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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Constructing metric terms for high order elements

Given 27 $(x,y,z)$ coordinates in 3D space which describe a generally curved quadratic hexahedron, which correspond to the HEXA_27 reference element figure with planar faces in $(\xi, \eta, \zeta)$ ...
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Finite difference problem

I have a problem to resolve with the Finite Difference method in $[a,b]$: $$-\frac{d}{dx}(\alpha(x)\frac{du}{dx})= g(x),$$ with $\alpha(x) \in L^{\infty}$ continuous in $]a,c[$ and $]c,b[$ and ...
290 views

Does the weighted residual method not use energy minimization in any form?

I've come across several texts/papers utilizing the concept of a minimum potential energy state corresponding to an equilibrium state, and I know that it is used in FEM formulations that are based on ...
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1 vote
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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 ...
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243 views

Errors imposing boundary conditions weakly with DG

I am using interior penalty discontinuous Galerkin to solve a simple Laplace problem: \begin{align*} \nabla u=0 \end{align*} with prescribed 0 and 1 Dirichlet boundary conditions on opposite edges of ...
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$u_t+a u_x=0$ solved by Discontinuous Galerkin

If $u_t+a \cdot u_x=0$ under a periodic boundary condition (to mimic an infinite domain) is solved by Discontinuous Galerkin (DG), how to implement periodic boundary condition and the other details in ...
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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 ...
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1 vote
137 views

Integration of (d-1)-dimensional functions on finite element surfaces

I am trying to integrate a function $\hat u$ on the common surface of discontinuous finite elements. The function $\hat u$ lives in a $d-1$-dimensional space of functions defined on the element ...
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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 ...
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52 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 ...
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1 vote
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How to Impose nonhomogeneous Neumann Boundary Condition in the DG Formulation

Consider the following partial differential equation \begin{align} \frac{\partial u}{\partial t}+\frac{\partial f}{\partial x} &= g(x,t), \ \ x\in \Omega = [x_{L},x_{R}] \\ u(x,0) &= u_{0}(x) ...
183 views

DG method for solving Hyperbolic Partial Differential Equation with Dirichlet Boundary Conditions

Consider the following partial differential equation \begin{align} \frac{\partial u}{\partial t}+\frac{\partial f}{\partial x} &= g(x,t), \ \ x\in \Omega = [x_{L},x_{R}] \\ u(x,0) &= u_{0}(x) ...
1 vote
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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 ...
153 views

Analysis of nonlinear finite element methods

I have been doing a lot of reading on the development of finite element methods and their analysis using, e.g., functional analysis. I am clear on the formulation of the weak form of a PDE and ...
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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 ...
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discretizing surface integral using nodal DG method

I am currently learning nodal DG methods, primarily through the book by Warburton, and am a bit confused on how to handle surface integrals using straight edged elements. On page 187 (and on page 214)...
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Gauss-Lobatto quadrature and nodal points for FEM

By using the Legendre-Gauss-Lobatto (LGL) quadrature formula (QF) and LGL nodal points one achives a diagonal mass-matrix for finite element problems. (More specifically, the spectral element method.) ...
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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 ...
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$L^\infty$ stability property of an ODE

Suppose we have the initial-value problem on $(0,L)$: $$\frac{d u(x)}{d x} = f(x) u(x),\, \qquad x\in\Omega,\,~~ u(0) = u_0,$$ I am reading a claim that says if we multiply the ODE by $u$ and ...
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