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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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3 votes
0 answers
62 views

Is AMG supposed to work with discontinuous Galerkin discretizations?

As the question says, are algebraic multigrid methods well suited to be used as preconditioners for problems discretised with Discontinuous Galerkin methods (say $p=1$)? I've always used AMG (actually,...
0 votes
1 answer
95 views

Discontinuous Galerkin for Stokes flow

Greetings fellow members, I'm trying to implement a Discontinuous Galerkin scheme for a Stokes flow (Poiseuille). While I get very satisfactory results on the velocity, I'm suprised with negative ...
0 votes
1 answer
99 views

Constructing metric terms for high order elements

Given 27 $(x,y,z)$ coordinates in 3D space which describe a generally curved quadratic hexahedron, which correspond to the HEXA_27 reference element figure with planar faces in $(\xi, \eta, \zeta)$ ...
2 votes
1 answer
176 views

Galerkin projection in AMG

In the context of Classical AMG for elliptic problems discretised with finite elements (DG or CG), one has the (fine) matrix of the problem, say $A_0$, and the coarser operators of the hierarchy $\{...
0 votes
1 answer
114 views

Intergrid transfer operator for Discontinuous Galerkin method (multigrid)

I am studying DG and I've seen that it's possible to define a multigrid method. Let's consider two nested grids $\Omega_{l-1}$ and $\Omega_{l}$ and $V_{l-1}$ and $V_l$ classical DG spaces, both of ...
3 votes
1 answer
211 views

Iterative solver for high order DG methods (3D Laplace problem)

I have a 3D Laplace problem on quite a complicated geometry where I am using Discontinuous Galerkin method. My mesh is composed by hexas, hence I am employing classical tensor product basis functions $...
3 votes
1 answer
278 views

Finite difference problem

I have a problem to resolve with the Finite Difference method in $[a,b]$: $$-\frac{d}{dx}(\alpha(x)\frac{du}{dx})= g(x),$$ with $\alpha(x) \in L^{\infty}$ continuous in $]a,c[$ and $]c,b[$ and ...
3 votes
1 answer
183 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\...
8 votes
1 answer
508 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 ...
5 votes
1 answer
293 views

Does the weighted residual method not use energy minimization in any form?

I've come across several texts/papers utilizing the concept of a minimum potential energy state corresponding to an equilibrium state, and I know that it is used in FEM formulations that are based on ...
2 votes
1 answer
1k 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. ...
10 votes
2 answers
1k 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 ...
12 votes
1 answer
579 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
1 answer
1k 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 ...
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{...
2 votes
1 answer
137 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} = ...
0 votes
0 answers
61 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 $f$...
0 votes
1 answer
118 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}...
2 votes
2 answers
250 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 ...
1 vote
0 answers
136 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 ...
0 votes
1 answer
115 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 ...
3 votes
1 answer
1k 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}]_{...
1 vote
2 answers
455 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 ...
2 votes
0 answers
93 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 ...
1 vote
1 answer
141 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 ...
0 votes
0 answers
155 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 ...
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 ...
0 votes
0 answers
53 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 ...
1 vote
1 answer
94 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) ...
2 votes
1 answer
189 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) ...
1 vote
0 answers
101 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 ...
5 votes
1 answer
154 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 ...
3 votes
0 answers
72 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 ...
4 votes
1 answer
250 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{...
2 votes
1 answer
121 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{...
2 votes
1 answer
202 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)...
6 votes
1 answer
2k 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.) ...
3 votes
0 answers
99 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
0 answers
72 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 ...
14 votes
0 answers
524 views

Sequential approach to solving coupled PDEs

I'm dealing with a coupled system of three transient, non-linear convection-diffusion equations. Let's just say to simplify the problem that they take the following form: $$ -\nabla\cdot(D_{1}(u_{2},...
0 votes
1 answer
184 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
1 answer
219 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 ...
1 vote
2 answers
267 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 ...
2 votes
1 answer
239 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
0 answers
54 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 ...
2 votes
1 answer
324 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 <...
4 votes
1 answer
346 views

Quadrature and quadrature-free discontinuous galerkin method for non-linear PDE

Quadrature-free DG method using nodal Lagrangian basis are computationally very efficient. I have seen many papers using this method for linear PDE but almost no literature for non-linear PDE like ...
4 votes
1 answer
333 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 ...
1 vote
0 answers
77 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
1 answer
187 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-...