Analysis of the truncation error and barrier-function technique for a Bakhvalov-type mesh

We use a barrier-function technique to prove the parameter-uniform convergence for singularly perturbed convection-diffusion problems discretized on a Bakhvalov-type mesh. This is the first proof of this kind in the research literature, the barrier-function approach having only been applied so far to Shishkin-type meshes

Using the Kellogg-Tsan Solution Decomposition in NumericalMethods for Singularly Perturbed Convection-Diffusion Problems

The linear one-dimensional singularly perturbed convection-diffusion problem is solved numerically by a second-order method that is uniform in the perturbation parameter . The method uses the Kellogg-Tsan decomposition of the continuous solution. This increases the accuracy of the numerical results and simplifies the proof of their -uniformit

A note on a generalized Shishkin-type mesh

The one-dimensional linear singularly perturbed convection-diffusion problem is discretized using the upwind scheme on a mesh which is a mild generalization of Shishkin-type meshes. The generalized mesh uses the transition point of the Shishkin mesh, but it does not require any structure of its fine and course parts. Convergence uniform in the perturbation parameter is proved by the barrier-function technique, which, because of the unstructured mesh, does not rely on any mesh-generating function. In this way, the technical requirements needed in the existing barrier-function approaches are simplified

A closed-form solution to the inverse problem in interpolation by a Bézier-spline curve

A geometric construction of a Bézier curve is presented by a unifiable way from the mentioned literature with some modification. A closed-form solution to the inverse problem in cubic Bézier-spline interpolation will be obtained. Calculations in the given examples are performed by a Maple procedure using this solution

Cooperative Learning Activities with a Focus on Geometry Applications in a Basic Math & Pre-Algebra Class

In this communication, we report on cooperative learning group activities with an emphasis on geome- try applications conducted in a Basic Math & Pre-Algebra Class (MATH-192) in Ohlone College. These collaborative strategies are used as an e cient tool, among others, to improve the class climate nature of MATH-192 where the majority students are typically adult working learners and returning students who are in needs to be motivated

A Uniform Convergence Analysis for a Bakhvalov-Type Mesh with an Explicitly Defined Transition Point

For singularly perturbed convection-diffusion problems, the truncation error and barrier-function technique for proving parameter-uniform convergence is well-known for finite-difference methods on Shishkin-type meshes (Roos and Linß in Computing, 63 (1999), 27–45). In this paper, we show that it is also possible to generalize this technique to a modification of the Bakhvalov mesh, such that the transition point between the fine and crude parts of the mesh only depends on the perturbation parameter and is defined explicitly. We provide a complete analysis for 1D problems for the simplicity of the present paper, but the analysis can be easily extended to 2D problems. With numerical results for 2D problems we show that the finite-difference discretization on the Bakhvalov-type mesh performs better than the Bakhvalov-Shishkin mesh

Robust hybrid schemes of higher order for singularly perturbed convection-diffusion problems

A class of linear singularly perturbed convection-diffusion problems in one dimension is discretized on the Shishkin mesh using hybrid higher-order finite-difference schemes. Under appropriate conditions, pointwise convergence uniform in the perturbation parameter ε is proved for one of the discretizations. This is done by the preconditioning approach, which enables the proof of ε-uniform stability and ε-uniform consistency, both in the maximum norm. The order of convergence is almost 3 when ε is sufficiently small

A numerical method for stationary shock problems with monotonic solutions

Numerical methods are considered for singularly perturbed quasilinear problems having interior-shock solutions. It is shown that the direct discretization on a layer-adapted mesh is ineffective for these problems. A special method is proposed for the case when the solution is monotonic: the problem is transformed by interchanging the dependent and independent variables, and it is then discretized on a uniform mesh. The method is analyzed both theoretically and numerically. It is shown that it can be effective, but that it is not entirely without problems. An approach for improving the method is suggested

An improved Kellogg-Tsan solution decomposition in numerical methods for singularly perturbed convection-diffusion problems

We consider the Kellogg-Tsan decomposition of the solution to the linear one-dimensional singularly perturbed convection-diffusion problem and improve it by including the solution of the corresponding reduced problem as a component. The upwind scheme on a modified Shishkin-type mesh is used to approximate the unknown component of the decomposition. It is proved that the error is O ( ε ( ln ε ) 2 N − 1 ), where ε is the perturbation parameter and N is the number of mesh steps. The high accuracy of the method is illustrated by numerical examples

Uniform Convergence via Preconditioning

The linear singularly perturbed convection-diffusion problem in one dimension is considered and its discretization on the Shishkin mesh is analyzed. A new, conceptually simple proof of pointwise convergence uniform in the perturbation parameter is provided. The proof is based on the preconditioning of the discrete system