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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).

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1answer
63 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 ...
0
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1answer
76 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 ...
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1answer
107 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 ...
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1answer
171 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
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1answer
171 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
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0answers
76 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
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1answer
87 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
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0answers
88 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
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1answer
176 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
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1answer
284 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 ...
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1answer
141 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 ...
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2answers
229 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
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1answer
90 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 ...
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1answer
127 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 ...
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0answers
38 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-...
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1answer
229 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. ...
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0answers
82 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}...
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0answers
153 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 (...
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0answers
72 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
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1answer
152 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
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0answers
368 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
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1answer
337 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
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1answer
185 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 ...
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1answer
266 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
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1answer
395 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 ? ...
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0answers
204 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 ...
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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
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0answers
74 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
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2answers
666 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
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1answer
73 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 $...
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1answer
295 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
388 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
177 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
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1answer
153 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
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1answer
268 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
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1answer
237 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
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1answer
175 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 ...
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0answers
99 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 ...
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0answers
192 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
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2answers
488 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
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1answer
793 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
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2answers
407 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 ...
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1answer
435 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: $$ \...
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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: ...
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0answers
452 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
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1answer
428 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
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1answer
342 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%...
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1answer
214 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 ...
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0answers
981 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{...