Vladimir Liseikin’s scientific contributions

The book describes algorithms for generating numerical grids for solving singularly-perturbed problems by using layer-damping coordinate transformations eliminating singularities of solutions to some required order n. The layer-damping coordinate transformations are generated explicitly by certain procedures with four basic univariate locally contracting mappings. The form of any contracting function depends on the qualitative behavior of the solution. The information about the qualitative solution structure is obtained either from a theoretical analysis of simpler model equations, specifically, the ordinary differential equations discussed in Chaps. 3--5, which simulate the qualitative features of the solutions, or from a preliminary numerical calculation for similar problems on coarse grids.The theoretical analysis has revealed new forms of layer-damping functions eliminating singularities of solutions to singularly--perturbed problems and corresponding layer-resolving grids -- functions and grids above and beyond those already well known and having broad acceptance, namely, those developed by Bakhvalov and Shishkin. The grids developed by Bakhvalov and Shishkin have been applied to diverse problems, but only to problems with exponential-type layers , typically represented by functions $\exp(-bx/\varepsilon^k)$, occurring in problems for which the solutions of reduced $(\varepsilon=0)$ problems do not have singularities. Such grids are not suitable for tackling other, wider, layers, and also require knowledge of the constant b affecting the width of the exponential layer -- when such knowledge is not always available, for example, for boundary layers in fluid-dynamics problems modeled by Navier-Stokes equations, or for interior layers in solutions to quasilinear nonautonomous problems. One spectacular example of the new layer-resolving grids being presented in the book, engendered by a function $\varepsilon^{rk}/(\varepsilon^k +x)^r$, $r>0$, is suitable for dealing not only with exponential layers having arbitrary widths, but with power of first type layers occurring in problems for which the solutions of reduced problems have singularities as well. Other examples of new layer--resolving grids are aimed at dealing with power of type 2 layers represented by functions $(\varepsilon^k +x)^r$, $0< r < 1$; logarithmic layers represented by functions $\ln(\varepsilon^k +x)/\ln\varepsilon^k$; and mixed layers. Numerical experiments are carried out in Chap. 8 for problems having various types of layers. It seems that the new layer-resolving grids described in the book should empower researchers to solve broader and more important classes of problems having not only exponential-, but power-, logarithmic-, and mixed-type boundary and interior layers.