15 research outputs found
Discontinuous Petrov-Galerkin method based on the optimal test space norm for one-dimensional transport problems
We revisit the finite element analysis of convection dominated flow problems within the recently developed Discontinuous Petrov-Galerkin (DPG) variational framework. We demonstrate how test function spaces that guarantee numerical stability can be computed automatically with respect to the so called optimal test space norm by using an element subgrid discretization. This should make the DPG method not only stable but also robust, that is, uniformly stable with respect to the P'eclet number in the current application. The effectiveness of the algorithm is demonstrated on two problems for the linear advection-diffusion equation. © 2011 Published by Elsevier Ltd
Wave number-Explicit Analysis for Galerkin Discretizations of Lossy Helmholtz Problems
We present a stability and convergence theory for the lossy Helmholtz
equation and its Galerkin discretization. The boundary conditions are of Robin
type. All estimates are explicit with respect to the real and imaginary part of
the complex wave number , ,
. For the extreme cases and , the estimates
coincide with the existing estimates in the literature and exhibit a seamless
transition between these cases in the right complex half plane.Comment: 29 pages, 1 figur
On the natural stabilization of convection dominated problems using high order BubnovâGalerkin finite elements
In the case of dominating convection, standard Bubnov–Galerkin finite elements are known to deliver oscillating discrete solutions for the convection–diffusion equation. This paper demonstrates that increasing the polynomial degree (p-extension) limits these artificial numerical oscillations. This is contrary to a widespread notion that an increase of the polynomial degree destabilizes the discrete solution. This treatise also provides explicit expressions as to which polynomial degree is sufficiently high to obtain stable solutions for a given PeÌclet number at the nodes of a mesh
Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales: proofs
We consider a coupled system of two singularly perturbed reaction-diffusion
equations, with two small parameters , each
multiplying the highest derivative in the equations. The presence of these
parameters causes the solution(s) to have \emph{boundary layers} which overlap
and interact, based on the relative size of and . We
construct full asymptotic expansions together with error bounds that cover the
complete range . For the present case of analytic
input data, we derive derivative growth estimates for the terms of the
asymptotic expansion that are explicit in the perturbation parameters and the
expansion order