The weighted error estimates of the functional-discrete methods for solving boundary value problems

Abstract

The monograph is devoted to the construction and study of the approximate methods for solving the problems of mathematical physics. It presents obtaining the weighted accuracy estimates of these methods with taking into account the influence of boundary and initial conditions. The boundary effect means that due to the Dirichlet boundary condition for a differential equation in a canonical domain, the accuracy of the approximate solution near the boundary of the domain is higher compared to the accuracy away from the boundary. A similar situation is observed for non-stationary equations in the mesh nodes where the initial condition is given. The boundary and initial effects are quantitatively described by means of weighted estimates with a suitable weight function that characterizes the distance of a point to the boundary of the domain. The idea of such estimates was first announced by the first coauthor for the elliptic equation in the case of generalized solutions from Sobolev spaces and then expanded to quasilinear stationary and non-stationary equations. The monograph develops the aforementioned approach and presents the new research into the impact of the initial and boundary conditions on the accuracy of the finite-difference method for elliptic and parabolic equations, the grid method for solving equations with fractional derivatives, and the Cayley transform method for abstract differential equations in Hilbert and Banach spaces. The proposed methodology of obtaining weighted estimates can be further employed for investigating exact and approximate solutions of many new problems. At the same time, taking into account the boundary and initial effects is not only of theoretical but also of practical value because it justifies, for example, the use of a coarser mesh (i.e. a larger mesh step) near the boundary of the domain. Moreover, the presented discrete approximations and methods without saturation of accuracy can be utilized for solving a wide range of applied problems in physics, engineering, chemistry, biology, finance, etc. The book is intended for scientists, university teachers, graduate and postgraduate students who specialize in the field of numerical analysis.