25 research outputs found
Superconvergence Using Pointwise Interpolation in Convection-Diffusion Problems
Considering a singularly perturbed convection-diffusion problem, we present
an analysis for a superconvergence result using pointwise interpolation of
Gau{\ss}-Lobatto type for higher-order streamline diffusion FEM.
We show a useful connection between two different types of interpolation,
namely a vertex-edge-cell interpolant and a pointwise interpolant. Moreover,
different postprocessing operators are analysed and applied to model problems.Comment: 19 page
Error estimates for linear finite elements on Bakhvalov-type meshes
summary:For convection-diffusion problems with exponential layers, optimal error estimates for linear finite elements on Shishkin-type meshes are known. We present the first optimal convergence result in an energy norm for a Bakhvalov-type mesh
On Bakhvalov-type meshes for a linear convection-diffusion problem in 2D
For singularly perturbed two-dimensional linear convection-diffusion problems, although optimal error estimates of an upwind finite difference scheme on Bakhvalov-type meshes are widely known, the analysis remains unanswered (Roos and Stynes in Comput. Meth. Appl. Math. 15 (2015), 531--550). In this short communication, by means of a new truncation error and barrier function based analysis, we address this open question for a generalization of Bakhvalov-type meshes in the sense of Boglaev and Kopteva. We prove that the upwind scheme on these mesh modifications is optimal first-order convergence, uniformly with respect to the perturbation parameter, in the discrete maximum norm. Furthermore, we derive a sufficient condition on the transition point choices to guarantee that our modified meshes can preserve the favorable properties of the original Bakhvalov mesh
On Bakhvalov-type meshes for a linear convection-diffusion problem in 2D
For singularly perturbed two-dimensional linear convection-diffusion problems, although optimal error estimates of an upwind finite difference scheme on Bakhvalov-type meshes are widely known, the analysis remains unanswered (Roos and Stynes in Comput. Meth. Appl. Math. 15 (2015), 531--550). In this short communication, by means of a new truncation error and barrier function based analysis, we address this open question for a generalization of Bakhvalov-type meshes in the sense of Boglaev and Kopteva. We prove that the upwind scheme on these mesh modifications is optimal first-order convergence, uniformly with respect to the perturbation parameter, in the discrete maximum norm. Furthermore, we derive a sufficient condition on the transition point choices to guarantee that our modified meshes can preserve the favorable properties of the original Bakhvalov mesh
Numerical analysis of a singularly perturbed convection diffusion problem with shift in space
We consider a singularly perturbed convection-diffusion problem that has in
addition a shift term. We show a solution decomposition using asymptotic
expansions and a stability result. Based upon this we provide a numerical
analysis of high order finite element method on layer adapted meshes. We also
apply a new idea of using a coarser mesh in places where weak layers appear.
Numerical experiments confirm our theoretical results.Comment: 17 pages, 1 figur
A balanced finite-element method for an axisymmetrically loaded thin shell
We analyse a finite-element discretisation of a differential equation
describing an axisymmetrically loaded thin shell. The problem is singularly
perturbed when the thickness of the shell becomes small. We prove robust
convergence of the method in a balanced norm that captures the layers present
in the solution. Numerical results confirm our findings