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'...
- 12k
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 ...
- 54.1k
9
votes
Accepted
Does the weighted residual method not use energy minimization in any form?
The answer to your question depends on the problem.
For example, consider the diffusion equation for a field $q$, with diffusivity $k$ and sources $f$.
The variational form of this problem states that,...
- 10.1k
8
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 ...
- 3,102
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=...
- 54.1k
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 ...
- 3,102
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 ...
- 3,042
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
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 ...
- 3,102
5
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 (...
- 54.1k
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
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 ...
- 2,332
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, ...
- 597
4
votes
$L^\infty$ stability property of an ODE
The $L^\infty$-stability would be the stability in the sense that the $L^\infty$ norm of the state $u$ of the system is smaller than the $L^\infty$ norm of the initial data $u_0$ :
$$\lVert u \rVert_{\...
4
votes
Accepted
DG-FEM integration by parts
We are considering the one-dimensional scalar conservation law,
$$
\frac{\partial u}{\partial t} + \frac{\partial f(u)}{\partial x}= 0, \quad x \in \Omega, \quad t > 0,
$$
subject to appropriate ...
- 182
4
votes
Accepted
Differentiation Matrix In DG-FEM - Hesthaven/Warburton
Taking your questions in order, the definition of $D_r$ as the operator converting the nodal values of $u_h$ into derivatives follows immediately from the definition, although the notation is a little ...
- 2,229
4
votes
Accepted
calculation of the right hand side of DG FEM (with code)
We can break down the code
rhsu = -a*rx.*(Dr*u) + LIFT*(Fscale.*(du));
into the two parts
-a*rx.*(Dr*u)
and
...
- 489
4
votes
Accepted
Slope limiting for discontinuous Galerkin (DG) method
The paper of Cockburn and Shu [1] explains this. If the solution is
$$
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 ...
- 1,208
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\...
- 1,208
4
votes
Finite difference problem
Since this is apparently a homework problem, let's just illustrate the idea on a small simple example. Let's take the domain [0,1], with the discontinuity at $x=0.5$, and assume $\alpha$=1 to the left ...
- 2,440
3
votes
Accepted
Discontinuous Galerkin FEM : Control points are mid-points of edges instead of nodes
For the discontinuous Galerkin method, the choice of control points is entirely unimportant because they conceptually lie in the interior of the cell (even if they are physically on the boundary of ...
- 54.1k
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 ...
- 3,263
3
votes
Orthonormal basis for hexahedron
On quad/hex you can use tensor product polynomials. For example in 2d, you map the cell to a reference cell $[-1,1] \times [-1,1]$ and if $N$ is the degree, you would use
$$
P_i(\xi) P_j(\eta), \qquad ...
- 2,993
3
votes
Accepted
Plot 2D piecewise constant in matlab in a finite elements mesh
I use typically the following approach. The idea is to make a new mesh where every vertex is duplicated so that each triangle has its own copy. Then you can use standard trisurf command to the ...
- 2,041
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