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).
0
votes
1answer
47 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
90 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
vote
1answer
142 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
158 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
72 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
81 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 ...
2
votes
0answers
82 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 ...
0
votes
1answer
164 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}\...
10
votes
1answer
262 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
136 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
211 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
84 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 ...
1
vote
1answer
118 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
36 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
206 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
81 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
121 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
66 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 ...
4
votes
1answer
149 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
326 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}]_{...
10
votes
1answer
318 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 ...
6
votes
1answer
178 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
256 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
381 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
190 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
66 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 ...
3
votes
0answers
69 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 ...
13
votes
1answer
613 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 ...
4
votes
1answer
72 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
vote
1answer
280 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}(\...
4
votes
1answer
382 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-...
6
votes
1answer
166 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 ...
5
votes
1answer
146 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 $\frac{1}{2k+1}$...
6
votes
1answer
249 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:
...
0
votes
1answer
215 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 ...
1
vote
1answer
2k 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 ...
1
vote
1answer
167 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 ...
1
vote
0answers
98 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 ...
1
vote
0answers
179 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 ...
5
votes
2answers
467 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{\...
2
votes
1answer
755 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 ...
3
votes
2answers
398 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 ...
0
votes
1answer
416 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:
$$
\...
4
votes
2answers
1k views
The meaning of conservative discretization in Galerkin FEM and Discontinuous Galerkin
I do understand the meanning of "conservative discretization" within the FVM/FDM framework, indeed it is well explained in this post.
Now, according to the table in this slide (pp.8), it concludes:
...
1
vote
0answers
435 views
Jacobian-Free Newton-Krylov vs explicitly forming jacobian in DG
For a given discontinuous galerkin (DG) implementation for Navier-Stokes, targeting 10,000 to 1,000,000 4th order cells in 3D, I'm using PETSc's suite of linear/non-linear solvers on the back-end. It ...
6
votes
1answer
416 views
How to project a vector into the H(div) space (in the context of finite elements)?
Say I have a simple elliptic PDE:
$$
-\nabla\cdot(K\nabla p) = f \;\;\;\text{in}\;\Omega
$$
with the appropriate boundary conditions. I solve for $p$ using a FEM (a discontinuous Galerkin method to ...
2
votes
1answer
333 views
Shallow Water Equations Boundary Conditions
I am trying to solve shallow water equations using DG methods. Flow over a bump is a common problem that comes up in this context. For example (http://loki.udc.es/info/asignaturas/calculo_ii/Finite%...
1
vote
1answer
210 views
Discontinuous Galerkin, residual orthogonal to test functions?
I am a little confused about where does the mass and stiff matrix come from. In Discontinuous Galerkin we divide the domain in elements, $\Omega = \cup^K_{k=1} D^k$. Then assume the solution $u$ can ...
1
vote
0answers
954 views
CFL condition and Lax-Friedrich numerical flux
I got confused when trying to implement a scheme using Lax-Friedrichs numerical flux for a system of equations in 1D.
According to my notes Lax-Friedrichs numerical flux is
$$f_{LF}(u_l,u_r) = \frac{...
2
votes
2answers
315 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 $v_j^k$ ...
