Book ChapterDOI
Numerical Study of Maximum Norm a Posteriori Error Estimates for Singularly Perturbed Parabolic Problems
Natalia Kopteva,Torsten Linβ +1 more
- pp 50-61
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TLDR
A second-order singularly perturbed parabolic equation in one space dimension is considered and computable a posteriori error estimates in the maximum norm for two semidiscretisations in time and a full discretisation using P 1 FEM in space are given.Abstract:
A second-order singularly perturbed parabolic equation in one space dimension is considered. For this equation, we give computable a posteriori error estimates in the maximum norm for two semidiscretisations in time and a full discretisation using P 1 FEM in space. Both the Backward-Euler method and the Crank-Nicolson method are considered. Certain critical details of the implementation are addressed. Based on numerical results we discuss various aspects of the error estimators in particular their effectiveness.read more
Citations
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Journal ArticleDOI
Maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions
Natalia Kopteva,Torsten Linss +1 more
TL;DR: A semilinear second-order parabolic equation is considered in a regular and a singularly perturbed regime and computable a posteriori error estimates in the maximum norm are given.
Journal ArticleDOI
Improved maximum-norm a posteriori error estimates for linear and semilinear parabolic equations
Natalia Kopteva,Torsten Linß +1 more
TL;DR: For second-order parabolic equations, it is shown that logarithmic dependence on the time step size can be eliminated and a posteriori error estimates in the maximum norm are given that improve upon recent results in the literature.
Journal ArticleDOI
A posteriori error estimation for arbitrary order FEM applied to singularly perturbed one-dimensional reaction-diffusion problems
TL;DR: FEM discretizations of arbitrary order r are considered for a singularly perturbed one-dimensional reaction-diffusion problem whose solution exhibits strong layers and a posteriori error bounds of interpolation type are derived in the maximum norm.
Proceedings ArticleDOI
On biorthogonal approximation of solutions of some boundary value problems on Shishkin mesh
E. Kulikov,A. Makarov +1 more
TL;DR: In this paper, a piecewise-uniform Shishkin mesh is considered, and a local approximation scheme is implemented, minimal splines are used as basis functions, and the coefficients are calculated as de Boor-Fix type functional values.
Applications of Mathematics
TL;DR: In this paper, the authors considered a singularly perturbed one-dimensional reaction-diffusion problem whose solution exhibits strong layers and derived a posteriori error bounds of interpolation type in the maximum norm.
References
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Book
Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems
TL;DR: In this paper, the Incompressible Navier-Stokes Equations are used to describe the existence and uniqueness of solutions to the problem of second-order boundary value problems.
Book
Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems
TL;DR: In this paper, the authors propose a finite element and finite volume method for solving one-dimensional and two-dimensional convection-diffusion problems, where the analytical behavior of solutions is analyzed.
Journal ArticleDOI
Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems
TL;DR: Energy techniques are combined with an appropriate pointwise representation of the error based on an elliptic reconstruction operator which restores the optimal order (and regularity for piecewise polynomials of degree higher than one).
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