On Asymptotically Optimal Meshes by Coordinate Transformation

We study the problem of constructing asymptotically optimal meshes with respect to the gradient error of a given input function. We provide simpler proofs of previously known results and show constructively that a closed-form solution exists for them. We show how the transformational method for obtaining meshes, as is, cannot produce asymptotically optimal meshes for general inputs. We also discuss possible variations of the problem definition that may allow for some forms of optimality to be proved.Engineering and Applied Science

Exact asymptotics of the uniform error of interpolation by multilinear splines

The question of adaptive mesh generation for approximation by splines has been studied for a number of years by various authors. The results have numerous applications in computational and discrete geometry, computer aided geometric design, finite element methods for numerical solutions of partial differential equations, image processing, and mesh generation for computer graphics, among others. In this paper we will investigate the questions regarding adaptive approximation of C2 functions with arbitrary but fixed throughout the domain signature by multilinear splines. In particular, we will study the asymptotic behavior of the optimal error of the weighted uniform approximation by interpolating and quasi-interpolating multilinear splines

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

Mean-square stability and error analysis of implicit time-stepping schemes for linear parabolic SPDEs with multiplicative Wiener noise in the first derivative

In this article, we extend a Milstein finite difference scheme introduced in [Giles & Reisinger(2011)] for a certain linear stochastic partial differential equation (SPDE), to semi- and fully implicit timestepping as introduced by [Szpruch(2010)] for SDEs. We combine standard finite difference Fourier analysis for PDEs with the linear stability analysis in [Buckwar & Sickenberger(2011)] for SDEs, to analyse the stability and accuracy. The results show that Crank-Nicolson timestepping for the principal part of the drift with a partially implicit but negatively weighted double It\^o integral gives unconditional stability over all parameter values, and converges with the expected order in the mean-square sense. This opens up the possibility of local mesh refinement in the spatial domain, and we show experimentally that this can be beneficial in the presence of reduced regularity at boundaries