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).

24 questions with no upvoted or accepted answers
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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},...
11
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1answer
461 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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0answers
173 views

How to compute numerical fluxes in the local discontinuous galerkin method for poisson equation 1D

Some days ago I began to study the local discontinuous galerkin (LDG) method, this is my first time working with a discontinuous method, so I decided to solve the poisson equation in 1D to learn the ...
5
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0answers
202 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 ...
3
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0answers
62 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) $$ \Delta t \approx \frac{h}{N^2} \frac{C}{|u| + |c| + \frac {N^2 \mu}{h}} $$ ($...
3
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0answers
111 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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0answers
691 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}]_{...
3
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0answers
136 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 ...
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 ...
2
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0answers
588 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 ...
1
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0answers
63 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 ...
1
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0answers
39 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 ...
1
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0answers
54 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
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0answers
477 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 ...
1
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0answers
100 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 ...
1
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0answers
48 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
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0answers
97 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
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0answers
294 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
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0answers
127 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 ...
1
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0answers
299 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
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0answers
74 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 ...
1
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0answers
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 ...
1
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0answers
341 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 ...
1
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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{...