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

Uniform convergence on a Bakhvalov-type mesh using the preconditioning approach: Technical report

The linear singularly perturbed convection-diffusion problem in one dimension is considered and its discretization on a Bakhvalov-type mesh is analyzed. The preconditioning technique is used to obtain the pointwise convergence uniform in the perturbation parameter.Comment: 12 page

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

On Bakhvalov-type meshes for a linear convection-diffusion problem in 2D

For singularly perturbed two-dimensional linear convection-diffusion problems, although optimal error estimates of an upwind finite difference scheme on Bakhvalov-type meshes are widely known, the analysis remains unanswered (Roos and Stynes in Comput. Meth. Appl. Math. 15 (2015), 531--550). In this short communication, by means of a new truncation error and barrier function based analysis, we address this open question for a generalization of Bakhvalov-type meshes in the sense of Boglaev and Kopteva. We prove that the upwind scheme on these mesh modifications is optimal first-order convergence, uniformly with respect to the perturbation parameter, in the discrete maximum norm. Furthermore, we derive a sufficient condition on the transition point choices to guarantee that our modified meshes can preserve the favorable properties of the original Bakhvalov mesh

On Bakhvalov-type meshes for a linear convection-diffusion problem in 2D

For singularly perturbed two-dimensional linear convection-diffusion problems, although optimal error estimates of an upwind finite difference scheme on Bakhvalov-type meshes are widely known, the analysis remains unanswered (Roos and Stynes in Comput. Meth. Appl. Math. 15 (2015), 531--550). In this short communication, by means of a new truncation error and barrier function based analysis, we address this open question for a generalization of Bakhvalov-type meshes in the sense of Boglaev and Kopteva. We prove that the upwind scheme on these mesh modifications is optimal first-order convergence, uniformly with respect to the perturbation parameter, in the discrete maximum norm. Furthermore, we derive a sufficient condition on the transition point choices to guarantee that our modified meshes can preserve the favorable properties of the original Bakhvalov mesh

On the Implementation of an Accurate and Efficient Solver for Convection-Diffusion Equations

In this dissertation, we examine several different aspects of computing the numerical solution of the convection-diffusion equation. The solution of this equation often exhibits sharp gradients due to Dirichlet outflow boundaries or discontinuities in boundary conditions. Because of the singular-perturbed nature of the equation, numerical solutions often have severe oscillations when grid sizes are not small enough to resolve sharp gradients. To overcome such difficulties, the streamline diffusion discretization method can be used to obtain an accurate approximate solution in regions where the solution is smooth. To increase accuracy of the solution in the regions containing layers, adaptive mesh refinement and mesh movement based on a posteriori error estimations can be employed. An error-adapted mesh refinement strategy based on a posteriori error estimations is also proposed to resolve layers. For solving the sparse linear systems that arise from discretization, goemetric multigrid (MG) and algebraic multigrid (AMG) are compared. In addiiton, both methods are also used as preconditioners for Krylov subspace methods. We derive some convergence results for MG with line Gauss-Seidel smoothers and bilinear interpolation. Finally, while considering adaptive mesh refinement as an integral part of the solution process, it is natural to set a stopping tolerance for the iterative linear solvers on each mesh stage so that the difference between the approximate solution obtained from iterative methods and the finite element solution is bounded by an a posteriori error bound. Here, we present two stopping criteria. The first is based on a residual-type a posteriori error estimator developed by Verfurth. The second is based on an a posteriori error estimator, using local solutions, developed by Kay and Silvester. Our numerical results show the refined mesh obtained from the iterative solution which satisfies the second criteria is similar to the refined mesh obtained from the finite element solution