On convergence of implicit finite element method scheme for a solution of a parabolic equation with double degeneration and nonlocal space operator

Abstract

© Published under licence by IOP Publishing Ltd. We consider the initial-boundary value problem for a parabolic equation. The space operator from this equation depends on the sought function, its derivatives and the nonlocal (integral) solution characteristic. This equation belongs to the class of parabolic equations with double nonlinearity and double degeneration: the nonlinearity and degeneration may appear in the space operator and the function, which is under the sign of the derivative with respect to a variable t, may not be separated from zero. The problem under consideration has applied nature. These type of equations arise, for example, in the case of modelling the bacteria population propagation process. In the present paper we propose and investigate the approximate method, which is constructed with the use of the semi-discretization with respect to a variable t and the finite element method (FEM) in the space variables with a descent the nonlocal characteristic to the lower layer. We proved an existence of approximate solution. Also we obtained a priori estimates, proved a convergence of a constructed algorithm.