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).
116
questions
1
vote
0
answers
42
views
Discontinuous Galerkin for transport equation with non-constant advection
This question is mainly an inquiry about the usefulness of Discontinuous Galerkin (DG) for the time independent transport equation of the form
$$\sigma u+\beta\cdot\nabla u =f,\;\;\;\text{on }\Omega\...
- 111
2
votes
1
answer
94
views
Cell-based vs face-based finite element methods
Notation:
Denote $T_{h} = \left\{K\right\}$ to be a face-conforming triangulation of a domain $\Omega$ such that $K_{i} \cap K_{j} = \emptyset$ for $i \neq j.$ Additionally, denote $\mathcal{V}_{h} = ...
- 21
0
votes
0
answers
50
views
solve a coupled PDE system with some discontinuity by a mixed FEM
$$
\begin{cases}
u_t=f(u,v,\nabla{v}),\\
\Delta{[g(u)v]}=0.
\end{cases}$$
I want to solve the above PDE system by a mixed FEM, that is, $u_t=f(u,v,\nabla{v})$ by discontinuous Galerkin (DG), where $f$...
- 207
0
votes
1
answer
91
views
How can I solve this PDE system by discontinuous Galerkin method?
As is known to all, the discontinuous Galerkin method (DG) was first used to solve the equation $u_t+u_x=0$. Now I have the following system of PDEs:
$$
\begin{cases}
u_t=f(u,v,\nabla{v}),\\
\Delta{v}...
- 207
1
vote
0
answers
125
views
Are both centered and upwind numerical fluxes correct in DGTD?
In discontinuous Galerkin time domain (DGTD) method, a critical concept is the numerical flux that is used to link neighbouring elements. The numerical flux is however not unique. The popular choices ...
- 302
2
votes
2
answers
181
views
Errors imposing boundary conditions weakly with DG
I am using interior penalty discontinuous Galerkin to solve a simple Laplace problem:
\begin{align*}
\nabla u=0
\end{align*}
with prescribed 0 and 1 Dirichlet boundary conditions on opposite edges of ...
- 306
0
votes
1
answer
89
views
$u_t+a u_x=0$ solved by Discontinuous Galerkin
If $u_t+a \cdot u_x=0$ under a periodic boundary condition (to mimic an infinite domain) is solved by Discontinuous Galerkin (DG), how to implement periodic boundary condition and the other details in ...
- 207
2
votes
0
answers
67
views
Modal representations of nodal tensor product Galerkin elements
Nodal discontinuous Galerkin methods on simplices, like those described in Hesthaven and Warburton, have the nice property that the number of nodes is equal to the minimum number needed to represent a ...
- 2,273
1
vote
1
answer
100
views
Integration of (d-1)-dimensional functions on finite element surfaces
I am trying to integrate a function $\hat u$ on the common surface of discontinuous finite elements. The function $\hat u$ lives in a $d-1$-dimensional space of functions defined on the element ...
- 63
0
votes
0
answers
145
views
Discontinuous Galerkin failing to converge Euler equations under p-refinement
I am solving the steady state compressible Euler equations in conservation form in 2D in a rectangular domain with a discontinuous Galerkin (DG) code I have developed. As a test, the boundary ...
- 63
0
votes
0
answers
46
views
Can you describe the Galerkin numerical method to solve the wave equation?
How would you describe the Galerkin method to solving the 3D wave equation
$$u_{tt}= c^2\Delta u$$
to someone who wants to implement it immediately?
More precisely, we want to solve the Cauchy problem
...
- 151
1
vote
1
answer
80
views
How to Impose nonhomogeneous Neumann Boundary Condition in the DG Formulation
Consider the following partial differential equation
\begin{align}
\frac{\partial u}{\partial t}+\frac{\partial f}{\partial x} &= g(x,t), \ \ x\in \Omega = [x_{L},x_{R}] \\
u(x,0) &= u_{0}(x) ...
- 83
2
votes
1
answer
147
views
DG method for solving Hyperbolic Partial Differential Equation with Dirichlet Boundary Conditions
Consider the following partial differential equation
\begin{align}
\frac{\partial u}{\partial t}+\frac{\partial f}{\partial x} &= g(x,t), \ \ x\in \Omega = [x_{L},x_{R}] \\
u(x,0) &= u_{0}(x) ...
- 83
1
vote
0
answers
99
views
Is it possible to use a fixed point iteration for solving this nonlinear system?
Consider the following differential equation
\begin{align}
\frac{\partial f(u)}{\partial x} &= g(x), \ \ x\in [x_{L},x_{R}] \label{Eq2.2} \\
u(x_{L}) &= g_{1}
\end{align}
where $f(u)$ is a ...
- 83
5
votes
1
answer
142
views
Analysis of nonlinear finite element methods
I have been doing a lot of reading on the development of finite element methods and their analysis using, e.g., functional analysis. I am clear on the formulation of the weak form of a PDE and ...
- 63
3
votes
0
answers
68
views
Typo in a-priori error estimate in a Discontinuous Galerkin paper
I'm looking at this famous paper which is available in the link below:
Franco Brezzi, LD Marini, Endre Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Mathematical Models ...
- 211
2
votes
1
answer
115
views
Discontinuous Galerkin order of convergence on arbirary refined mesh: step-12 deal.ii tutorial
I'm learning DG methods and in order to practice a little bit I'm using the deal.ii library. In particular, I'm looking at step-12, where they solve
$$\operatorname{div}(\beta u) = 0$$
$$u = g_D \text{...
- 211
4
votes
1
answer
202
views
Discontinuous Galerkin: confusion about the weak formulation for linear advection equation
In an introduction to Discontinuous galerkin methods, I have some problems in checking the weak formulation. I'm looking at page 16 here
The context is the advection reaction equation: $$\operatorname{...
- 211
2
votes
1
answer
162
views
discretizing surface integral using nodal DG method
I am currently learning nodal DG methods, primarily through the book by Warburton, and am a bit confused on how to handle surface integrals using straight edged elements. On page 187 (and on page 214)...
- 23
6
votes
1
answer
1k
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.)
...
- 128
3
votes
0
answers
96
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)
$$
\Delta t \approx \frac{h}{N^2} \frac{C}{|u| + |c| + \frac
{N^2 \mu}{h}}
$$
($...
- 141
1
vote
0
answers
71
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 ...
- 810
2
votes
1
answer
159
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 ...
user36293
2
votes
1
answer
179
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 ...
- 75
1
vote
2
answers
229
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 ...
- 621
1
vote
0
answers
49
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 ...
- 621
0
votes
1
answer
172
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 ...
- 81
2
votes
1
answer
246
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 <...
- 155
4
votes
1
answer
247
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 ...
- 81
1
vote
0
answers
73
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/...
- 11
1
vote
1
answer
163
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-...
- 121
1
vote
1
answer
82
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 $|u(k)|=\sqrt{2E(k)}$. ...
- 173
1
vote
1
answer
236
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, ...
- 173
3
votes
1
answer
236
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
...
- 295
2
votes
1
answer
162
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(...
- 295
6
votes
1
answer
326
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:
$$\frac{\partial u }{\...
- 295
4
votes
1
answer
353
views
DG-FEM integration by parts
I am going through the book of Hesthaven and Warburton on discontinuous Galerkin methods. I have difficulties understanding some basic steps in the calculations.
Consider the PDE:
$$\frac{\partial u}...
- 295
1
vote
1
answer
939
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 ...
- 141
8
votes
1
answer
342
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 ...
- 642
1
vote
1
answer
230
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 $$\...
- 75
8
votes
1
answer
308
views
Size of jump for piecewise discontinuous approximations
If one has a sufficiently smooth function $u$ that is approximated by a piecewise constant function $u_h=\Pi^0_h u$ on a mesh of cell size $h$ (where $\Pi^0_h$ is the $L_2$ projection onto the ...
- 52.4k
1
vote
1
answer
229
views
Penalization parameter for DG with jump penalization
I adapted this FEniCS code for my problem and I'm wondering if there is any good resource about how to choose the penalty parameter $\alpha$? Best case would be, if I can define it through some ...
- 13
1
vote
1
answer
134
views
How is nonlinear flux interface term assembled for Discontinuous Galerkin method for hyperbolic conservation laws?
For example, for 1D Burgers equation
$$
u u_x = 0 \\
$$
equivalently,
$$
\frac{dF(u)}{dx} = 0\\
F(u)= \frac{u^2}{2}
$$
If I want to obtain $A_{ij},i\ne j$ for two DOFs ($U_i$ and $U_j$) of two ...
- 131
0
votes
1
answer
345
views
Discontinuous Galerkin - Inhomogeneous Dirichlet B.C. for 1D Poisson Equation
I am trying to get some code working for the 1D Poisson equation using the textbook: Nodal Discontinuous Galerkin Methods Algorithms, Analysis, and Applications.
I use the following formulation (for ...
- 1
2
votes
1
answer
1k
views
Discontinuous Galerkin method VS Continuous Galerkin method Degrees of freedom
I was looking into the Book of Riviere " Discontinuous Galerkin Methods for solving Elliptic and Parabolic Equations". In the comparaison of section 2.12 (copied below), the example of rectangular ...
7
votes
1
answer
334
views
Solving the Advection Equation with Forcing using the Discontinuous Galerkin Method
I've been learning about the Discontinous Galkerin Method by reading the book by Hesthaven and Warburton and have ran into a problem with the advection equation with forcing
$u_t + u_x = g(x,t)$
...
- 397
3
votes
0
answers
123
views
Piecewise Constant Enrichment of the continuous galerkin method
I am interested to study crack propagation in a hyperelastic material in a variational setting. The crack surface exhibits a jump discontinuity. The function space for displacement field should ...
- 95
3
votes
1
answer
173
views
Discontinuous Galerkin FEM : Control points are mid-points of edges instead of nodes
I am thinking to use discontinuous galerkin FEM (DGFEM) method to estimate discontinuous displacement field $u: \Omega \rightarrow \mathbb{R}^2$ at the crack surface of a material.
The domain is ...
- 95
1
vote
0
answers
108
views
Do DG methods for the Helmholtz equation always return positive quantities?
Helmholtz Diffusion equation with reaction term:
$$
k\Delta u + u = f ~ \text{in} ~\Omega
$$
$$
\nabla u \cdot \mathbf{n} = 0 ~ \text{in} ~\partial \Omega
$$
For sufficiently small $k$ (relative to ...
- 591
0
votes
1
answer
621
views
Interior penalty discontinuous Galerkin Matlab implementation
I want to solve the 2D poisson problem using the interior penalty discontinuous Galerkin methods:
−∇a(x)(∇u)=0 in Ω.
The variational formulation is such that :
$$a_{\epsilon}(u,v)=\sum_{K\in T_h}\...