Superconvergence for convection-diffusion problems with low regularity
Some properties of the discontinuous Galerkin method for one-dimensional singularly perturbed problems.
A comparison of four- and five-point difference approximations for stabilizing the one-dimensional stationary convection-diffusion equation.
A uniformly accurate finite volume discretization for a convection-diffusion problem.
Error estimates for linear finite elements on Bakhvalov-type meshes
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.
Uniformly enclosing discretization methods and grid generation for semilinear boundary value problems with first order terms
The paper deals with uniformly enclosing discretization methods of the first order for semilinear boundary value problems. Some fundamental properties of this discretization technique (the enclosing property, convergence, the inverse-monotonicity) are proved. A feedback grid generation principle using information from the lower and upper solutions is presented.
Die Konstruktion asymptotischer Fundamentalsysteme für lineare Differentialgleichungen mit Wendepunkten
Ein singulär gestörtes Problem für eine lineare parabolische Differentialgleichung beliebiger Ordnung
Analysis of finite element methods on Bakhvalov-type meshes for linear convection-diffusion problems in 2D
So far optimal error estimates on Bakhvalov-type meshes are only known for finite difference and finite element methods solving linear convection-diffusion problems in the one-dimensional case. We prove (almost) optimal error estimates for problems with exponential boundary layers in two dimensions.
Necessary conditions for uniform convergence of finite difference schemes for convection-diffusion problems with exponential and parabolic layers
Singularly perturbed problems of convection-diffusion type cannot be solved numerically in a completely satisfactory manner by standard numerical methods. This indicates the need for robust or -uniform methods. In this paper we derive new conditions for such schemes with special emphasize to parabolic layers.
An enclosure generating modification of the method of discretization in time
The combination technique for a two-dimensional convection-diffusion problem with exponential layers
Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a sparse grid Galerkin finite element method with Shishkin meshes. Writing for the maximum number of mesh intervals in each coordinate direction, our “combination” method simply adds or subtracts solutions that have been computed by the Galerkin FEM on , and meshes. It is shown that the combination FEM yields (up to a factor ) the same order of accuracy in the associated...
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