Questions tagged [discontinuous-galerkin]
Questions about analysis, implementation or application of Galerkin methods for partial differential equations using piecewise functions that are not globally continuous (and hence require surface terms on element boundaries in addition to the usual volume terms occurring in finite element methods).
98
questions
6
votes
1answer
153 views
Gauss-Lobatto quadrature and nodal points for FEM
By using the Legendre-Gauss-Lobatto (LGL) quadrature formula (QF) and LGL nodal points one achives a diagonal mass-matrix for finite element problems. (More specifically, the spectral element method.)
...
2
votes
0answers
55 views
Explicit DG time step restriction for compressible Navier-Stokes equations
Hesthaven's book 1 mentions the following time step restriction for Navier-Stokes equations (see (7.32) in 2008 edition)
$$
\Delta t \approx \frac{h}{N^2} \frac{C}{|u| + |c| + \frac
{N^2 \mu}{h}}
$$
($...
1
vote
0answers
62 views
Solution predictors for accelerating convergence in nonlinear FEM
I am looking for the details of commonly-used predictors for accelerating the convergence of iterations using Newton-Raphson scheme for nonlinear problems in FEM. I am looking specifically for static ...
3
votes
1answer
86 views
Pros of Fourier-Galerkin spectral methods
What are the pros of Fourier-Galerkin spectral methods while solving PDEs?
Here's the one that came in my mind first:
Easy implementation: using this method, differentiation operator computation is ...
2
votes
1answer
95 views
Calculate stable time step DG method for advection-diffusion
For stable time steps for the RKDG method for transport equations we require that
$$
\Delta t \le \frac{\Delta x CFL}{(2k + 1)|\lambda|},
$$
where $\lambda$ is the eigenvalue of our conservation law ...
1
vote
2answers
156 views
$P0$ elements for $H1$
Are there $P0$ (zero degree/constant element) nonconforming methods for approximating solutions in $H1$? More specifically, I have the equation:
$$u-f - T\Delta u = 0$$
Which can be interpreted as ...
1
vote
0answers
37 views
Cell-centered DG extension to the two-point flux approximation scheme
A current problem that I am working on requires me to compute the solution from the heat diffusion evolution on a discontinuous function. More precisely - I have a Delaunay triangulation and within ...
1
vote
1answer
121 views
Normalized legendre and quadrature basis for discontinous Galerkin method
I've successfully implemented a 1D DG code with non-normalized Legendre basis and I've now moved onto developing a 2D code using tensor products. For my 2D code I've chosen to have normalized ...
2
votes
1answer
141 views
Gradient-jump penalty term in FEM
I am slightly confused regarding the meaning of the $i-th$ gradient-jump term $[\nabla \phi_i]$ in the context of finite element methods, used in the assembly of the stiffness matrix (an example with <...
2
votes
1answer
125 views
Slope limiting for discontinuous Galerkin (DG) method
I had a question regarding the implementation of the TVB limiter for the RKDG method by Cockburn. I have seen that some implementations of the DG method use normalized Legendre polynomials such that ...
5
votes
0answers
169 views
How to compute numerical fluxes in the local discontinuous galerkin method for poisson equation 1D
Some days ago I began to study the local discontinuous galerkin (LDG) method, this is my first time working with a discontinuous method, so I decided to solve the poisson equation in 1D to learn the ...
1
vote
0answers
51 views
Question regarding 1D implementation of the DG method
I'm pretty new to the DG method and have been writing a 1D code to help me understand the coding aspect. With respect a reference, I've been following these notes https://www3.nd.edu/~zxu2/...
1
vote
1answer
111 views
Are there any commercial CFD codes that implement a Discontinuous Galerkin scheme?
I've been reading about the Discontinuous Galerkin discretization scheme and it's application to CFD for fluid flow. It seems to be a promising method for simulating turbulent flows, by using higher-...
1
vote
1answer
56 views
Obtain velocity from imposed energy spectrum using the inverse FFT
I am trying to obtain the spatial representation of $u(x)$ (e.g. velocity) from its energy spectrum $E(k)=k^4\exp(-(k/k_0)^2)$, which is given in the frequency domain, provided $|u(k)|=\sqrt{2E(k)}$. ...
1
vote
1answer
131 views
How to compute turbulent energy cascade
I need to compute the kinetic energy cascade using a finite volume solution in an equally spaced grid. I wonder if it is more correct to first compute the kinetic energy in the space (or time) domain, ...
3
votes
1answer
185 views
calculation of the right hand side of DG FEM (with code)
I got stuck with Hestaven/Warburton's dG-FEM Matlab code.
Starting with the file AdvecRHS1D.m, we see in line 11
...
2
votes
1answer
119 views
Differentiation Matrix In DG-FEM - Hesthaven/Warburton
In the book of Hesthaven and Warburton on discontinual Galerkin methods the authors give motivation to the differentiation matrix (page 52), referred to as $D_r(i,j)=\frac{dl_j}{dr}|_{r_i}$ where $l_i(...
5
votes
1answer
265 views
Stability of hyperbolic PDE and DG-FEM
In the book of Hesthaven and Warburton on discontinuous Galerlkin methods in example 2.3 (regarding solutions of the wave equation), the authors regard the following PDE:
$$\frac{\partial u }{\...
4
votes
1answer
250 views
DG-FEM integration by parts
I am going through the book of Hesthaven and Warburton on discontinuous Galerkin methods. I have difficulties understanding some basic steps in the calculations.
Consider the PDE:
$$\frac{\partial u}...
1
vote
0answers
390 views
Library for Discontinuous Galerkin method: FEniCS vs deal.ii
I am aware that both FEniCS and deal.ii are capable of solving problems with Discontinuous Galerkin (DG) method. I would like to specifically know if any of these two softwares can cater these ...
4
votes
1answer
98 views
$L^\infty$ stability property of an ODE
Suppose we have the initial-value problem on $(0,L)$:
$$
\frac{d u(x)}{d x} = f(x) u(x),\, \qquad x\in\Omega,\,~~ u(0) = u_0,
$$
I am reading a claim that says if we multiply the ODE by $u$ and ...
1
vote
1answer
189 views
Normalization of polynomials for discontinuous Galerkin methods (DGM)
I was curious if someone could share their opinion on this matter. I have noticed that some people in literature normalize their Legendre polynomials, i.e. divide or multiply the polynomial by $$\...
7
votes
1answer
242 views
Size of jump for piecewise discontinuous approximations
If one has a sufficiently smooth function $u$ that is approximated by a piecewise constant function $u_h=\Pi^0_h u$ on a mesh of cell size $h$ (where $\Pi^0_h$ is the $L_2$ projection onto the ...
1
vote
1answer
148 views
Penalization parameter for DG with jump penalization
I adapted this FEniCS code for my problem and I'm wondering if there is any good resource about how to choose the penalty parameter $\alpha$? Best case would be, if I can define it through some ...
1
vote
1answer
119 views
How is nonlinear flux interface term assembled for Discontinuous Galerkin method for hyperbolic conservation laws?
For example, for 1D Burgers equation
$$
u u_x = 0 \\
$$
equivalently,
$$
\frac{dF(u)}{dx} = 0\\
F(u)= \frac{u^2}{2}
$$
If I want to obtain $A_{ij},i\ne j$ for two DOFs ($U_i$ and $U_j$) of two ...
0
votes
1answer
256 views
Discontinuous Galerkin - Inhomogeneous Dirichlet B.C. for 1D Poisson Equation
I am trying to get some code working for the 1D Poisson equation using the textbook: Nodal Discontinuous Galerkin Methods Algorithms, Analysis, and Applications.
I use the following formulation (for ...
2
votes
1answer
715 views
Discontinuous Galerkin method VS Continuous Galerkin method Degrees of freedom
I was looking into the Book of Riviere " Discontinuous Galerkin Methods for solving Elliptic and Parabolic Equations". In the comparaison of section 2.12 (copied below), the example of rectangular ...
7
votes
1answer
296 views
Solving the Advection Equation with Forcing using the Discontinuous Galerkin Method
I've been learning about the Discontinous Galkerin Method by reading the book by Hesthaven and Warburton and have ran into a problem with the advection equation with forcing
$u_t + u_x = g(x,t)$
...
3
votes
0answers
107 views
Piecewise Constant Enrichment of the continuous galerkin method
I am interested to study crack propagation in a hyperelastic material in a variational setting. The crack surface exhibits a jump discontinuity. The function space for displacement field should ...
3
votes
1answer
139 views
Discontinuous Galerkin FEM : Control points are mid-points of edges instead of nodes
I am thinking to use discontinuous galerkin FEM (DGFEM) method to estimate discontinuous displacement field $u: \Omega \rightarrow \mathbb{R}^2$ at the crack surface of a material.
The domain is ...
1
vote
0answers
99 views
Do DG methods for the Helmholtz equation always return positive quantities?
Helmholtz Diffusion equation with reaction term:
$$
k\Delta u + u = f ~ \text{in} ~\Omega
$$
$$
\nabla u \cdot \mathbf{n} = 0 ~ \text{in} ~\partial \Omega
$$
For sufficiently small $k$ (relative to ...
1
vote
1answer
411 views
Interior penalty discontinuous Galerkin Matlab implementation
I want to solve the 2D poisson problem using the interior penalty discontinuous Galerkin methods:
āāa(x)(āu)=0 in Ī©.
The variational formulation is such that :
$$a_{\epsilon}(u,v)=\sum_{K\in T_h}\...
11
votes
1answer
437 views
DG local equation, how to interpret mean-averaged test function
In the paper http://www.sciencedirect.com/science/article/pii/S0045782509003521, an HDG element-local equation is described on page 584 equation (4), with one of the equations taking the following ...
1
vote
1answer
199 views
Viewing HDG FEM edge variables in vtk / paraview
For a 2D HDG code, I would like to be able to visualize the solution on the edge space between elements. Basically, this amounts to plotting the solution on the "green" nodes below.
Is there a ...
1
vote
2answers
474 views
Vandermonde matrix DG Hestaven
I am trying to understand the nodal and modal basis formulation from the book of Hesthaven (Nodal Discontinuous Galerkin Methods, Hesthaven, Jan S., Warburton, Tim). For $N=2$, I get the Vandermonde ...
2
votes
1answer
153 views
Higher order interpolation in DWR method
Based on page $35$ of the book:
(W. Bangerth and R. Rannacher, "Adaptive Finite Element Methods for Solving Differential Equations", BirkhƤuser, 2003,)
for computing the error in dual weighted ...
1
vote
1answer
194 views
Orthonormal basis for hexahedron
Orthogonal polynomials are often preferred as basis functions. Recently I learned selecting orthonormal basis further simplifies the mass matrix from diagonal to simply the identity matrix when used ...
1
vote
0answers
46 views
How do we implement Parameter free generalised Moment limiter in 1D Case in Discontinuous Galerkin methods?
I am referring to this paper:-
"A Parameter-Free Generalized Moment Limiter for High-
Order Methods on Unstructured Grids " by Michael Yang and Z.J. Wang.
http://dept.ku.edu/~cfdku/papers/AIAA-2009-...
1
vote
1answer
570 views
Plot 2D piecewise constant in matlab in a finite elements mesh
I need to generate a discontinuous plot (piecewise in each triangle) in matlab, something like:
This plot is from http://www.alecjacobson.com/weblog/?p=3616, but I don't understand how generate it.
...
1
vote
0answers
95 views
Discontinuos Galerkin Method - inhomogeneous advection problem
I'm currently trying to get into this topic. I've learned that the basic scheme for the advection problem ($D_{x}u+a*D_{t}u=0$) can be solved in a scheme like $$ M^{k}\frac{d}{dt}u^{k}_{h}-(S^{k})^{T}...
1
vote
0answers
281 views
How to perform Taylor series expansion consisting of cell averaged derivatives in a computational element?
I have encountered the term cell "Cell averaged derivatives" and "Taylor series expansion using cell averaged derivatives about centroid" in the context of polynomial representation in any cell (...
1
vote
0answers
122 views
How do we derive the elemental equation for Discontinuous Galerkin method using Centered Numerical Flux?
If we take a 1st order polynomial approximation in each cell, we can find the (2x2) -mass matrix, differentiation matrix, and flux matrix through the integration of Lagrange polynomials.
However, I ...
3
votes
1answer
204 views
Is it possible to solve Euler equation numerically without using any flux limiter (in DG scheme)?
I have recently learned about Discontinuous Galerkin method to solve differential equations and I was trying to implement it to solve Euler equation. For now, consider the standard Sod Shock Tube Case....
3
votes
0answers
659 views
1D Discontinuous Galerkin - Lagrange vs Legendre Basis
Consider the 1D advection equation in its strong and weak forms
$$\ u_t + a u_x = 0 $$
$$\ \int_{x_{j-0.5}}^{x_{j+0.5}} w u_t \ dx - a \int_{x_{j-0.5}}^{x_{j+0.5}} w_x u \ dx + a [w(x)\hat{u}]_{...
9
votes
1answer
626 views
CFL condition in Discontinuous Galerkin schemes
I have implemented an ADER-Discontinuous Galerkin scheme for the resolution of linear systems of conservation laws of the type of $\partial_t U + A \partial_x U + B \partial_y U=0 $ and observed that ...
5
votes
1answer
283 views
Numerical quadrature in Discontinuous Galerkin
I would like to know which is the best way to integrate numerically Legendre polynomials. I am building up a Discontinuous Galerkin CFD code for which Legendre polynomials are used as basis functions ...
1
vote
1answer
361 views
Library for generating Discontinuous Galerkin FEM mesh
There are a number of packages available for generating Continuous Galerkin (CG) FEM meshes (gmsh, tetgen, netgen, etc.), but I have been unable to find one that generates Discontinuous Galerkin (DG) ...
2
votes
1answer
615 views
Spectral methods, Spectral Volume methods, Spectral Difference methods
Could someone explain the link (if any) between the spectral methods (SM), as presented for example here
and the so called spectral volume methods (SV) and spectral difference methods (SD) for CFD ?
...
1
vote
0answers
295 views
Package for Discontinuous Galerkin method
I am trying to find some package of Discontinuous Galerkin (DG) method for solving hyperbolic and parabolic equations.
In my research, I focus on designing new schemes for some very simple equations ...
1
vote
0answers
73 views
Time integration for elastodynamics
I'd like to solve the elastodynamics equation for a problem with fast scales from an impact (although no shocks).
$\rho\ddot{\mathbf{u}} - \nabla \cdot \sigma(\mathbf{u}) = \mathbf{f}$
In structural ...
