11 votes
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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 ...
Tristan Montoya's user avatar
10 votes
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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'...
Paul's user avatar
  • 12k
9 votes
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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 ...
Wolfgang Bangerth's user avatar
9 votes
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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,...
Daniel Shapero's user avatar
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=...
Wolfgang Bangerth's user avatar
8 votes
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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 ...
Jesse Chan's user avatar
  • 3,142
6 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 ...
Will P.'s user avatar
  • 821
6 votes
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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 ...
davidhigh's user avatar
  • 3,127
5 votes
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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 ...
cfdlab's user avatar
  • 3,028
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
Guido Kanschat's user avatar
5 votes
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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:\...
Will P.'s user avatar
  • 821
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 ...
Mithridates the Great's user avatar
5 votes

Finite difference problem

Since this is apparently a homework problem, let's just illustrate the idea on a simple small example. Let's take the domain [0,1], with the discontinuity at $x=0.5$, and assume $\alpha$=1 to the left ...
Maxim Umansky's user avatar
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 ...
bigge's user avatar
  • 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, ...
user107904's user avatar
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_{\...
Kenneth Assogba's user avatar
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 ...
Tristan Montoya's user avatar
4 votes
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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 ...
origimbo's user avatar
  • 2,249
4 votes
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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 ...
Vikram's user avatar
  • 489
4 votes
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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 -...
cfdlab's user avatar
  • 3,028
4 votes
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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 ...
ConvexHull's user avatar
  • 1,287
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\...
ConvexHull's user avatar
  • 1,287
3 votes
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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 ...
Wolfgang Bangerth's user avatar
3 votes

Numerical quadrature in Discontinuous Galerkin

Which basis you use for your finite element space does not matter for quadrature in general. If you use, for example, polynomials of degree $k$ (whether the Lagrange basis, or the Legendre basis, or ...
Wolfgang Bangerth's user avatar
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 ...
cfdlab's user avatar
  • 3,028
3 votes
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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 ...
knl's user avatar
  • 2,096
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 ...
DanielRch's user avatar
  • 472
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 ...
Reid.Atcheson's user avatar
3 votes

Obtain velocity from imposed energy spectrum using the inverse FFT

I recommend you first read answer here to find out why $\tilde{E}(\mathbf{k}) \neq \frac{1}{2} \tilde{\mathbf{u}}(\mathbf{k}) \cdot \tilde{\mathbf{u}}(\mathbf{k})$. In order to find velocity profile ...
Mithridates the Great's user avatar
3 votes

Are there any commercial CFD codes that implement a Discontinuous Galerkin scheme?

I believe Numeca is developing a Flux Reconstruction (related to DG) extension of their solver. I am sure other companies are doing so as well. However, commercial codes have a somewhat different set ...
Vikram's user avatar
  • 489

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