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
12
votes
1
answer
574
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 ...
- 642
1
vote
1
answer
240
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 ...
- 642
1
vote
2
answers
668
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 ...
- 13
2
votes
1
answer
196
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", Birkhäuser, 2003,)
for computing the error in dual weighted ...
- 523
1
vote
1
answer
221
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 ...
- 233
1
vote
0
answers
52
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-...
- 163
2
votes
1
answer
879
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.
...
- 515
1
vote
0
answers
106
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 $$ M^{k}\frac{d}{dt}u^{k}_{h}-(S^{k})^{T}...
- 11
1
vote
0
answers
360
views
How to perform Taylor series expansion consisting of cell averaged derivatives in a computational element?
I have encountered the term cell "Cell averaged derivatives" and "Taylor series expansion using cell averaged derivatives about centroid" in the context of polynomial representation in any cell (...
- 163
1
vote
0
answers
149
views
How do we derive the elemental equation for Discontinuous Galerkin method using Centered Numerical Flux?
If we take a 1st order polynomial approximation in each cell, we can find the (2x2) -mass matrix, differentiation matrix, and flux matrix through the integration of Lagrange polynomials.
However, I ...
- 163
3
votes
1
answer
233
views
Is it possible to solve Euler equation numerically without using any flux limiter (in DG scheme)?
I have recently learned about Discontinuous Galerkin method to solve differential equations and I was trying to implement it to solve Euler equation. For now, consider the standard Sod Shock Tube Case....
- 163
3
votes
1
answer
948
views
1D Discontinuous Galerkin - Lagrange vs Legendre Basis
Consider the 1D advection equation in its strong and weak forms
$$\ 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}]_{...
- 31
10
votes
2
answers
906
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 ...
- 173
5
votes
1
answer
405
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 ...
- 173
1
vote
1
answer
435
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) ...
- 674
1
vote
1
answer
701
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 ?
...
- 55
1
vote
2
answers
395
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 ...
- 246
1
vote
0
answers
87
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 ...
- 601
2
votes
0
answers
118
views
FEM libraries with trace spaces
To implement hybridizable discontinuous Galerkin methods, one needs finite element spaces defined on the skeleton of the mesh. deal.II has support for HDG through FE_FaceQ class which provides ...
- 2,993
15
votes
3
answers
1k
views
Role of the numerical flux in DG-FEM
I am learning the theory behind DG-FEM methods using the Hesthaven/Warburton book and I am a bit confused about the role of the 'numerical flux.' I apologize if this is a basic question, but I have ...
- 642
3
votes
1
answer
112
views
In mixed elliptic formulation, what are the weakest requirements to ensure the flux is in $H^1$?
In the book Mixed Finite Element Methods and Applications by Boffi, Brezzi, and Fortin there is a pretty long discussion about why the Raviart-Thomas (RT) projection is only defined for functions in $...
- 1,000
1
vote
1
answer
666
views
Discretization of lifting operator in BR2 scheme
The lifting operator $\mathbf{r}(\mathbf{v_h})$ for the '2nd version' of Bassi-Rebay scheme for elliptic problems in $d$ dimensions is defined as
$$
\int_{\Omega_h} \mathbf{w}_h \cdot \mathbf{r}(\...
- 405
3
votes
1
answer
505
views
Intro to DG Finite Element methods
I wrote a number of 1D/2D FE and FD programs as a bachelor student, but the main problem I continually came into contact with was gradient shocks related to convection/diffusion problems in convection-...
- 674
6
votes
1
answer
329
views
Velocity-Stress formulation of Elastodynamics/Wave Equation for beginner
I'm used to displacement forumlation of elastic wave equation:
$$
\nabla \cdot \sigma (u) + F = \rho \ddot{ u }
$$
where $u$ is the primary variable. Recenty I started experimenting with DG and in ...
- 219
5
votes
1
answer
364
views
Proof of CFL condition for RKDG scheme
The cfl condition for linear advection equation
$$
u_t + a u_x = 0
$$
using a DG method of degree $k$ polynomials, upwind flux and an RK scheme of $k+1$ stage/accuracy is stated to be $\frac{1}{2k+1}$...
- 2,993
5
votes
1
answer
558
views
Choosing the penalty for Discontinuous Galerkin
So I am attempting to solve a 3D poisson problem with Discontinuous Galerkin (interior penalty method). The weak form (written in FEniCS) is as following:
...
- 781
0
votes
1
answer
778
views
Order of accuracy of DGFEM or FEM
I know that it is possible to determine the theoretical order of accuracy (B) of numerical solutions in FVM (for instance for a steady problem in which only the central differencing scheme (CDS) is ...
- 301
3
votes
1
answer
4k
views
what is the difference between non-conformal and conformal?
So far what I understand is that two neighbouring elements are conformal if their edges and faces match exactly, whereas with non-conformal elements this is not the case. For instance, h-refinement ...
- 301
1
vote
1
answer
241
views
Are FEM or DGFEM methods based on integrals or PDEs?
I know that FVM is based on the integral form of conservation laws, and FDS is based on PDEs. What I'm confused by, is whether FEM and DGFEM formulations are based on integral or pde form of ...
- 301
1
vote
0
answers
128
views
understanding interior penalty jump of basis function
If cardinal basis functions are used (i.e. $\psi_{ij}=1$ iff $i=j$, and 0 otherwise) in interior penalty methods for elliptic equations such as SIPG & NIPG, shoudn't the jump in basis functions ...
- 233
1
vote
0
answers
385
views
Solving the Jump function in discontinuous Galerkin method
I am trying to solve the elasticity equation using Crouzeix-Raviart element. Since this can not be solved using continuous FEM I am trying to solve this by discontinuous Galerkin method. In the ...
- 11
5
votes
2
answers
789
views
Absorbing boundary conditions for acoustics in Discontinuous Galerkin
Note: I'm trying to implement a Discontinuous Galerkin method, as kind of a way to learn about these things.
As of now, I've taken the acoustic wave equation $c^2 \nabla \cdot \nabla u(x,t) - \frac{\...
- 301
2
votes
1
answer
1k
views
Matlab implementation of 2D Interior penalty discontinuous Galerkin poisson problem
Basically, I am trying to solve the 2D poisson problem in order to learn implementation of IPDG methods. The problem states $-\nabla a(x)(\nabla u)=0\ \text{in} \ \Omega$ with $U=0$ on Dirichlet ...
- 21
3
votes
2
answers
738
views
What's the difference between C0 penalty methods and Discontinuous Galerkin methods?
I am trying to understand the DG FEM methods, but I got lost in their definitions. In some papers I read that the "C0 penalty method" is one example of the DG method, but sometimes they are separated ...
- 97
0
votes
1
answer
849
views
How to impose Neumann boundary conditions in interior penalty DG method
Consider the following two point BVP:
$$
-u''(x)=f(x),~~~u(0)=u(1)=0.
$$
An interior penalty DG method for this BVP that weakly imposes homogeneous Dirichlet boundary conditions is of the form:
$$
\...
- 587
4
votes
2
answers
2k
views
The meaning of conservative discretization in Galerkin FEM and Discontinuous Galerkin
I do understand the meanning of "conservative discretization" within the FVM/FDM framework, indeed it is well explained in this post.
Now, according to the table in this slide (pp.8), it concludes:
...
- 593
2
votes
0
answers
623
views
Jacobian-Free Newton-Krylov vs explicitly forming jacobian in DG
For a given discontinuous galerkin (DG) implementation for Navier-Stokes, targeting 10,000 to 1,000,000 4th order cells in 3D, I'm using PETSc's suite of linear/non-linear solvers on the back-end. It ...
- 2,293
6
votes
1
answer
744
views
How to project a vector into the H(div) space (in the context of finite elements)?
Say I have a simple elliptic PDE:
$$
-\nabla\cdot(K\nabla p) = f \;\;\;\text{in}\;\Omega
$$
with the appropriate boundary conditions. I solve for $p$ using a FEM (a discontinuous Galerkin method to ...
- 937
2
votes
1
answer
669
views
Shallow Water Equations Boundary Conditions
I am trying to solve shallow water equations using DG methods. Flow over a bump is a common problem that comes up in this context. For example (http://loki.udc.es/info/asignaturas/calculo_ii/Finite%...
- 322
1
vote
1
answer
304
views
Discontinuous Galerkin, residual orthogonal to test functions?
I am a little confused about where does the mass and stiff matrix come from. In Discontinuous Galerkin we divide the domain in elements, $\Omega = \cup^K_{k=1} D^k$. Then assume the solution $u$ can ...
- 1,019
2
votes
1
answer
2k
views
CFL condition and Lax-Friedrich numerical flux
I got confused when trying to implement a scheme using Lax-Friedrichs numerical flux for a system of equations in 1D.
According to my notes Lax-Friedrichs numerical flux is
$$f_{LF}(u_l,u_r) = \frac{...
- 1,019
2
votes
2
answers
568
views
How to prove the strong formulation for Discontinuous Galerkin is conservative?
I've been trying to figure out how to prove that the following equation is conservative
$$\int_{D^k} \partial_t(u^k) v_j^k + \partial_x(f) v_j^k dx = [(f -f^*) v^k_j]_{x^k}^{x^{k+1}}$$
where $v_j^k$ ...
- 1,019
1
vote
2
answers
223
views
Discontinuous Galerkin energy method
I am studying DG for conservation laws from this book.
Local inner product is defined like
$$(u,v)_{D^k} = \int_{D^k} uv dx$$
and the $L^2(D^k)$-norm as
\begin{equation}
(u,u)_{D^k} = ||u||^2_{...
- 1,019
2
votes
1
answer
332
views
Problem with implementing linear advection using DG-method
I am trying to implement a second order DG-method using a monomial basis and explicit Euler in time. I have written down some of the theory, which I present below:
Theory
Consider the linear ...
- 43
5
votes
1
answer
368
views
Are additional penalty terms necessary to solve elliptic PDE's with DG-FEM?
After a cursory glance at several references to discontinuous galerkin finite element methods for the elliptic poisson PDE, i notice that all of them emphasize using penalty methods where an ...
- 11.9k
3
votes
0
answers
159
views
Mixed DG for Poisson with mixed BC's
I am trying to find a good reference on a proper weak formulation for mixed DG (Raviart Thomas and DG) formulation for a Poisson equation with mixed boundary conditions. Can anyone suggest a good ...
- 81
2
votes
1
answer
210
views
Optimal Discontinuous Galerkin (DG) solver on a parallel system
I am seeking the optimal method for implementing DG on a parallel system. For my research, I come across two types of problems.
For the first problem, I am solving a time-independent (steady-state) ...
1
vote
1
answer
367
views
Evaluation of interface terms in Discontinuous Galerkin method
I would like to ask how is the evaluation of integrals over inter-element interfaces implemented in a typical DG code. I can think of two basic approaches (I assume 2D mesh here):
1) Perform 1D ...
- 405
3
votes
1
answer
569
views
Recommendations on FEM software for implementing Nitsche's method on interfaces between matching meshes?
Suppose:
I have two domains, $\Omega_{1} = [0, 1/2] \times [0, 1]$ and $\Omega_{2} = [1/2, 1] \times [0, 1]$.
The domains share an interface $\Gamma = \{1/2\} \times [0, 1] = \partial\Omega_{1} \cap \...
- 30.1k
5
votes
1
answer
1k
views
Which finite-element framework allows the easy implementation of HDG methods?
I have tinkered around with the implementation of hybridizable Discontinuous Galerkin (HDG) methods in DUNE-PDELab and DUNE-FEM for my PhD but since then I have not had the time to work on these codes ...
- 1,074