scholarly journals Analysis of Galerkin and streamline-diffusion FEMs on piecewise equidistant meshes for turning point problems exhibiting an interior layer

2018 ◽  
Vol 123 ◽  
pp. 121-136
Author(s):  
Simon Becher
Keyword(s):  
Turning Point ◽  
Interior Layer ◽  
Streamline Diffusion

A Singularly Perturbed Convection Diffusion Turning Point Problem with an Interior Layer

2012 ◽  
Vol 12 (2) ◽  
pp. 206-220 ◽  
Author(s):  
Eugene O'Riordan ◽  
Jason Quinn
Keyword(s):  
Finite Difference ◽  
Turning Point ◽  
Singularly Perturbed ◽  
Difference Operators ◽  
Point Problem ◽  
Convection Coefficient ◽  
Interior Layer ◽  
Convection Diffusion ◽  
Fine Mesh ◽  
Uniform Grids

Abstract A finite difference scheme on special piecewise-uniform grids condensing in tA linear singularly perturbed interior turning point problem with a continuous convection coefficient is examined in this paper. Parameter uniform numerical methods composed of monotone finite difference operators and piecewise-uniform Shishkin meshes, are constructed and analysed for this class of problems. A refined Shishkin mesh is placed around the location of the interior layer and we consider disrupting the centre point of this fine mesh away from the point where the convection coefficient is zero. Numerical results are presented to illustrate the theoretical parameter-uniform error bounds established.

scholarly journals Acoustic trapped modes in a three-dimensional waveguide of slowly varying cross section

2013 ◽  
Vol 469 (2149) ◽  
pp. 20120384 ◽  
Author(s):  
Simon N. Gaulter ◽  
Nicholas R. T. Biggs
Keyword(s):  
Cross Section ◽  
Turning Point ◽  
Harmonic Wave ◽  
Three Dimensional ◽  
Longitudinal Direction ◽  
Interior Layer ◽  
Oscillatory Behaviour ◽  
Trapped Modes ◽  
Trapped Mode ◽  
Time Harmonic

In this paper, we develop an asymptotic scheme to approximate the trapped mode solutions to the time harmonic wave equation in a three-dimensional waveguide with a smooth but otherwise arbitrarily shaped cross section and a single, slowly varying ‘bulge’, symmetric in the longitudinal direction. Extending previous research carried out in the two-dimensional case, we first use a WKBJ-type ansatz to identify the possible quasi-mode solutions that propagate only in the thicker region, and hence find a finite cut-on region of oscillatory behaviour and asymptotic decay elsewhere. The WKBJ expansions are used to identify a turning point between the cut-on and cut-off regions. We note that the expansions are non-uniform in an interior layer centred on this point, and we use the method of matched asymptotic expansions to connect the cut-on and cut-off regions within this layer. The behaviour of the expansions within the interior layer then motivates the construction of a uniformly valid asymptotic expansion. Finally, we use this expansion and the symmetry of the waveguide around the longitudinal centre, x =0, to extract trapped mode wavenumbers, which are compared with those found using a numerical scheme and seen to be extremely accurate, even to relatively large values of the small parameter.

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