# 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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### 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 ...

### 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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### Are additional penalty terms necessary to solve elliptic PDE's with DG-FEM?

The penalty term does indeed act as a stabilization - coercivity is usually shown using norms which depend on this penalty, and you typically need a "sufficiently large penalty" to show convergence of ...

### 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., ...

### 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 (...
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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). ...

### 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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### 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 ...

### 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 ...

### 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 ...

### Discontinuous Galerkin, residual orthogonal to test functions?

Some answers to your questions: since the test functions are discontinuous, all test functions contain those with only element support. in matrix-vector form, you would write the discrete variational ...
When you apply an energy method, multiply by the solution $u$ and integrate over the domain $D^k$ you obtain  \displaystyle\int_{D^k} \left(\frac{\partial u}{\partial t} + \frac{\partial (au)}{\...