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).
27
questions with no upvoted or accepted answers
14
votes
0
answers
497
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:
$$
-\nabla\cdot(D_{1}(u_{2},...
- 937
5
votes
0
answers
212
views
Discontinuous Galerkin for flow through porous media
I am struggling with DG methods for 2 phase flow through porous media. I managed to get the global pressure, total flux equations to work with an unconditionally stable mixed FE DG formulation as ...
- 81
3
votes
0
answers
129
views
Discontinuous Galerkin for transport equation with non-constant advection
This question is mainly an inquiry about the usefulness of Discontinuous Galerkin (DG) for the time-independent transport equation of the form
$$\sigma u+\beta\cdot\nabla u =f,\;\;\;\text{on }\Omega\...
- 131
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 ...
- 251
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)
$$
\Delta t \approx \frac{h}{N^2} \frac{C}{|u| + |c| + \frac
{N^2 \mu}{h}}
$$
($...
- 141
3
votes
0
answers
133
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 ...
- 95
3
votes
0
answers
161
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
0
answers
88
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 ...
- 2,345
2
votes
0
answers
120
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
2
votes
0
answers
631
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 ...
- 2,345
1
vote
0
answers
134
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 ...
- 302
1
vote
0
answers
100
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 ...
- 83
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 ...
- 865
1
vote
0
answers
50
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 ...
- 971
1
vote
0
answers
76
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/...
- 11
1
vote
0
answers
108
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 ...
- 601
1
vote
0
answers
53
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
1
vote
0
answers
107
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
369
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
152
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
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
1
vote
0
answers
129
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
402
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
0
votes
0
answers
63
views
Is there a FreeFEM++ code for the Modified Weak Galerkin Finite Element Method to solve the Poisson Problem?
I wrote a FreeFEM++ code to solve the Poisson Problem -∆u=f in Ω u=g on ∂Ω using the Modified Weak Galerkin Finite Element Method based on the paper:
[1] “A modified weak Galerkin finite element ...
0
votes
0
answers
56
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}$$
I want to solve the above PDE system by a mixed FEM, that is, $u_t=f(u,v,\nabla{v})$ by discontinuous Galerkin (DG), where $f$...
- 227
0
votes
0
answers
149
views
Discontinuous Galerkin failing to converge Euler equations under p-refinement
I am solving the steady state compressible Euler equations in conservation form in 2D in a rectangular domain with a discontinuous Galerkin (DG) code I have developed. As a test, the boundary ...
- 63
0
votes
0
answers
50
views
Can you describe the Galerkin numerical method to solve the wave equation?
How would you describe the Galerkin method to solving the 3D wave equation
$$u_{tt}= c^2\Delta u$$
to someone who wants to implement it immediately?
More precisely, we want to solve the Cauchy problem
...
- 151