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Uniform Convergence Analysis of the Discontinuous Galerkin Method on layer-adapted meshes for Singularly Perturbed Problem

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Wuhan University Journal of Natural Sciences 2023年第5期28卷 411-420页

作者： SHI Jiamin LU Zhongshu ZHANG Luyi LU Sunjia CHENG Yao School of Mathematical Sciences Suzhou University of Science and TechnologySuzhou 215009JiangsuChina

This paper concerns a discontinuous Galerkin(DG)method for a one-dimensional singularly perturbed problem which possesses essential characteristic of second order convection-diffusion problem after some simple *** derive an optimal convergence of the DG method for eight layer-adapted meshes in a general *** convergence rate is valid independent of the small ***,we establish a sharper L^(2)-error estimate if the true solution has a special regular *** experiments are also given.

关键词： layer-adapted meshes singularly perturbed problem uniform convergence discontinuous Galerkin method

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GRADED meshes FOR HIGHER ORDER FEM

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Journal of Computational Mathematics 2015年第1期33卷 1-16页

作者： Hans-Gorg Roos Ljiljana Teofanov Zorica Uzelac Institute of Numerical Mathematics Technical University of Dresden Dresden D-01062 Germany Department for Fundamental Disciplines Faculty of Technical Sciences University of Novi Sad Trg D. Obradovida 6 21000 Novi Sad Serbia

A singularly perturbed one-dimensional convection-diffusion problem is solved numeri- cMly by the finite element method based on higher order polynomials. Numerical solutions are obtained using S-type meshes with special emphasis on meshes which are graded （based on a mesh generating function） in the fine mesh region. Error estimates in the s-weighted energy norm are proved. We derive an ＇optimal＇ mesh generating function in order to min- imize the constant in the error estimate. Two layer-adapted meshes defined by a recursive formulae in the fine mesh region are also considered and a new technique for proving er- ror estimates for these meshes is presented. The aim of the paper is to emphasize the importance of using optimal meshes for higher order finite element methods. Numerical experiments support all theoretical results.

关键词： Singular perturbation Boundary value problem layer-adapted meshes Gradedmeshes Finite element method.

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Superconvergence of a Galerkin FEM for Higher-Order Elements in Convection-Diffusion Problems

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Numerical Mathematics(Theory,Methods and Applications) 2014年第3期7卷 356-373页

作者： Sebastian Franz H.-G.Roos Institut für Numerische Mathematik Technische Universit¨at Dresden01062 DresdenGermany

In this paper we present a first supercloseness analysis for higher-order Galerkin FEM applied to a singularly perturbed convection-diffusion *** a solution decomposition and a special representation of our finite element space,we are able to prove a supercloseness property of p+1/4 in the energy norm where the polynomial order p≥3 is odd.

关键词： Singular perturbation layer-adapted meshes superconvergence postprocessing

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Numerical Analysis of a System of Singularly Perturbed Convection-Diffusion Equations Related to Optimal Control

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Numerical Mathematics(Theory,Methods and Applications) 2011年第4期4卷 562-575页

作者： Hans-Görg Roos Christian Reibiger Technische Universität Dresden Institut für Numerische Mathematik01062 DresdenGermany

We consider an optimal control problem with an 1D singularly perturbed differential state *** solving such problems one uses the enhanced system of the state equation and its adjoint ***,we obtain a system of two convectiondiffusion *** linear finite elements on adapted grids we treat the effects of two layers arising at different boundaries of the *** proof uniform error estimates for this method on meshes of Shishkin *** present numerical results supporting our analysis.

关键词： Convection-diffusion linear finite elements a priori analysis layer-adapted meshes singular perturbed optimal control

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