Cholesky factorisation of linear systems coming from finite difference approximations of singularly perturbed problems

We consider the solution of large linear systems of equations that arise when two-dimensional singularly perturbed reaction-diffusion equations are discretized. Standard methods for these problems, such as central finite differences, lead to system matrices that are positive definite. The direct solvers of choice for such systems are based on Cholesky factorisation. However, as observed by MacLachlan and Madden (SIAM J. Sci. Comput. 35-5 (2013), pp. A2225-A2254), these solvers may exhibit poor performance for singularly perturbed problems. We provide an analysis of the distribution of entries in the factors based on their magnitude that explains this phenomenon, and give bounds on the ranges of the perturbation and discretization parameters where poor performance is to be expected.Comment: 9 pages, 2 figure

On Ɛ-uniform convergence of exponentially fitted methods

A class of methods constructed to numerically approximate solution of two-point singularly perturbed boundary value problems of the form $varepsilon u\u27\u27 + b u\u27 + c u = f$ use exponentials to mimic exponential behavior of the solution in the boundary layer(s). We refer to them as exponentially fitted methods. Such methods are usually exact on polynomials of certain degree and some exponential functions. Shortly, they are exact on exponential sums. It is often possible that consistency of the method follows from the convergence of interpolating function standing behind the method. Because of that, we consider interpolation error for exponential sums. A main result of the paper is an error bound for interpolation by exponential sum to the solution of singularly perturbed problem that does not depend on perturbation parameter $varepsilon$ when $varepsilon$ is small with the respect to mesh width. Numerical experiment implies that the use of dense mesh in the boundary layer for small meshwidth results with $varepsilon$-uniform convergence

Stability of an upwind Petrov Galerkin discretization of convection diffusion equations

We study a numerical method for convection diffusion equations, in the regime of small viscosity. It can be described as an exponentially fitted conforming Petrov-Galerkin method. We identify norms for which we have both continuity and an inf-sup condition, which are uniform in mesh-width and viscosity, up to a logarithm, as long as the viscosity is smaller than the mesh-width or the crosswind diffusion is smaller than the streamline diffusion. The analysis allows for the formation of a boundary layer.Comment: v1: 18 pages. 2 figures. v2: 22 pages. Numerous details added and completely rewritten final proof. 8 pages appendix with old proo

On supraconvergence phenomenon for second order centered finite differences on non-uniform grids

In the present study we consider an example of a boundary value problem for a simple second order ordinary differential equation, which may exhibit a boundary layer phenomenon. We show that usual central finite differences, which are second order accurate on a uniform grid, can be substantially upgraded to the fourth order by a suitable choice of the underlying non-uniform grid. This example is quite pedagogical and may give some ideas for more complex problems.Comment: 26 pages, 2 figures, 2 tables, 37 references. Other author's papers can be downloaded at http://www.denys-dutykh.com