Numerical Methods for Partial Differential Equations · 2002 · 63 citations · 13 references
ComputingComputational EngineeringComputational PhysicsFinite Element AnalysisFinite Element MethodsBoundary LayersBoundary-layer TheoryConvergence AnalysisError EstimationFinite Element DiscretizationNumerical ConvergenceEnergy Norm Error AnalysisError BoundsError Estimation For Finite Element SolutionsLower Bounds Of ErrorSingular Perturbations In Partial Differential EquationsSingularly Perturbed Boundary Value ProblemsSuperconvergence
Abstract Superconvergence approximations of singularly perturbed two‐point boundary value problems of reaction‐diffusion type and convection‐diffusion type are studied. By applying the standard finite element method of any fixed order p on a modified Shishkin mesh, superconvergence error bounds of ( N −1 ln ( N + 1)) p +1 in a discrete energy norm in approximating problems with the exponential type boundary layers are established. The error bounds are uniformly valid with respect to the singular perturbation parameter. Numerical tests indicate that the error estimates are sharp; in particular, the logarithmic factor is not removable. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 374–395, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10001
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