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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2
votes
1answer
62 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) ...
1
vote
1answer
68 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
votes
1answer
154 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 ...
4
votes
1answer
167 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
votes
2answers
99 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 ...
6
votes
5answers
230 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.
4
votes
1answer
103 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: $$ ...
6
votes
0answers
118 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: $$ ...
3
votes
0answers
65 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 ...
3
votes
2answers
196 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} = ...
3
votes
1answer
154 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, ...
11
votes
2answers
527 views

Discontinuous Galerkin: Nodal vs Modal advantages and disadvantages

There are two general approaches to representing solutions in the discontinuous galerkin method: nodal and modal. Modal: Solutions are represented by sums of modal coefficients multiplied by a set ...
6
votes
2answers
182 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 ...
7
votes
2answers
121 views

Has a uniform estimate in k of the inf-sup constant for hp-DG methods for the Stokes problem been established?

In Theorem 6.2 of their 2003 paper on "Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal., 40(6), 2171–2194", D. Schötzau, Ch. Schwab, and A. Toselli prove a bound of the inf-sup constant ...
7
votes
2answers
93 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 ...
3
votes
2answers
173 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
votes
0answers
101 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
votes
1answer
112 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 ...
4
votes
3answers
190 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 ...
6
votes
1answer
580 views

Visualizing discontinuous Galerkin/finite element data

I would like to visualize simulation results, obtained using the discontinuous Galerkin (DG) approach, within ParaView. Similarly to finite volume methods, the problem domain is divided into ...