Roos, Hans-Görg; Schopf, Martin Finite elements on locally uniform meshes for convection-diffusion problems with boundary layers. (English) Zbl 1228.65129 Computing 92, No. 4, 285-296 (2011). Summary: The layer-adapted meshes used to achieve robust convergence results for problems with layers are not locally uniform. We discuss concepts of almost robust convergence and some realizations of locally-uniform meshes. Cited in 1 Document MSC: 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations 34E15 Singular perturbations for ordinary differential equations 65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 35B25 Singular perturbations in context of PDEs 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65L20 Stability and convergence of numerical methods for ordinary differential equations Keywords:finite element method; singular perturbation; convection-diffusion problem; Bakhvalov-type meshes; locally uniform meshes; layer-adapted meshes; boundary layers; numerical examples; convergence PDFBibTeX XMLCite \textit{H.-G. Roos} and \textit{M. Schopf}, Computing 92, No. 4, 285--296 (2011; Zbl 1228.65129) Full Text: DOI References: [1] Apel T (1999) Anisotropic finite elements: local estimates and applications. In: Advances in numerical mathematics. B.G. Teubner, Stuttgart · Zbl 0917.65090 [2] Duran RG, Lombardi AL (2006) Finite element approximation of convection diffusion problems using graded meshes. Appl Numer Math 56: 1314–1325. doi: 10.1016/j.apnum.2006.03.029 · Zbl 1104.65109 · doi:10.1016/j.apnum.2006.03.029 [3] Dobrowolski M, Roos H-G (1997) A priori estimates for the solution of convection-diffusion problems and interpolation on Shishkin meshes. Z Anal Anwend 16: 1001–1012 · Zbl 0892.35014 · doi:10.4171/ZAA/801 [4] Franz S (2008) Singularly perturbed problems with characteristic layers. Dissertation, TU Dresden · Zbl 1302.35002 [5] Linß T (2000) Analysis of a Galerkin finite element method on a Bakhvalov-Shishkin mesh for a linear convection-diffusion problem. IMA J Numer Anal 20: 621–632. doi: 10.1093/imanum/20.4.621 · Zbl 0966.65083 · doi:10.1093/imanum/20.4.621 [6] Linß T (2000) Uniform superconvergence of a Galerkin finite element method on Shishkin-type meshes. Numer Methods Partial Differ Equ 16(5): 426–440. doi: 10.1002/1098-2426(200009)16:5<426::AID-NUM2>3.0.CO;2-R · Zbl 0958.65110 · doi:10.1002/1098-2426(200009)16:5<426::AID-NUM2>3.0.CO;2-R [7] Linß T (2010) Layer-adapted meshes for reaction-convection-diffusion problems. In: Lecture notes in mathematics, vol 1985. Springer, Berlin · Zbl 1056.65076 [8] Mekchay K, Nochetto RH (2005) Convergence of adaptive finite element methods for general second order linear elliptic PDEs. SIAM J Numer Anal 43(5): 1803–1827. doi: 10.1137/04060929X · Zbl 1104.65103 · doi:10.1137/04060929X [9] Miller JJH, O’Riordan E, Shishkin GI (1996) Fitted numerical methods for singular perturbation problems. World Scientific, Singapore [10] Roos H-G, Linß T (1999) Sufficient conditions for uniform convergence on layer-adapted grids. Computing 63: 27–45. doi: 10.1007/s006070050049 · Zbl 0931.65085 · doi:10.1007/s006070050049 [11] Roos H-G (2006) Error estimates for linear finite elements on Bakhvalov-type meshes. Appl Math 51: 63–72. doi: 10.1007/s10492-006-0005-y · Zbl 1164.65486 · doi:10.1007/s10492-006-0005-y [12] Roos H-G, Stynes M, Tobiska L (2008) Robust numerical methods for singularly perturbed differential equations. Springer, Berlin · Zbl 1155.65087 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.