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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129 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 ...
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1answer
210 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....
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0answers
702 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}]_{...
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1answer
676 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 ...
5
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1answer
303 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
379 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) ...
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1answer
632 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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303 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
76 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 ...
2
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0answers
107 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 ...
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2answers
683 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{\...
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1answer
496 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}(\...
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1answer
99 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 $...
3
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1answer
466 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-...
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1answer
300 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 $\frac{1}{2k+1}$...
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1answer
467 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: ...
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1answer
530 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 ...
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1answer
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 ...
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1answer
230 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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121 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
345 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 ...
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2answers
624 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
726 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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1answer
533 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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0answers
589 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 ...
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1answer
596 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 ...
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1answer
271 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
1k 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{...
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2answers
467 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 $v_j^k$ ...
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2answers
204 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_{...
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1answer
312 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 ...
5
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1answer
326 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 ...
3
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0answers
137 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 ...
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1answer
193 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) ...
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1answer
324 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 ...
3
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1answer
499 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 \...
5
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1answer
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 ...
3
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2answers
496 views

Nodal DG method and limiters for hyperbolic conservation laws

All the papers I have seen on DG methods for hyperbolic conservation laws together with limiters to compute discontinuous solutions make use of Taylor polynomial basis (Pk basis) or Legendre ...
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5answers
1k views

Is discontinuous Galerkin really any more parallelizable than continuous Galerkin?

I've always heard that easy parallelization was one of the advantages of DG methods, but I don't really see why any of those reasons don't also apply to continuous Galerkin.
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1answer
616 views

How to handle inflow and outflow boundaries for a non-linear convection-diffusion equation (DGFEM)

Following "A conservative DGM for Convection-Diffusion and Navier-Stokes Problems" (Oden and Baumann), if we have a linear convection-diffusion equation of the following form: $$ \nabla\cdot(\mathbf{b}...
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2answers
1k views

Slow convergence of Newton's method for finite elements

The application is a simple non-linear advection diffusion problem (steady state) using DGFEM. My error at each iteration is given by $$ e_{n+1} = ||\mathbf{J}^{-1}(\mathbf{u}_{n})\mathbf{F}(\mathbf{u}...
3
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1answer
350 views

Convergence of interior penalty DG methods

I’m currently having some issues with my routine for the linear advection-diffusion problem. The model problem is as follows: $$ \nabla\cdot(\mathbf{s} u) - \nabla\cdot(\kappa\nabla u) = f, \;\;\;\...
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2answers
136 views

Convergence of adaptive finite elements with inexact solves

I'm working on some adaptive discontinuous Galerkin codes for time harmonic wave propagation, currently just Helmholtz, but will be branching out once I have a working prototype in this case. There ...
6
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2answers
810 views

Local inversion of small matrices on GPUs?

I don't know much about GPU computing at the moment, so please pardon the simple question. Can one invert local matrices in parallel on the GPU? CUBLAS doesn't seem to support factorization, and most ...
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2answers
1k views

How to properly use polynomial projection to get values at visualization nodes?

I am trying to implementing a nodal discontinuous Galerkin spectral element method for linear and non-linear systems of equations. The solution at each time step is given at ...
5
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0answers
203 views

Discontinuous Galerkin for flow through porous media

I am struggling with DG methods for 2 phase flow through porous media. I managed to get the global pressure, total flux equations to work with an unconditionally stable mixed FE DG formulation as ...
8
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1answer
149 views

Testing and visualizing large index arrays

I will be implementing nodal discontinuous Galerkin method soon, and having done this before I know the basic indexing arrays I will need to compute, given a mesh and polynomial data. The problem I ...
5
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3answers
266 views

How to efficiently structure simulation data in memory for cells with varying degrees of freedom?

For a discontinuous Galerkin-based simulation I need to store cell-based simulation data in memory. Since the order of the polynomial approximation $N_p$ may vary between cells, I wonder what the most ...

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