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).

learn more… | top users | synonyms (1)

4
votes
2answers
94 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) - ...
1
vote
0answers
53 views

Matlab implementation of 2D Interior penalty discontinuous Galerkin poisson problem

Basically, I am trying to solve the 2D poisson problem in order to learn implementation of IPDG methods. The problem states $-\nabla a(x)(\nabla u)=0\ \text{in} \ \Omega$ with $U=0$ on Dirichlet ...
3
votes
2answers
134 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 ...
0
votes
1answer
101 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: $$ ...
2
votes
2answers
180 views

The meaning of conservative discretization in Galerkin FEM and Discontinuous Galerkin

I do understand the meanning of "conservative discretization" within the FVM/FDM framework, indeed it is well explained in this post. Now, according to the table in this slide (pp.4), it concludes: ...
2
votes
0answers
152 views

Help implementing 1D (ODE) discontinuous Galerkin method

I think I made a mistake in calculating the system matrix $\bf H$ described below, I need help figuring out what went wrong. I'm trying to apply a discontinuous Galerkin method to approximate the ...
1
vote
0answers
89 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 ...
6
votes
1answer
157 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 ...
1
vote
1answer
82 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 ...
1
vote
1answer
98 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 ...
0
votes
0answers
163 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) = ...
1
vote
2answers
123 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 $v_j^k$ ...
1
vote
2answers
91 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} = ...
1
vote
1answer
83 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
votes
1answer
159 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 ...
2
votes
0answers
66 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
votes
1answer
97 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
116 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
258 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
233 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
130 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 ...
7
votes
5answers
292 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
177 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
152 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
92 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
314 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
188 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, ...
12
votes
2answers
831 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
213 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
127 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
108 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
281 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
129 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
119 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
199 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
823 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 ...