# Tag Info

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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 ...
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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'...
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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 ...
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• 791

### 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 ...

### 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., ...
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### 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
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### 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). ...
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### 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 ...
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### 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, ...
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### 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 ...
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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 ...
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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 ...
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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 -...
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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 ...
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### 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\...
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### 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 ...
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### 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 ...
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