead to numerical solutions for conservation laws that violate many of the physically reasonable constraints (e.g., ...
- 52.4k
4
votes
Are FEM or DGFEM methods based on integrals or PDEs?
Basically, you have the following set of derivations:
Strong form of the PDE -> Finite differences
Integral form of the PDE -> Finite volumes
Weak (variational) form of the PDE -> Galerkin methods (...
- 52.4k
4
votes
Accepted
What's the difference between C0 penalty methods and Discontinuous Galerkin methods?
DG methods are a more general description, referring to finite element methods which represent the solution in a discontinuous manner (usually with coupling between elements through a numerical flux). ...
- 3,052
4
votes
Proof of CFL condition for RKDG scheme
As far is I know, a proof for the general case has not been presented so far. The most detailed analysis has probably been performed by Krivodonova & Qin 2013 in this work. Quoting from the ...
- 293
4
votes
Interior penalty discontinuous Galerkin Matlab implementation
If you are adamant on using MATLAB, chapter 14 of the following book walks through a 2D IPDG Poisson problem using piecewise linear basis functions on triangles:
The Finite Element Method: Theory, ...
- 587
4
votes
Accepted
DG-FEM integration by parts
We are considering the one-dimensional scalar conservation law,
$$
\frac{\partial u}{\partial t} + \frac{\partial f(u)}{\partial x}= 0, \quad x \in \Omega, \quad t > 0,
$$
subject to appropriate ...
- 182
4
votes
Accepted
Differentiation Matrix In DG-FEM - Hesthaven/Warburton
Taking your questions in order, the definition of $D_r$ as the operator converting the nodal values of $u_h$ into derivatives follows immediately from the definition, although the notation is a little ...
- 2,199
4
votes
Accepted
calculation of the right hand side of DG FEM (with code)
We can break down the code
rhsu = -a*rx.*(Dr*u) + LIFT*(Fscale.*(du));
into the two parts
-a*rx.*(Dr*u)
and
...
- 499
4
votes
Accepted
Slope limiting for discontinuous Galerkin (DG) method
The paper of Cockburn and Shu [1] explains this. If the solution is
$$
u_h(x) = u_i + u_{xi} \phi(x), \qquad \phi(x) = \frac{x - x_i}{\Delta x/2}
$$
Then the limiter is
$$
u_{xi} = minmod(u_{xi}, u_i -...
- 2,993
4
votes
Accepted
Pros of Fourier-Galerkin spectral methods
Pros:
With trigonometric basis functions your problem size is $N \text{log}(N)$ instead of $N^2$.
Stabilization techniques are easy to implement and cheap:
Filtering in the modal space.
Zero padding ...
- 875
4
votes
Errors imposing boundary conditions weakly with DG
I'm not familiar with deal.II. However, to show that DG is able to reproduce
the constant gradient solution I will post some results with a different tool using IP. The penalty is about $\sigma\...
- 875
3
votes
Role of the numerical flux in DG-FEM
Very loosely speaking there are two things most discretization techniques need in order to converge to the actual solution of your PDE as you increase their approximation quality, regardless if you're ...
- 3,243
3
votes
Velocity-Stress formulation of Elastodynamics/Wave Equation for beginner
The stress-velocity formulation has been used extensively in DG context on account of the fact that it can lead to a first order system form of the elastic equation. The latter proved to lend itself ...
- 472
3
votes
What's the difference between C0 penalty methods and Discontinuous Galerkin methods?
From a mathematical perspective, the term 'DG method' by itself refers only to the class of methods that use a piecewise discontinuous basis. This choice of basis necessitates the introduction of some ...
3
votes
Absorbing boundary conditions for acoustics in Discontinuous Galerkin
There exist Absorbing Boundary Conditions for the wave equation that are stable and that go up to any order of accuracy (limited only by the accuracy of discretization of your model), so that they are ...
- 472
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